Graphing Calculator and Spreadsheet Masters · 2017. 6. 15. · and be used solely in conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited without

Graphing Calculatorand

Spreadsheet Masters

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

Glencoe Algebra 2ISBN: 0-07-828020-6 Graphing Calculator and Spreadsheet Masters

2 3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

RESOURCE MATERIALSTI-83 Plus Graphing Calculator Template ........................................................ 1Spreadsheet Template ....................................................................................... 3

TI-83 Plus Graphing Calculator HandbookIntroduction to the Graphing Calculator ................................................................ 5Graphing Functions .............................................................................................. 8Analyzing Functions..............................................................................................11Graphing Inequalities............................................................................................16Matrices ................................................................................................................19Graphing Trigonometric Functions .......................................................................21Graphing Special Functions..................................................................................22Statistics and Statistical Graphs ...........................................................................23

GRAPHING CALCULATOR INVESTIGATIONSUse with Lesson Title Page

1-3 Solving Equations and Checking Solutions...........................272-6 Transformations: Greatest Integer Function..........................293-4 Linear Programming..............................................................314-4 Matrices for 30°, 45°, and 60° Rotations ...............................335-4 Using Tables to Factor by Grouping......................................356-7 Quadratic Inequalities and the Test Menu.............................377-6 Rational Root Theorem .........................................................398-3 Matrices and Equations of Circles.........................................419-3 Horizontal Asymptotes and Tables........................................43

10-1 Regression Equation Lab ......................................................4511-6 Recursion and Iteration .........................................................4712-5 Probabilities...........................................................................4913-4 Law of Sines: Ambiguous Case ............................................5114-2 Sinusoidal Equations.............................................................53

SPREADSHEET INVESTIGATIONS Use with Lesson Title Page

1-4 Absolute Value Statements ...................................................282-4 Using Linear Equations .........................................................303-1 Break-Even Point ..................................................................324-5 Cramer’s Rule .......................................................................345-7 Appreciation and Depreciation ..............................................366-5 Approximating the Real Zeros of Polynomials ......................387-7 Operations on Functions .......................................................408-2 Parabolas ..............................................................................429-4 Variation ................................................................................44

10-6 Net Present Value .................................................................4611-2 Sequences and Series ..........................................................4812-2 Permutations And Combinations...........................................5013-1 Cofunctions ...........................................................................5214-3 Trigonometric Identities .........................................................54

Answers ..............................................................................................................55

iii

CONTENTS

iv

Teacher’s Guide to Using the Graphing Calculator and

Spreadsheet Masters

This booklet contains a TI-83 Plus Graphing CalculatorHandbook, which summarizes many of the graphing calculatorskills students might use in pre-algebra. This is a referencetool and does not contain additional exercises for students.

This booklet also includes a Graphing Calculator Investigationand a Spreadsheet Investigation for each chapter in GlencoeAlgebra 2.• The graphing calculator activities are written with TI-83

Plus keystrokes provided. If your students use anothercalculator, they will need to modify their keystrokes tocomplete the activity.

• The spreadsheet masters were developed for use withMicrosoft® Excel. If you use a different spreadsheetapplication software, you may need to alter the commandsgiven in the activity.

When to Use Each activity should be used as an extension ofthe lesson to which it is referenced. Because these activitiescan be done independently of classroom time, they may beused as extra credit or as enrichments for those students whohave completed their assignments in a timely manner.

vGlencoe Division, Macmillan/McGraw-Hill Geometry

TI–83 Plus Graphing Calculator Template

NAME ______________________________________________ DATE ____________ PERIOD _____

© Glencoe/McGraw-Hill 1 Glencoe Algebra 2

Spreadsheet Template

NAME ______________________________________________ DATE ____________ PERIOD _____


TI–83 Plus Graphing Calculator HandbookIntroduction to the Graphing Calculator

NAME ______________________________________________ DATE ____________ PERIOD _____

11


This section introduces you to some commonly-used keys and menus ofthe calculator.

MODE The key allows you to select your preferences in manyaspects of calculation and graphing. Many of these settings are rarelychanged in common usage. This screen shows the default mode settings.

← type of numeric notation← number of decimal places in results← unit of angle measure used← type of graph (function, parametric, polar, sequence)← whether to connect graphed points

← real, rectangular complex, or polar complex number system← graph occupies full screen, top of screen with

HOME screen below, or left side of screen with TABLE onright

To change the preferences, use the arrow keys to highlight your choiceand press .

FORMAT The FORMAT menu is the second function of andsets preferences for the appearance of your graphing screen. The defaultscreen is shown below.

← rectangular or polar coordinate system← whether to display the cursor coordinates on screen← whether to show a grid pattern on screen← whether to show the axes← whether to label the axes← whether to show the equation being graphed

You can change your preferences in the FORMAT menu in the sameway you change settings.

Many keys on the calculator access menus from which you can select afunction, command, or setting. Some keys access multiple menus. Youcan use the right and left arrow keys to scroll through the differentmenu names located at the top of the screen. As each menu name ishighlighted, the choices on the screen change. The screens on the nextpage show various menus accessed by using .MATH

MODE

ZOOM

ENTER

MODESettingPreferences

Using Menus

To select a choice in a menu, either use the arrow keys to highlight yourchoice and press or simply press the number or letter of yourselection. Notice that entry 7 in the first screen has a down arrow insteadof a colon after the 7. This signifies there are more entries in the menu.

Above most keys are one or two additional labels representing commands,menus, letters, lists, or operational symbols. These are accessed by using

or .

• accesses the commands on the left above the key. Note thatthese commands and are the same color.

• accesses the commands on the right above each key. These commands and are also the same color.

• Pressing engages the [A-LOCK] or Alpha Lock command.This enables you to select consecutive commands without pressing

before each command. This is especially useful when enteringprograms.

Each letter accessed by using can be used to enter words or labelson the screen, but can also be used as a variable. A value can be storedto each variable.

A graphing calculator is also a scientific calculator. That is, it follows the order of operations when evaluating entries. Unlike some scientificcalculators, the graphing calculator displays every entry in the expression.

Before pressing to evaluate the expression, you can use the arrowkeys to scroll through the expression to make corrections. Correctionscan be made in three ways.

• Use to delete any unwanted entries.

• Use [INS] to insert omitted entries.

• “Type” over an incorrect entry. This overprints any entries and doesnot shift the entries to the right as a word processor does.

2nd

DEL

ENTER

ALPHA

ALPHA

ALPHA

ALPHA2nd

ALPHA

ALPHA

2nd

2nd

ALPHA2nd

ENTER


AlternateFunction Keys

Whenever analternate functionis indicated in thekeystrokes of thisappendix, we willuse brackets toshow that thefunction is listedabove a key.

Computation

Math menu Number menu

Complex Number menu Probability menu

If you have an expression that you wish to evaluate repeatedly with a change in one part of the expression, you can press [ENTRY]after you have pressed and the expression will reappear. You can edit it for your next computation. The ENTRY command always repeatsthe last entered expression. You cannot scroll back through previousexpressions you have evaluated.

Evaluate .

Press: [��] 3 4 6

5 12 3 6

Note that the square root function automaticallyincludes a left parenthesis. You must enter theright parenthesis to indicate the end of theexpression under the radical sign. If you havethe decimal in the Float mode, as many as 10digits may appear in the answer.

Evaluate each expression if a � 4, b � �5, c � 2, d � �23�, and e � �1.5.

a. abc � 3de4 b.�ce

2��

48ab�

For a series of expressions that use the same values for the variables, itis often helpful to store the value for each variable into the calculator.You can combine several commands in one line by using the colon aftereach command. The following commands save the values for variablesa, b, c, d and e.

Press: 4 [A] [:] 5 [B] [:] 2

[C] [:] 2 3 [D] [:] 1.5

[E]

a. Method 1: Using stored values

[A] [B] [C] 3

[D] [E] 4

Method 2: Entering computations

4 5 2 3 2 3

1.5 4

b. Method 1: Using stored values

[E] 4 [A]

[C] 8 [B]

Method 2: Entering computations

1.5 4 4 2

8 5 ENTER)(–)�

+x 2(�)�+(–)(

ENTER)ALPHA+x 2ALPHA

(�)ALPHA+ALPHA(

ENTER)

(–)(��—�(–)�

ENTERALPHA

ALPHA—ALPHAALPHAALPHA

ENTERALPHA

STO(–)ALPHAALPHASTO�ALPHAALPHA

STOALPHAALPHASTO(–)ALPHAALPHASTO

ENTER�))(–)—

(+)(—x 22nd

�32 � 4�(6) ��[5 � (��12)]�3��6

ENTER

2nd


The minus keyand the negativekey are differentkeys.

Example 1

2


Most functions can be graphed by using the key. The viewing window most often used for non-trigonometric functions is the standardviewing window [�10, 10] scl:1 by [�10, 10] scl:1, which can be accessedby selecting 6:ZStandard on the ZOOM menu. Then the window can beadjusted so that a complete graph can be viewed. A complete graph isone that shows the basic characteristics of the parent graph.

Linear Functions A complete linear graph shows the x- and y-intercepts.

a. Graph y � 3x � 4 in the standard viewing window.

Press: 3 4 6

If your calculator is already set for thestandard viewing window, press instead of 6.

Both the x- and y-intercepts of the lineargraph are viewable in this window, so thegraph is complete.

b. Graph y � �2(x � 5) � 2.

Press: 2 5 2

When this equation is graphed in the standard viewing window(Figure 1), a complete graph is not visible. The graph indicates thatthe y-intercept is less than �10. You can experiment with the Yminsetting or you can rewrite the equation in y � mx � b form, whichwould be y � �2x � 12. The y-intercept is �12, so Ymin should beless than �12. Remember that Xmax and Ymax can be less than 10so that your screen is less compressed. Use the WINDOW menu tochange the parameters, or settings, and press to view theresult. There are many windows that will enable you to view thecomplete graph (Figure 2).

Quadratic Functions When graphing quadratic functions, acomplete graph includes the vertex of the parabola and enough of thegraph to determine if it opens upward or downward.

(continued on the next page)

GRAPH

GRAPH—)+((–)Y=

ZOOM

GRAPH

ZOOM—Y=

Y=

TI–83 Plus Graphing Calculator HandbookGraphing Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Example 1

2

Figure 1

[�10, 10] scl:1 by [�10, 10] scl:1

Figure 2

[�10, 10] scl:1 by [�15, 5] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


Graph y � 4(x � 3)2 � 4.

Press: 4 3 4

While the standard viewing window shows a complete graph, you may want to change the viewing window to see more of the graph.

Polynomial Functions The graphs of other polynomial functionsare complete when their maxima, minima, and x-intercepts are visible.

a. Graph y � 5x3 � 4x2 � 2x � 4.

Press: 5 3 4 2

4

A complete graph is shown in the standard viewing window. You may want to redefine your window to observe the intercepts,maximum, and minimum points more closely.

b. Graph y � x4 � 13x2 � 36.

Press: 4 13 36

The standard viewing window (Figure 1) does not show a completegraph. It seems that only the y parameters need to be adjusted.Experiment to find a window that is suitable. Figure 2 shows asample.

Exponential Functions A complete graph of an exponentialfunction shows the curvature of the graph and the y-value that itapproaches.

Graph y � 92 � x.

Press: 9 2

Note that you must use parentheses to group theterms that make up the exponent.

A complete graph seems to appear in the secondquadrant of the standard viewing window. Vary the WINDOW settings to view the graph moreclosely.

GRAPH)+(Y=

GRAPH+x 2—Y=

GRAPH+

—x 2+Y=

GRAPH+x 2)—(Y=

Example 3

4

[�7, 1] scl:1 by [�1, 9] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

Figure 1

[�10, 10] scl:1 by [�10, 10] scl:1

Figure 2

[�10, 10] scl:1 by [�8, 40] scl:2


Logarithmic Functions A complete graph of a logarithmic functionshows the curvature of the graph and the values, or locations, of theasymptotes that the curve approaches.

a. Graph y � log (x � 6).

Press: 6

An entire graph appears in the standardviewing window, but is very small. Redefine the WINDOW parameters for y, so that the graph is more visible.

b. Graph y � log4 x.

To graph a logarithmic function with a base other than 10, you must first change the function by using the change of base formula,loga x � �

lloogg

ax

�.

Press:4

An entire graph appears in the standardviewing window, but is very small. Redefine the WINDOW parameters, so that the graph is more visible.

You can graph multiple functions on a single screen. Each function isdenoted by Y1�, Y2�, Y3�, and so on, in the Y� menu. To graph morethan one function, press at the end of each function you are entering and the cursor will move to the next function to be entered.

Systems of Equations

Graph y � 0.5x � 4 and y � 2x2 � 5x � 1.

Press: 0.5 4 2

5 1

The standard viewing window shows that the line and parabola intersect in two points.

GRAPH+—

x 2ENTER+Y=

ENTER

GRAPH)

LOG�)LOGY=

GRAPH

)+LOGY=

Example 5

Example 6

[�10, 10] scl:1 by [�2, 2] scl:1

[�1, 5] scl:1 by [�3, 3] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


In addition to graphing a function, you can use other tools on a graphingcalculator to analyze functions. One of those tools is a function table.

How to Use a Table You may complete a table manually orautomatically. To create a table for one or more functions, you mustfirst enter each function into the Y� list. Then set up and create thetable.

a. Use a table to evaluate the function y � 4x2 � 2x � 7 for {�9, �4, 0, 1, 5}.

In this case you only need to evaluate the function for selectedvalues, so use the TBLSET menu to have the calculator ask for thevalues of the independent variable (domain) and find the functionvalue (range) automatically.

Press: 4 27 [TBLSET]

[TABLE] 9 4

0 1 5

b. Use a table to evaluate the functions y � 5x2 � x � 1 and y � 6 � x3 for the integers from �3 to 3, inclusive.

When you want to evaluate a function for a range of values, have thecalculator find both the values of the independent variable and thefunction values automatically. In Table Setup, enter the initialnumber of the domain as the TblStart value and the incrementbetween the values of the independent variable as �Tbl. Enteringmore than one function in the Y� list allows you to evaluate all ofthe functions in one table.

Press: 5 1 6 3[TBLSET] 3 1

[TABLE]

Once you create a table, you can scroll through the values using the arrow keys.

2ndENTER

ENTERENTER(–)

2nd—ENTER

+—x 2Y=

ENTERENTERENTERENTER

(–)ENTER(–)2nd

ENTER2nd

+—x 2Y=

TI–83 Plus Graphing Calculator HandbookAnalyzing Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

33

The functionvalues are thedependentvariable values.

Example 1

allows you to quickly adjust the viewing window of a graph indifferent ways. The effect of each choice on the ZOOM menu is shown onthe next page.

1: ZBox Allows you to draw a box to define the viewing window2: Zoom In Magnifies the graph around the cursor3: Zoom Out Views more of a graph around the cursor4: ZDecimal Sets �X and �Y to 0.15: ZSquare Sets equal-sized pixels on the x-and y-axes6: ZStandard Sets the standard viewing window, [�10, 10] scl:1 by

[�10, 10] scl:17: ZTrig Sets the built-in trig window, ��

4274�

π, �4274�

π� scl: �π2� by [�4, 4]

scl:1 for radians or [�352.5. 352.5] scl:90 by [�4, 4] scl: 1for degrees

8: ZInteger Sets integer values on both the x-and y-axes9: ZoomStat Sets values for displaying all of the data in the current

stat lists0: ZoomFit Fits Ymin and Ymax to show all function values for Xmin

to Xmax

Using to Graph in the Standard and Square Windows

Graph the circle with equation x2 � y2 � 16 in the standardviewing window. Then use ZSquare to view the graph in asquare screen.

First solve the equation for y in order to enter it into the Y� list.

x2 � y2 � 16 → y � � �16 � x�2��

The two pieces of the graph can be entered at one time using {�1, 1}.This expression tells the calculator to graph �1 and 1 times thefunction.

Press: [ { ] 1 1 [ } ] [√__

] 16

6

The circle is distorted when viewed in the standard viewing window.

Press: 5

Using ZSquare makes the circle appear as a circle.

ZOOM

ZOOM

)x 2—2nd2nd,(–)2ndY=

ZOOM

ZOOM

Example 2


[�10, 10] scl:1 by [�10, 10] scl:1

[�15.16, 15.16] scl:1 by [�10, 10] scl:1

Make sure thatCoordOn ishighlighted in theFORMAT menuto display thecursor coordinatesas you trace.

Using to Zoom In and Out Graph y � 0.5x3 � 3x2 � 12in the standard viewing. Zoom out to view a complete graph.Then zoom in to approximate the y-intercept of the graph tothe nearest whole number.

Press: 0.5 3 3 12 6

The complete graph is not shown in the standard viewing window.(Figure 1) When you zoom out or in, the calculator allows you to choosethe point around which it will zoom. Zooming out around the originonce allows a complete graph to be shown. (Figure 2)

Press: 3

Now zoom in to approximate the y-intercept. Choose a point close to theintercept by using the arrow keys.

Press: 2

The y-intercept appears to be about �12.Zooming in again may allow you to makea closer approximation.

The feature allows you to move the cursor along a graph anddisplay the coordinates of the points on the graph.

Using Graph y � 4x � 2 and y � �3x2 � x � 5. Use theTRACE feature to approximate the coordinates of theintersection of the graphs in the first quadrant. Then evaluatey � �3x2 � x � 5 for x � 1.7.

Press: 4 2 3 5

Move the cursor along the graphs using and .

Pressing or moves the cursor more quickly. If your cursor moves offof the screen, the calculator will automaticallyupdate the viewing window so that the cursoris visible. Use and to move from onefunction to the other. The intersection is atabout (0.4, 4).

To evaluate a function for a value and move tothat point, place the cursor on the function graph.Then enter the value and press . When x � 1.7, y � �5.37 for y � �3x2 � x � 5.

ENTER

2nd2nd

TRACE+—x 2(–)ENTER+Y=

TRACE

TRACE

ENTERZOOM

ENTERZOOM

ZOOM—x 2—Y=

ZOOMExample 3

Example 4


Figure 1

[�10, 10] scl:1 by [�10, 10] scl:1

Figure 2

[�40, 40] scl:1 by [�40, 40] scl:1

[�10, 10] scl:1 by [�24.19, �4.19] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


The intersectionof the graphsmust appear onthe screen to findthe coordinateswhen usingintersect.

Using to locate the intersection points of the graphs of twofunctions gives you an approximation of the coordinates. For more accuratecoordinates, you can use the intersect option on the CALC menu.

Finding Intersection Points Use 5:intersection on the CALCmenu to find the coordinates of the intersection of the graphs ofy � 4x � 2 and y � �3x2 � x � 5.

If you do not have the functions graphed, enter the functions into theY� list and press . Then find the coordinates of the intersection.

Press: [CALC] 5

Place the cursor on one graph and press . Then move the cursor to the other

graph and press . To guess at theintersection or enter an x-value and press

. If there is more than one intersectionpoint, the caculator will find the one closestto your guess. The cursor will move to theintersection point and the coordinates will be displayed.

The CALC menu also allows you to find the zeros of a function.

Finding Zeros Find the zeros of f(x) � �2x4 � 3x2 � 2x � 5.

Press: 2 4 3 2 5 [CALC] 2

The calculator can find one zero at a time. Usethe arrow keys or enter a value to choose the left bound for the interval in which the calculator will search for the zero and press

. Choose the right bound and press

. Select a point near the zero using the arrow keys or by entering a value and press

. Repeat with another interval to find the other zero. The zeros of this function are about �1.42 and 1.71.

Real-world application problems often require you to find the relativeminimum or maximum of a function. You can use 3:minimum or4:maximum features on the CALC menu of a graphing calculator tosolve these problems.

ENTER

ENTER

ENTER

2nd++x 2+(–)Y=

ENTER

ENTER

ENTER

2nd

GRAPH

TRACE

Example 5

Example 6

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


Finding Maxima and Minima Determine the relativeminimum and the relative maximum for the graph of f(x) � 4x3 � 6x � 5.

First graph the function.

Press: 4 3 6 5 6

To find the relative minimum press [CALC] 3.

Similar to finding a zero, choose the left and right bound of the interval and guess theminimum or the maximum. The point at about(0.71, 2.17) is a relative minimum.

Use a similar method to find the relative maximum, by pressing [CALC] 4. The point at about (�0.71, 7.83) is a relativemaximum.

2nd

2nd

ZOOM+—Y=

Example 7

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


NAME ______________________________________________ DATE ____________ PERIOD _____

44 TI–83 Plus Graphing Calculator HandbookGraphing Inequalities

Most linear and nonlinear inequalities can be graphed using the key and selecting the appropriate graph style in the Y� editor. To select theappropriate graph style, select the graph style icon in the first column of theY� editor and press repeatedly to rotate through the graph styles.

• To shade the area above a graph, select the Above style icon, .

• To shade the area below a graph, select the Below style icon, .

Before graphing an inequality, clear any functions in the Y� list bypressing and then using the arrow keys and the key to selectand clear all functions. If you do not wish to clear a function, you can turnthat particular graph off by using the arrow keys to position the cursorover that function’s � sign and then pressing to change theselection status.

Linear Inequalities

a. Graph y � 2x � 3 in the standard viewing window.

First enter the boundary equation y � 2x � 3 into the Y� list.

Press: 2 3

Next, press the key until the icon before � flashes. Press until the icon changes to the Below style icon, , for “y �”.Finally, if your calculator is not already set forthe standard viewing window, press 6.Otherwise, press .

b. Graph y � �4x � 5 in the standard viewing window.

Press: 4 5

Next, press the key until the icon before � flashes.Then press until the icon changes to the Above style icon, , since the inequality asks for “y ”. Finally, press .GRAPH

ENTER

+(–)Y=

GRAPH

ZOOM

ENTER

—Y=

ENTER

CLEARY=

ENTER

Y=

Example 1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

Nonlinear Inequalities The procedure for graphing nonlinearinequalities is the same as that of graphing linear inequalities.

a. Graph y � 0.25x2 � 4 in the standard viewingwindow.

Press: 0.25 4

Next, select the Below style icon, , since theinequality asks for “y �”, and then press

.

b. Graph y � 0.2x4 � 3x2 � 4.

Press: 0.2 4

2 4

Next, select the Above style icon, , since the inequality asks for “y ”. Then press

.

c. Graph y � �x � 2� � 4.

Press: [�� ] 2 4

Next, select the Below style icon, , since the inequality asks for “y �”. Then press

.

d. Graph y � 3x � 5.

Press: 3 5

Next, select the Above style icon, , since the inequality asks for “y ”. Then press

.GRAPH

—Y=

GRAPH

+)+2ndY=

GRAPH

+

—Y=

GRAPH

—x 2Y=

Example 2


[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

The Shade(command canonly be used withtwo inequalitieswhich can bewritten with “y �”in one inequalityand “y ” in theother.

Graphing systems of inequalities on a graphing calculator is similar tographing systems of equations.

Graph the system of inequalities.y � 2x � 5 y � x2 � 4x � 1

Method 1: Shading Options in Y�

Press: 2 5

4 1

Select the Above style icon, , for y 2x � 5 and the Below style icon, , for y � x2 � 4x � 1.Then press .

Notice that the first inequality is indicated using vertical lines and the second inequalityuses horizontal lines. The solution to the systemis shown by the intersection of the shaded areas.

Method 2: Using the Shade Command

Some systems of inequalities can be graphed by using the Shade(command and entering a function for a lower boundary and a functionfor the upper boundary of the inequality. The calculator first graphsboth functions and then shades above the first function entered andbelow the second function entered.

Before graphing an inequality using the Shade( command, clear any graphics from the viewing window by pressing [DRAW] 1 .Also clear any equations in the Y� list. If not already there, return tothe home screen by pressing [QUIT].

Press: [DRAW] 7 2 5

4 1 ENTER)+—x 2

,—2nd

2nd

ENTER2nd

GRAPH

+—

x 2ENTER—Y=

Example 3


[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

TI–83 Plus Graphing Calculator HandbookMatrices

NAME ______________________________________________ DATE ____________ PERIOD _____

55


A graphing calculator can perform operations with matrices. It can alsofind determinants and inverses of matrices. The MATRX menus areaccessed using [MATRX].

There are three menus in the MATRX menu.

• The NAMES menu lists the matrix locations available. There are tenmatrix variables, [A] through [J].

• The MATH menu lists the matrix functions available.

• The EDIT menu allows you to define matrices.

A matrix with dimension 2 3 indicates a matrix with 2 rows and 3columns. Depending on available memory, a matrix may have up to 99rows or columns.

Entering a Matrix Enter matrix A � � �.To enter a matrix into your calculator, choose the EDIT menu andselect the matrix name. Then enter the dimensions and elements of thematrix.

Press: [MATRX] 22 1 3 2 (�) 2

Press [QUIT] to return to the HOME screen. Then press [MATRX]

to display the matrix.

You can find the determinant and inverse of a matrix very quickly witha graphing calculator.

Determinant and Inverse of a Matrix

a. Find the determinant of matrix A.

Press: [MATRX] 1 [MATRX]1

The determinant of matrix A is �8.

b. Find the inverse of matrix A.

Press: [MATRX] 1

A�1� � �0.375�0.125

0.250.25

ENTERx –12nd

ENTER

2nd2nd

ENTER

ENTER2nd

2nd


ENTERENTER2nd

3�2

12

2nd

Example 1

Example 2

Operations with Matrices

Enter matrix B � � �. Then perform each operation.

a. �12

�B

First, enter matrix B.

Press: [MATRX] 2 2

3 2 3 64 8 5 [QUIT]

Then find �12

�B.

Press: .5 [MATRX] 2

�12

�B � � �

b. AB

Press: [MATRX] 1 [MATRX]

2

AB � � �

c. A2

Press: [MATRX] 1

A2 � � �

d. AB � B

Press: [MATRX] 1 [MATRX]

2 [MATRX] 2

AB � B � � �277

�1814

160

ENTER2nd+

2nd2nd

�310

7�2

ENTERx 22nd

212

�2122

14�4

ENTER

2nd2nd

32.5

1.5�4

12

ENTER2nd

2ndENTER(–)ENTER


ENTER2nd

65

3�8

24

Example 3


TI–83 Plus Graphing Calculator HandbookGraphing Trigonometric Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

66


Trigonometric functions and the inverses of trigonometric functions canbe graphed using . The functions and their inverses can be graphedin degrees or radians. You must set the calculator in Radian or Degreemode. The standard viewing window for trigonometric functions can beset by pressing 7:Trig, which automatically adjusts the x- and y-axes scales for degrees or radians.

a. Using Degrees Graph y � cos x.

First, set the calculator in degree modeby pressing .

Now enter and graph the function. Press 7.

b. Using Radians Graph y � sin x.

Change to radian mode by pressing

. Press to delete

the function entered in part a. Then press

to enter the new function.

Next, press 7 to set the viewing window to accommodate radian mode and graph the function.

c. Amplitude, Period, and Phase Shift Graph y � 2 sin ��12

� x � 60°�using the viewing window [�540, 540] scl:90 by [�3, 3] scl:1.Then state the amplitude, period, and phase shift.

Make sure the calculator is in degree mode.

Press: 2 2 60

The amplitude of the function is �|�22� 2|�

or 2. The period is or 720°, and

the phase shift is or �120°.��

160�

�2

360��

�12�

GRAPH

)—�SINY=

ZOOM

)SIN

CLEARY=ENTER

MODE

ZOOM)COSY=

ENTERMODE

ZOOM

Y=

Example 1

[�352.5, 352.5] scl:90 by [�4, 4] scl:1

[�2 , 2 ] scl:�2

� by [�4, 4] scl:1

[�540, 540] scl:90 by [�3, 3] scl:1


Most special functions can be graphed using the key. The absolutevalue function abs( and the greatest integer function int( can be found inthe MATH NUM menu.

Absolute Value Graph y � 2|x � 4|.

Press 2 1 4

6 to graph the function in the standard viewing window.

Greatest Integer Function Graph y � [[x � 1.5]].

First, make sure the calculator is set for dotplotting rather than the connected plotting usedin most other functions. Press , highlightDot, and press .

Then, enter the function. Press

5 1.5 6. If your

calculator is already set for the standard

viewing window, press instead of 6.

The TEST menu allows you to graph other piecewise functions. Enter thepieces of the function as a sum of the products of each piece of the

function and its domain. For example, y � � is entered as

(5)(X<2) � (4X)(X>2) in the Y� menu.

Piecewise Function Graph y � � .

Place the calculator in Dot mode. Then enter the function in the Y�list using the TEST menu options.

Press: 3 [TEST] 6 3 1

3 [TEST] 5

[TEST] 6 29 2

[TEST] 3 2 6ZOOM)

2nd()—(

+)2nd(

)2nd(–)(

)+(+)(–)

2nd()(Y=

3 if x � �31 � x if �3 x � 29 � 2x if x 2

5 if x � 24x if x 2

ZOOM

GRAPH

ZOOM)—

MATHY=

ENTER

MODE

ZOOM

)—MATHY=

Y=

TI–83 Plus Graphing Calculator HandbookGraphing Special Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

77

Example 1

Example 3

2

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


TI–83 Plus Graphing Calculator HandbookStatistics and Statistical Graphs

NAME ______________________________________________ DATE______________ PERIOD _____

99

A graphing calculator allows you to enter a set of data and generatestatistics and statistical graphs. Before you enter data values, make sureyou clear the Y� list, L1 and L2, and the graphics screen. Clear the Y�list by pressing . Use the key to select additional equations and clear them also. To clear L1 and L2, press 4[L1] [L2] . If you need to clear the graphics screen, press [DRAW] 1 .

Enter Data into Lists Enter the following data into a graphingcalculator.

49 53 54 54 56 55 57 61 51 58 41 59 54 50 60 44

Press: 1 49 53 54 54

56 55 57 61

51 58 41 59

54 50 60 44

You can use the up and down arrow keys to scroll through the list.

Find Mean, Median, and Mode Find the mean, median, andmode of the data in Example 1.

Press 1 . This function displays many statistics aboutthe data. X–– denotes the mean. Scroll down to find the median.

The calculator does not have a function to determine the mode. You canfind the mode by examining the data. First sort the data to write themin order from least to greatest.

Press: 2 [L1]

Then scroll through the data by pressing 1 and using the and keys. You will find that the mode is 54.

STAT

ENTER)2ndSTAT

ENTERSTAT




ENTERENTERENTERENTERSTAT

ENTER

2ndENTER,2ndSTAT

CLEARY=

Example 1

2

mean

median

ENTER


Box-and-Whisker Plots

a. Draw a box-and-whisker plot for the data.

30.2 29.0 26.2 25.8 23.8 43.0 19.8 19.4 26.0 46.6 26.8 22.8 35.4 25.2 12.2 31.4

Set the viewing window. Next, set the plottype. Press [STAT PLOT] 1

to highlight and press. Make sure L1 is entered in Xlist:.

If not, move the cursor to highlight andpress [L1] . Then, enter the data into L1. Press 1 30.229 ... 31.4 .

b. Draw a box-and-whisker plot with outliers using the data.

Without clearing the lists or graphic screen,press [STAT PLOT] 2

, to highlight , and press .

Make sure L1 is entered in Xlist:.Press .

c. Find the upper and lower quartiles, the median, and the outliers.

Press and use and to move the cursor along the graph.The values will be displayed. For this data, the upper quartile is30.8, the lower quartile is 23.3, the median is 26.1, and the outliersare 43 and 46.6.

Histograms

a. Use the data on the number of public libraries in each stateand Washington, D.C., to make a histogram.

273 102 159 196 1030 235 244 30 27 428 366 49 141772 427 554 372 188 322 273 187 491 659 361 243 346110 283 78 238 455 92 1067 352 86 684 192 201 64074 180 134 284 753 96 204 308 309 174 451 74

Enter the data in L1. Press 1 273 102 ... 74 .

Set the viewing window. Choose Xmin, Xmax, and Xscl to determinethe number of bars in the histogram. For this data, the least value is27 and the greatest is 1067. If Xmin � 0, Xmax � 1100, and Xscl � 100, the histogram will have 11 bars each representing aninterval of 100.

Choose the type of graph. Press

[STAT PLOT] 1

[L1] 1 . Then press

to draw the histogram.GRAPH

ENTERENTER2nd

ENTERENTER

2nd

ENTERENTERENTERSTAT

TRACE

GRAPH

ENTER

ENTER2nd

GRAPHENTERENTER

ENTERSTAT

ENTER2nd

ENTER

ENTER2nd

Examples 3

4

[10, 50] scl:2 by [0, 10] scl:1

[0, 1100] scl:100 by [0, 12] scl:1

[10, 50] scl:2 by [0, 10] scl:1


Make sure youhave cleared theY= list, L1 andL2, and thegraphic screen.

b. Use the data in the table to draw ahistogram and its frequency polygram.

Enter the class marks as L1. Press

1 162.5 177.5 ... 312.5 .

Move the cursor to L2. Enter thefrequencies.

Press 7 15 ... 3[QUIT].

Set the viewing window. Use the minimum and maximum of theclass limits for Xmin and Xmax. Use the size of the intervals forXscl. Choose the y-axis values to show the complete histogram.

Set the plot type. Press [STAT PLOT] 1[L1] [L2] . Then press

.(Figure 1) Without clearing the lists or graphic screen,press [STAT PLOT] 1 to highlight , and press . Make sure L1 is entered in Xlist: and L2 is the Ylist.Choose as the mark. Press .

Scatter Plot, Connected Line Scatter Plot, and Regression Line

a. Use these data to draw a scatter plot:(20.0, 5.2), (10.2, 1.9), (7.3, 1.6), (6.8, 2.6),(5.9, 1.0), (2.6, 0.7), (2.8, 0.35), (2.7, 0.15).

Clear previous data and graphs and setthe viewing window. Enter the x-valuesinto L1 and the y-values into L2. Thendraw the scatter plot by pressing [STAT PLOT] 1 to highlight and press . Make sure L1 is the Xlist: and L2 is the Ylist:. Then press

.

b. Use the data to draw a line graph.

Press [STAT PLOT] 1 to highlight and press .GRAPHENTER

2nd

GRAPH

ENTER

ENTER

2nd

GRAPH

ENTER

ENTER2nd

GRAPH

ENTER2ndENTER2ndENTER

ENTER2nd

2ndENTERENTERENTER

ENTERENTERENTER

STAT

Example 5

Class Limits Frequency

155–170 7170–185 15185–200 34200–215 38215–230 42230–245 35245–260 33260–275 21275–290 18290–305 6305–320 3

[155, 320] scl:15 by [0, 50] scl:5

Figure 1

[155, 320] scl:15 by [0, 50] scl:5

Figure 2

[0, 25] scl:5 by [0, 6] scl:1

[0, 25] scl:5 by [0, 6] scl:1


c. Draw a regression line for the data in the table.

Set the plot to display a scatter plot by pressing [STAT PLOT] 1

[QUIT].

To calculate the coefficients of regression press 4 .

Then, write the equation of the regressionline. You can automatically enter theregression equation in the Y� list.

Press 5 1. Finally, graph the regression line by pressing .

There are also regression models for analyzing data that are not linearbuilt into the calculator.

Nonlinear Regression Find a sine regression equation tomodel the data in the table. Graph the data and the regressionequation.

Enter the data into lists L1 and L2. Press 1 2 ...12 39 42 ... 40 .

Find the regression statistics.

Press: [C]

Enter the regression equation into the Y� list.

Press: 5 1

Then format the scatter plot to graph the data by pressing [STAT PLOT] 1 .Make sure that L1 is chosen as the Xlist and L2 is chosen as the Ylist. Set the viewingwindow. Press to see the scatter plot and the graph of the regression equation.

GRAPH

ENTER2nd

VARSY=

ENTERALPHASTAT


ENTERENTERSTAT

GRAPH

VARSY=

ENTERSTAT

2nd

ENTER2nd

Example 6

[0, 25] scl:5 by [0, 6] scl:1

x 1 2 3 4 5 6 7 8 9 10 11 12

y 39 42 45 48 54 59 63 64 59 52 44 40

[0, 13] scl:1 by [30, 70] scl:5

Graphing Calculator InvestigationSolving Equations and Checking Solutions(Use with Lesson 1-3.)

NAME ______________________________________________ DATE ____________ PERIOD _____

11


When solving equations, checking the solutions is an important process. Agraphing calculator can be used to check the solution of an equation.

Solve �2(5y � 1) � y � �4(y � 3).

Graph the expression on the left side of the equation in Y1 and the expression on the right side of the equation in Y2. Choose anappropriate view window so that the intersection of the graphs isvisible. Then use the intersect command to find the coordinates ofthe common point.

Keystrokes: 2 5 1 4 3 6 8 [CALC] 5

[QUIT] .

The x-coordinate, ��

710�, is the solution to the equation. The y-coordinate

is the value of both sides of the equation when x � ��

710�.

ENTERENTERMATH2ndENTERENTERENTER

2ndENTERZOOMZOOM)—(

(–)ENTER—)—((–)Y=

Solve each equation.

1. �3(2w � 7) � 9 � 2(5w � 4) 2. 1.5(4 � x) � 1.3(2 � x) 3. �14�(a � 2) � �

16�(5 � a)

w � ��52

� x � 17 a � �45

�

4. 3(2z � 25) � 2(z � 1) � 78 5. �m

3� 4� � �

3m5� 1� � 1 6. �

x �2

5� � �

12� � 2x � �

x �8

3�

z � �14

� m � �8 x � �2111�

Example 1Example 1

Solve �x5� � �

x4� � �

12�(x � 2).

Graph the expression on the left side of the equation in Y1 and theexpression on the right side of the equation in Y2. Enter Y1 - Y2 inY3. Then graph the function in Y3. Use the zero function under theCALC menu to determine where the graph of Y3 equals zero. Thispoint will be the solution.

Keystrokes: 5 4 1 2 2

2 6 [CALC] 2.

Use arrow keys and enter to set the bound prompts. The solution is x � �2101�

.

2ndZOOMENTER

ENTERENTERVARS—ENTERENTER

VARSENTER)—()�(ENTER

)�(—)�(Y=

Example 2Example 2

ExercisesExercises

[�47, 47] scl:10 by [�31, 31] scl:10

[�10, 10] scl:1 by [�10, 10] scl:1


You can use a spreadsheet to try several different values in an equation tohelp you determine whether the statement is sometimes, always, or nevertrue. Remember that showing that a statement is true for some values doesnot prove that it is true for all values. However, finding one value for which astatement is false proves that it is not true for all values.

Spreadsheet InvestigationAbsolute Value Statements (Use with Lesson 1-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

11

Use a spreadsheet to determine whether each absolute value statement is sometimes, always, or never true.

1. For all real numbers a and b, a � 0, |ax � b| � 0. sometimes

2. If a and b are real numbers, then |a � b| � |a| � |b|. sometimes

3. If a and b are real numbers, then |a � b| � �x. never

4. If a and b are real numbers, then |a| � |b| � a � b. sometimes

5. If a and b are real numbers, then c|a � b| � c|a| � |b|. sometimes

ExercisesExercises

Determine whether c|a � b| � |ca � cb| is sometimes,always, or never true.

Try a number of values for a, b, and c to determine whether the statement istrue or false for each set of values.

Step 1 Use Columns A, B, and Cfor the values of a, b, andc. Choose several sets ofvalues including positiveand negative numbers,and zero.

Step 2 Use Column D to testthe equation. A formulasuch as C2*ABS(A2�B2)� ABS(C2*A2�C2*B2)in cell D2 returns TRUEif the equation is true.

Through observation of Column D, when c is negative the statement is nottrue. The absolute value statement, c|a � b| � |ca � cb| is sometimes true;it is true only if c � 0.

ExampleExample

Graphing Calculator InvestigationTransformations: Greatest Integer Function(Use with Lesson 2-6.)

NAME ______________________________________________ DATE ____________ PERIOD _____

22


A graphing calculator can be used to display transformations to the greatestinteger function. This is done by using the int( command under the MATH:NUM menu. When graphing the greatest integer function, it is important toset the calculator to Dot mode.

Graph each function. Evaluate it for x � 1, x � 1.3, and x � 2.Compare the graph of the function to the graph of f(x) � [[x]].

1. g(x) � [[x]] � 3 2. g(x) � [[x � 2]] 3. g(x) � [[x + 4]]

[�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1

g(1)� �2, g(1.3) � �2, g(2) � �1 g(1) � �1, g(1.3) � �1, g(2) �0 g(1) �5, g(1.3) �5, g(2) �6shifted down 3 units shifted right 2 units shifted left 4 units

4. g(x) � [[�x]] 5. g(x) � [[2x]] 6. g(x) � 3[[x]]

[�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1 [�4.7, 4.7] scl: 1 by [�3.1, 3.1] scl: 1

g(1)� �1, g(1.3) � �2, g(2) � �2 g(1) � 2, g(1.3) � 2, g(2) � 4 g(1) � 3, g(1.3) � 3, g(2) � 6reflected across the y-axis compressed by �

12

� horizontally expanded by 3 vertically

ExercisesExercises

Graph f(x) � [[x]] and g(x) � [[x]] � 2 on the same axes.Evaluate each function for x � 0.6, x � 1, x � 1.3, x � 2,

x � �0.5, and x � �1.1. Compare the graphs of the functions.Enter f(x) in Y1 and g(x) in Y2. Graph the functions.Keystrokes: 5 5 2

6 4.

Use TRACE to evaluate each function for the given values.Keystrokes: 0.6 1 1.3 2 0.5

1.1 0.6 1 1.3 2

0.5 1.1 .

f(0.6) � 0, f(1) � 1, f(1.3) � 1, f(2) � 2, f(�0.5) � �1, and f(�1.1) � �2g(0.6) � 2, g(1) � 3, g(1.3) � 3, g(2) � 4, g(�0.5) � �1, and g(�1.1) � 0

The graph of g(x) � [[x � 2]] is the same as the graph of f(x) � [[x]]shifted 2 units up.

ENTER(–)ENTER

(–)ENTERENTERENTERENTERENTER(–)ENTER

(–)ENTERENTERENTERENTERTRACE

ZOOMZOOM

ENTER+)MATHENTER)MATHY=

ExampleExample

[�4.7, 4.7] scl:1 by [�3.1, 3.1] scl:1


The slope intercept form of a linear equation is y � mx � b, where m is theslope and b is the y-intercept. Recall that the formula for the slope of a line through (x1, y1) and (x2, y2) is m � �

yx

22

��

yx

11

�.You can use the formula for slope and the slope-intercept form to find the value of b.

Spreadsheet InvestigationUsing Linear Equations (Use with Lesson 2-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

22

Use a spreadsheet to find the slope and y-intercept of the line through each pairof points. Then write an equation of the line in slope-intercept form.

1. (0, �5), (2, 5) 2. (4, 2), (�3, �5)5, �5; y � �5x � 5 1, �2; y � �x � 2

3. (�1, �4), (1, 3) 4. (�4, �9), (8, 3)

�72

�, ��12

�; y � ��72

�x � �12

� �1121�, �5�

13

�; y � �1121�x � 5�

13

�

5. (12, 9), (10, 10) 6. (�1.5, 3.1), (0.9, 1.9)

��12

�, 15; y � ��12

�x � 15 �0.5, 2.35; y � �0.5x � 2.35

7. Does the spreadsheet work when two points have the same x-coordinates? Explain.No; The slope is undefined.

ExercisesExercises

State the slope and y-intercept of the graph of the linethrough (5, 2) and (4, 1). Then write an equation of the

line in slope-intercept form.

Step 1 Use Columns A and B to represent the first point, and Columns Cand D to represent the second point on the line. Enter the formulafor slope in Column E.

Step 2 Substitute one of the given points into the slope-intercept from andsolve for b. Since we know the slope of the line, we can solve for b.

y � mx � b Slope-intercept formy1 � mx1 � b Substitute (x1, y1).

y1 � mx1 � b Solve for b.

Enter this formula into Column F using the names of the spreadsheet cells.

The slope of the line through (5, 2) and (4, 1) is 1 andthe y-intercept is �3. Thus, the equation of the lineslope-intercept form is y � 1x � (�3) or y � x � 3.

ExampleExample

Graph the system x � 3y � �7, 5x � y � 13, x � 6y � �9,3x � 2y � �7, and f(x, y) � 4x � 3y. Find the coordinates

of the feasible region. Then find the maximum and minimum valuesfor the system.Solve each inequality for y. Enter each boundary equation in the Y�screen. Find the vertices of the feasible region. Then find the valuesof f(x, y) to determine the maximum and minimum values.Keystrokes: 1 3 7 3

5 13 1 6 3

2 3 2 7 2 6

[CALC] 5 [QUIT] [ { ]

[Y] 4 3 [Y] [ } ]

[CALC] 5 [QUIT]

[ENTRY] [CALC] 5

[QUIT] [ENTRY]

[CALC] 5 [QUIT]

[ENTRY] .

The maximum value of the system is 18 and the minimum value is �10.

ENTER

2nd2ndENTERENTERENTER

2ndGRAPHENTER2nd2ndENTERENTER

ENTER2ndGRAPHENTER2nd

2ndENTERENTERENTER2ndGRAPH

ENTER2ndALPHA—,ALPHA,2nd2ndENTERENTERENTER2nd

ZOOM)�(+)�(ENTER)

�(—)�((–)ENTER+(–)

ENTER)�(+)�(Y=

ExampleExample

Graphing Calculator InvestigationLinear Programming (Use with Lesson 3-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

33


A graphing calculator can store the x- and y-coordinates when using theintersect command in the [CALC] menu. This can be displayed on thehome screen and used to evaluate an expression with x and y variables. Thisprocess is useful in finding the vertices of the feasible region and determiningthe maximum or minimum value for f(x, y).

[�10, 10] scl:1 by [�10, 10] scl:1

Graph each system. Find the coordinates of the vertices of the feasibleregion. Then find the maximum and minimum values for the system.

1. 2x � 3y � 6 2. y � 4x � 6 3. y � 16 � x3x � 2y � �4 x � 4y � 7 0 � 2y � 175x � y � 15 2x � y � 7 2x � 3y � 11f(x, y) � x � 3y x � 6y � 10 y � 3x � 1

f(x, y) � 2x � y y � 2x � 13y � 7 � 2xf(x, y) � 5x � 6y

[�10, 10] scl:1 by [�10, 10] scl: 1 [�10, 10] scl:1 by [�10, 10] scl: 1 [�10, 10] scl:1 by [�10, 10] scl: 1

(0, 2), (3, 0), (2, 5); (�1, 2), (�2, �2), (3, 1), (4, �1); (5.5, 0), (6.5, 0), (7.5, 8.5),min. � 3, max. � 17 min. � �4, max. � 9 (1.2, 4.6), (2.5, 2), (9.66, 6.33);

min. � 24.5, max. � 88.5

ExercisesExercises


Spreadsheet InvestigationBreak-Even Point (Use with Lesson 3-1.)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

1. If Carly could decrease her annual overhead to $14,000, what would thebreak-even point be? between 1400 and 1500 candles

2. Suppose Carly decreases her annual overhead to $14,000 and increasesthe price of a candle to $14.00. What is the new break-even point?between 1200 and 1300 candles

ExercisesExercises

Carly Ericson is considering opening a candle business. She estimates that she will have an annual

overhead of $15,000. It costs Carly $3.00 to make a jar candle, whichshe sells for $12.50. What is Carly’s break-even point?

Use Column A for the number of candles. Columns Band C are the cost and the income, respectively.

Extend the rows of the spreadsheet to find the point atwhich the income first exceeds the cost. The break-evenpoint occurs between this point and the previous point.In this case, the break even point occurs between 1500and 1600 candles.

The chart tool of the spreadsheet allows you to graphthe data. The graph verifies the solution.

ExampleExample

You have learned that the break-even point is the point at which the incomeequals the cost. You can use the formulas and charts in a spreadsheet to finda break-even point.

Graphing Calculator InvestigationMatrices for 30°, 45°, 60° Rotations (Use with Lesson 4-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

44


The rotation matrix for 90° counterclockwise about the origin is [0 �11 0 ].

The general rotation matrix for any angle counterclockwise about the

origin is [cos �sin].

Quadrilateral ABCD has vertices A(0, 0), B(4, 0), C(4, 6), and D(0, 6).Find the coordinates of the vertices of the image after each counterclockwise rotation. Round to the nearest tenth.

1. 45° 2. 30° 3. 60°A(0, 0), B(2.8, 2.8), A(0, 0), B(3.5, 2), A(0, 0), B(2, 3.5),C(�1.4, 7.1), D(�4.2, 4.2) C(0.5, 7.2), D(�3, 5.2) C(�3.2, 6.5), D(�5.2, 3)

4. 120° 5. 75° 6. 225°A(0, 0), B(�2, 3.5), A(0, 0), B(1, 3.9), A(0, 0), B(�2.8, 2.8),C(�7.2, 0.5),D(-5.2, �3) C(�4.8, 5.4), D(�5.8, 1.6) C(�7.1, �1.4), D(�4.2, �4.2)

ExercisesExercises

Find the coordinates of the image of �ABC with vertices A(0, 0), B(6, 0) and C(3, 4) after a

counterclockwise rotation of 30° about the origin.Enter the coordinates of the vertices of the in vertex matrix [A] andthe rotation matrix in matrix [B]. Be sure the calculator is set inDegree mode. Keystrokes: [MATRX] 2 3 0 6 3 0 0 4 [MATRX] 2 2 2 30

30 30 30 [QUIT] [MATRX] 2 [MATRX] 1 . Hold to scroll across to see the other coordinates.

The coordinates of the image of �ABC are A(0, 0), B(5.2, 3), andC(0.6, 5.0).

ENTER2nd2nd

2ndENTER)COS)SINENTER)SIN

(–)ENTER)COSENTERENTER

2ndENTERENTERENTERENTERENTERENTERENTER

ENTERENTER2nd

Example 1Example 1

Find the coordinates of the image of �ABC with vertices A(0, 0), B(6, 0) and C(3, 4) after a two

rotations of 45° counterclockwise about the origin.In order to rotate the image twice, store the vertex matrix of the first image.Keystrokes: [MATRX] 2 45

45 45 45

[QUIT] [MATRX] 2 [MATRX] 1

[ENTRY] [MATRX] 6 [ENTRY]

[ENTRY] [MATRX] 6 .

The vertices of �ABC are A(0, 0), B(0, 6), and C(�4, 3).

ENTER2nd

2nd2ndENTER2ndSTO

2ndENTER2nd2nd2nd

ENTER)COS)SINENTER)SIN(–)ENTER

)COSENTERENTER2nd

Example 2Example 2

sin cos


Spreadsheet InvestigationCramer’s Rule (Use with Lesson 4-5.)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

You have learned to solve systems of linear equations by using matrix equations and the inverse matrix. Another way to solve systems is to useCramer’s Rule. Study the spreadsheet below to discover Cramer’s Rule.

To use the spreadsheet to solve asystem of equations, write eachequation in the form below.

ax � by � c

The values for the system 6x � 3y� �12 and 5x � y � 8 are shown.In the spreadsheet, the values of a,b, and c for the first equation areentered in cells A1, B1, and C1,respectively. The values of a, b, andc for the second equation areentered in cells A2, B2, and C2,respectively.

The values in cells B10 and B11represent the solution for the system.

1. Study the formula in cell A4. Write a matrix whose determinant is found using this formula.

[A1 B1]A2 B22. Write matrices whose determinants are found using the formulas in cells A6 and A8.

[C1 B1] ; [A1 C1]C2 B2 A2 C23. Explain how the values of x and y are found using Cramer’s rule.

|C1 B1| |A1 C1|x �C2 B2

; y �A2 C2

|A1 B1| |A1 B1|A2 B2 A2 B2Use the spreadsheet to solve each system of equations.

4. 6x � 3y � �12 5. 5x � 3y � 19 6. 8x � 3y � 115x � y � 8 7x � 2y � 8 6x � 9y � 15(4, �12) (2, �3) (1.6, 0.6)

7. 0.3x � 1.6y � 0.44 8. 3y � 4x + 28 9. y � �0.5x � 40.4x � 2.5y � 0.66 5x � 7y � 8 y � 4x � 5(0.4, 0.2) (�4, 4) (2, 3)

ExercisesExercises

Graphing Calculator InvestigationUsing Tables to Factor by Grouping (Use with Lesson 5-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

55


The TABLE feature of a graphing calculator can be used to help factor apolynomial of the form ax2 � bx � c.

Factor 10x2 � 43x � 28 by grouping.

Make a table of the negative factors of 10 � 28 or 280. Look for a pairof factors whose sum is �43.

Enter the equation y � �28

x0

� in Y1 to find the factors of 280. Then,

find the sum of the factors using y � �28

x0

� � x in Y2. Set up the table

to display the negative factors of 280 by setting �Tbl = to �1.Examine the results.

Keystrokes: 280 [TBLSET] 1 1

[TABLE].

The last line of the table shows that �43x may be replaced with �8x +(�35x).

10x2 � 43x � 28 � 10x2 � 8x � (�35x) � 28� 2x(5x � 4) � (�7)(5x � 4)� (5x � 4)(2x � 7)

Thus, 10x2 � 43x � 28 � (5x � 4)(2x � 7).

2ndENTER(–)ENTER(–)2ndENTER

+ENTERENTERVARSENTER�Y=

Factor each polynomial.

1. y2 � 20y � 96 2. 4z2 � 33z � 35 3. 4y2 � y �18 4. 6a2 � 2a � 15(y � 4)(y � 24) (4z � 5)(z � 7) (4y � 9)(y � 2) prime

5. 6m2 � 17m � 12 6. 24z2 � 46z � 15 7. 36y2 � 84y � 49 8. 4b2 � 36b � 403(2m � 3)(3m � 4) (12z � 5)(2z � 3) (6y � 7)2 (2b � 31)(2b � 13)

Example 1Example 1

Factor 12x2 � 7x � 12.

Look at the factors of 12 � �12 or �144 for a pair whose sum is �7.Enter an equation to determine the factors in Y1 and an equation tofind the sum of factors in Y2. Examine the table to find a sum of �7.Keystrokes: 144

[TBLSET] 1 1 [TABLE].

12x2 � 7x � 12 � 12x2 � 9x � (�16x) � 12� 3x(4x � 3) � 4(4x � 3)� (4x � 3)(3x � 4)

Thus, 12x2 � 7x � 12 � (4x � 3)(3x � 4).

2ndENTERENTER2ndENTER+

ENTERENTERVARSENTER�(–)Y=

ExercisesExercises

Example 2Example 2


Spreadsheet InvestigationAppreciation and Depreciation (Use with Lesson 5-7.)

NAME ______________________________________________ DATE ____________ PERIOD _____

55

1. If Mr. Blackstock chooses another property in the neighborhood that costs$99,900, what are the expected values of that home in the same periods of time?$103,896.00, $105,953.55, $116,868.87, $130,178.88

2. What would Mr. Blackstock’s profit be on the $99,900 home if he sold itafter 9 years and 3 months? $143,589.89

3. If an antique chair worth $165.00 increases in value an average of 3�12�%

every year, how much will it be worth next year? $170.78

4. Often assets like cars decrease in value over time. This asset is said todepreciate. If the value decreases by a fixed percent each year, or otherperiod of time, the amount y of that quantity after t years is given by y � a(1 � r)t, where a is the initial amount and r is the percent of decreaseexpressed as a decimal. Use a spreadsheet to find the value of a car purchased for $18,500 after 2 years, 2 years and 6 months, and 4 yearsand 3 months if the car depreciates at a rate of 12% per year.$14,326.40, $13,439.35, $10,745.41

ExercisesExercises

Michael Blackstock is considering buying a piece ofinvestment property for $95,000. The homes in the

area are appreciating at an average rate of 4% per year. Find theexpected value of the home in 1 year, 1 year and 6 months, 4 years,and 6 years and 9 months.Use rows 1 and 2 to enter the initial amount and the rate ofincrease. Then use Column A to enter the amounts of time.Enter the numbers of months as a fraction of a year since tis measured in years. Column B contains the formulas forthe value of the home.

Format the cells containing the values as currency so thatthey are displayed as dollars and cents. The expected valueof the home after each amount of time is shown in thespreadsheet.

ExampleExample

When an asset such as a house increases in value over time, it is said toappreciate. If the value increases by a fixed percent each year, or other period of time, the amount y of that quantity after t years is given by

y � a(1 � r)t,

where a is the initial amount and r is the percent of increase expressed as adecimal. You can use a spreadsheet to investigate future values of an asset.

Graphing Calculator InvestigationQuadratic Inequalities and the Test Menu(Use with Lesson 6-7.)

NAME ______________________________________________ DATE ____________ PERIOD _____

66


The inequality symbols, called relational operators, in the TEST menu can beused to display the solution of a quadratic inequality. Another method that canbe used to find the solution set of a quadratic inequality is to graph each sideof an inequality separately. Examine the graphs and use the intersect functionto determine the range of values for which the inequality is true.

Solve each inequality.

1. �x2 � 10x � 21 � 0 2. x2 � 9 � 0 3. x2 � 10x � 25 � 0{x | x �7 or x � �3} {x | �3 x 3} {x | x � �5}

4. x2 � 3x � 28 5. 2x2 � x � 3 6. 4x2 � 12x � 9 0{x | �7 � x � �4 } {x | x � �1.5 or x � 1} {x | x �1.5 or x � �1.5}

7. 23 �x2 � 10x 8. x2 � 4x � 13 � 0 9. (x � 1)(x �3) 0{x | x � 3.58 or x � 6.41} {x | �2.12 � x � 6.12} {x | x �1 or x � 3}

ExercisesExercises

Solve x2 � x � 6.

Place the calculator in Dot mode. Enter the inequality into Y1.Then trace the graph and describe the solution as an inequality.Keystrokes: [TEST] 4 6 4.

Use TRACE to determine the endpoints of the segments.Theses values are used to express the solution of the inequality,{ x | x � � 3 or x � 2 }.

ZOOM2nd+x 2Y=

Example 1Example 1

Solve 2x2 � 4x � 5 � 3.

Place the left side of the inequality in Y1 and the right side in Y2.Determine the points of intersection. Use the intersection points to express the solution set of the inequality. Be sure to set the calculator to Connected mode.Keystrokes: 2 4 5 3

6.

Press [CALC] 5 and use the key to move the cursor to the left of the first intersection point. Press . Then move the cursor to the right of the intersection point and press

. One of the values used in the solution set is displayed.Repeat the procedure on the other intersection point.

The solution is { x | �3.24 � x � 1.24}.

ENTER

ENTER

ENTER

2nd

ZOOM

ENTERENTER—+x 2Y=

Example 2Example 2

[�4.7, 4.7] scl:1 by [�3.1, 3.1] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1

[�10, 10] scl:1 by [�10, 10] scl:1


You have learned the Location Principle, which can be used to approximatethe real zeros of a polynomial.

In the spreadsheet above, the positive real zero of ƒ(x) � x2 � 2 can beapproximated in the following way. Set the spreadsheet preference to manualcalculation. The values in A2 and B2 are the endpoints of a range of values.The values in D2 through J2 are values equally in the interval from A2 toB2. The formulas for these values are A2, A2 � (B2 � A2)�6, A2 � 2*(B2 � A2)/6,A2 � 3*(B2 � A2)/6, A2 � 4*(B2 � A2)/6, A2 � 5*(B2 � A2)/6, and B2,respectively.

Row 3 gives the function values at these points. The function ƒ(x) � x2 � 2 isentered into the spreadsheet in Cell D3 as D2^2 � 2. This function is thencopied to the remaining cells in the row.

You can use this spreadsheet to study the function values at the points incells D2 through J2. The value in cell F3 is positive and the value in cell G3is negative, so there must be a zero between �1.6667 and 0. Enter these values in cells A2 and B2, respectively, and recalculate the spreadsheet. (Youwill have to recalculate a number of times.) The result is a new table fromwhich you can see that there is a zero between 1.41414 and 1.414306.Because these values agree to three decimal places, the zero is about 1.414.This can be verified by using algebra.

By solving x2 � 2 � 0, we obtain x � ��2�. The positive root is x � ��2� � 1.414213. . . , which verifies the result.

Spreadsheet InvestigationApproximating the Real Zeros of Polynomials(Use with Lesson 6-5.)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

1. Use a spreadsheet like the one above to approximate the zero of ƒ(x) � 3x � 2 to threedecimal places. Then verify your answer by using algebra to find the exact value of theroot. The spreadsheet gives x � 0.667. By solving for x algebraically,x � �

23

�. So, the approximation is correct.

2. Use a spreadsheet like the one above to approximate the real zeros of f(x) � x2 � 2x � 0.5.Round your answer to four decimal places. Then, verify your answer by using the quadratic formula. The process gives �1.7071 and �0.2929 to the nearest ten-thousandth. The quadratic formula gives x � �1 � �

�22�

�. �1 � ��22�

� ��1.7071 and �1 � �

�22�

� � �0.2929.3. Use a spreadsheet like the one above to approximate the real zero of ƒ(x) � x3 � �

32�x2 � 6x � 2

between � 0.4 and � 0.3. �0.3781 to the nearest ten-thousandth

ExercisesExercises

Graphing Calculator InvestigationRational Root Theorem (Use with Lesson 7-6.)

NAME ______________________________________________ DATE ____________ PERIOD _____

77


The following program performs synthetic division and displays thedepressed polynomial coefficients in rational form. The program will allowthe testing of possible rational zeros of a polynomial function.

PROGRAM:SYNTHDIVDisp "DEGREE OF DIVIDEND" P�1→P Q→L2(P)Input M Disp "COEFFICIENT" P+1→PDisp "COEFFICIENTS?" Input A If P�M+1Disp "0�SAME" A→L1(P) Goto 3Disp "1�QUOTIENT" If PM�1 StopDisp "2�NEW" Goto 1 Lbl 4Input U Lbl 2 0→PDisp "POSSIBLE ROOT" 1→P Lbl 5Input R 0→S 1�P→PIf U�0 Lbl 3 L2(P)→L1(P)Goto 2 L1(P) →F If PM�1If U�1 F�S→Q Goto 5Goto 4 Disp Q � Frac Goto 20→P Pause Lbl 1 RQ→S

Find all of the rational zeros of f(x) � 2x3 � 11x2 � 12 x � 9.

Use the program to test possible zeros.Keystrokes: [SYNTHDIV] 3 2

1 2 11 12 9 .Press until the screen displays Done.

The column of numbers are the coefficients of the depressed polynomial.Since the last number is not zero, press 3 . Choose 0 for the same coefficients. Press 1 then until finished. Repeat this until a zero is found. Then press 2 for the degree of the depressed polynomial and 1 for the quotient.

The zeros are 3, 3, and �12�.

ENTER

ENTER

ENTER(–)ENTER

ENTERENTER

ENTER

ENTERENTERENTER(–)ENTERENTERENTER

ENTERENTERENTERPRGM

Find all the zeros of each function.

1. f(x) � x3 � 8x2 � 23x � 30 1, �3, 10 2. f(x) � x3 �7x2 � 2x � 40 �2, 4, 5

3. f(x) � 2x3 � x2 � 32x � 16 4, �4, �12

� 4. f(x) � x4 � x3 � 11x2 � 9x � 18 1, �2, 3, �3

5. p(x) � 3x4 � 11x3 � 11x2 � x � 2 �1,�2, �13

� 6. p(x) � x4 � 2x3 � x2 � 8x � 12 �1, 3, �2i

7. p(x) � 3x5 � x4 � 243x � 81 3, �3,��13

� 8. p(x) � 3x4 � 13x3 � 15x2 � 4 �2,��1 �6

�13��

ExampleExample

ExercisesExercises


Spreadsheet InvestigationOperations on Functions (Use with Lesson 7-7.)

NAME ______________________________________________ DATE ____________ PERIOD _____

77

Study and use the spreadsheet above.

1. Find k(x) � (3x � 2) � (x2 � 2x). How does it compare to h(x)?k(x) � x2 � x � 2 � h(x)

2. Change the functions in the spreadsheet to f(x) � �2x

�, g(x) � 1 � x2, and

h(x) � 1 � �2x

� � x2. How are these functions related? Is it true that

f(x) � g(x) � h(x)? (f � g)(x) � h(x); yes

3. Make a conjecture about (f � g)(x) for any functions f(x) and g(x).(f � g)(x) � f(x) � g(x)

4. Make a conjecture about (f � g)(x) for any functions f(x) and g(x). Use thespreadsheet to test your conjecture. Does it appear to be true? Explainyour answer. (f � g)(x) � f(x) � g(x); See students’ work.

Find (f � g)(x), (f � g)(x), for each f(x) and g(x). Use the spreadsheetto find function values to verify your solutions. 5-7. See students’spreadsheets.

5. f(x) � 6x � 8 6. f(x) � x2 � 1 7. f(x) � 10x2

g(x) � 9 � x g(x) � 3x � 4 g(x) � 6 � x2

7x � 17; 5x � 1 x2 � 3x � 3; x2 � 3x � 5 9x2 � 6; 11x2 � 6

ExercisesExercises

It is possible to perform operations on functions such as addition, subtrac-tion, multiplication and division. You can use a spreadsheet to investigatethe relationships among functions.

Consider the functions f(x) � 3x � 2, g(x) � x2 � 2x, and h(x) � x2 � x + 2.Find the function values of each function for several values of x.Does it appear that f(x) � g(x) � h(x)?

Use Column A for the chosen values of x.Columns B, C, and E are f(x), g(x), and h(x)respectively. Use Column D for f(x) � g(x).

For every value of x, f(x) � g(x) � h(x).

Graphing Calculator InvestigationMatrices and Equations of Circles (Use with Lesson 8-3.)

NAME ______________________________________________ DATE ____________ PERIOD _____

88


A graphing calculator can be used to write the equation of a circle in theform x2 � y2 � Dx � Ey � F � 0 given any three points on the circle.

Write the equation of the circle that passes through the given points. Identify the center and

radius of each circle.

a. A(5, 3), B(�2, 2), and C(�1, �5)

Substitute each ordered pair for (x, y) in x2 � y2 � Dx � Ey � F � 0 to formthe a system of equations.5D � 3E � F � �34 �2D � 2E � F � �8 �D � 5E � F � �26Solve the system using a matrix equation to find D, E, and F. Replace thecoefficients in the expanded form. Then, complete the square to write theequation in standard form to identify the center and radius.Keystrokes: [MATRX] 3 3 5

3 1 2 2 1 1 5 1 [MATRX] [EDIT] 2

3 1 34 8 26

[QUIT] [MATRX] [MATRX] 2

.Thus, D � �4, E � 2, and F � �20.The expanded form is x2 � y2 � 4x � 2y � 20 � 0.After completing the square, the standard form is (x � 2)2 � (y � 1)2 � 25.The center is ( 2, �1), and the radius is 5.

b. A(�2, 3), B(6, �5), and C(0, 7)

Find a system of equations. Then enter the equations into an augmentedmatrix. Reduce the matrix to row reduced echelon form using the rref(command. The row reduced echelon form of an augmented matrix willdisplay the solution to the system.�2D � 3E + F � �13 6D � 5E � F � �61 7E � F � �49Keystrokes: Enter the system of equations as [A], a 3 � 4 augmented matrix. Then use the reduced row echelon form by pressing [MATRX] [B] [MATRX] .The solution is D = �10, E = �4, and F = �21. The expanded formis x2 � y2 � 10x � 4y � 21 = 0, standard form is (x � 5)2 � (y � 2)2

� 50. The center is (5, 2) and the radius is 5�2�.

ENTER)ENTER2ndALPHA

2nd

ENTER

2nd x –1ENTER2nd2nd

ENTER(–)ENTER(–)ENTER(–)ENTERENTERENTER

2ndENTERENTER(–)ENTER

(–)ENTERENTERENTER(–)ENTERENTERENTER

ENTERENTERENTER2nd

ExampleExample

Write the equation of the circle that passes through the givenpoints. Identify the center and radius of each circle.

1. (0, �1), (�3, �2), and (�6, �1) 2. (7, �1), (11, �5), and (3, �5) 3. (�2, 7), (�9, 0), and (�10, �5)x2 � y2 � 6x � 6y � 7 � 0; x2 � y2 � 14x � 10y �58 � 0; x2�y2� 6x � 10y � 135 � 0;C(�3, 3), R � 5 C(7, �5), R � 4 C(3, �5), R � 13

ExercisesExercises


Spreadsheet InvestigationParabolas (Use with Lesson 8-2.)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

The spreadsheet below uses the equation of a parabola in the form y � a(x � h)2 � k or x � a(y � k)2 � h to find information about theparabola. x or y is entered in Column D and the values of a, h, and kare entered into Columns A, B, and C respectively.

1. Which row represents the equation y � 3x2 � 24x � 50? row 3

2. Write the standard form of the equation represented by row 2.

x � �14

� (y � 1)2 � 33. What formula should be used in cell F2? 1/ABS(A2)

4. Find the vertex, length of latus rectum, axis of symmetry, focus, directrix,and direction of opening of a parabola with equation (y � 8)2 � �4(x � 4).(8, 4); 4; y � 4; (7, 4); x � 9; left

ExercisesExercises

You have learned many of thecharacteristics of parabolaswith vertical and horizontalaxes of symmetry. The information is summarized in the table at the right. Youcan use what you know to create a spreadsheet to analyze given equations ofparabolas.

form of equation y � a(x � h)2 � k x � a(y � k)2 � hvertex (h, k) (h, k)axis of symmetry x � h y � kfocus (h, k � �

41a�) (h � �

41a�, k)

directrix y � k � �41a� x � h � �

41a�

direction of opening upward if a 0, right if a 0, leftdownward if a � 0 if a � 0

length of latus |�1a

�| units |�1a

�| unitsrectum

Graphing Calculator InvestigationHorizontal Asymptotes and Tables(Use with Lesson 9-3.)

NAME ______________________________________________ DATE ____________ PERIOD _____

99


The line y � b is a horizontal asymptote for the rational function f(x) if f(x) → b as x → � or as x → � �. The horizontal asymptote can be found byusing the TABLE feature of the graphing calculator.

ExercisesExercises

Find the horizontal asymptote for each function.

a. f(x) � �x2 � 41x � 5�

Enter the function into Y1. Place [TblSet] in the Ask mode. Enter thenumbers 10,000, 100,000, 1,000,000, and 5,000,000 and their opposites inthe x-list.Keystrokes: 1 4 5 [TBLSET] [TABLE]. Then enter thevalues for x.

Notice that as x increases, y approaches 0. Thus, when y � 0 is thehorizontal asymptote.

b. f(x) � �2x2 �3x

52

x � 6�

Enter the equation into Y1. Enter the numbers 10,000, 100,000,1,000,000, and 5,000,000 and their opposites in the x-list. Note thepattern. As x increases, y approaches 1.5. Thus, y � 1.5 is thehorizontal asymptote.

2ndENTER

2nd)—+x 2(�Y=

ExampleExample

Find the horizontal asymptote for each function.

1. f(x) � �x2�x

1� y � 2 2. f(x) � �2x2x�

2

7�x

1� 12� y � �

12

� 3. f(x) � �2x3 �62xx3

2 � 2� y � 3

4. f(x) � �3x2 �25xx � 1� y � 0 5. f(x) � �

15x2 �x3

3x � 7� y � 0 6. f(x) � y � 0

7. f(x) � �5xx2

��

23

� none 8. f(x) � �2x2 �6x

33

x � 6� none 9. f(x) � �2x

2� 4� none

x3 � 8x2 � 4x � 11��x4 � 3x3� 4x � 6

You have learned to solve problems involving direct, inverse, and joint variation.Many physical situations involve at least one of these types of variation. Forexample, according to Newton’s law of universal gravitation, the weight of amass near Earth depends on the distance between the mass and the centerof Earth. Study the spreadsheet below to determine the type of variationthat exists between the quantity of an astronaut’s weight and the distance of the astronaut from the center of Earth.

In the spreadsheet, the values for the astronaut’s weight in newtons areentered in the cells in column A, and the values for the astronaut’s distancein meters from the center of Earth are entered in cells in column B. Column Ccontains the astronaut’s distance from Earth’s surface.


Spreadsheet InvestigationVariation (Use with Lesson 9-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

99

1. Use the values in the spreadsheet to make a graph of the astronaut’s weight plotted against the astronaut’s distance from Earth’s center.

2. Based on your graph, is this an inverse or direct variation? inverse

3. Write an equation that represents this situation. LetW represent the astronaut’s weight, k the constant ofvariation, and R the distance from Earth’s center.

W � �RK

2�

4. Use the equation to find the weight of the astronaut at these distances from Earth’s surface. (Hint: Remember to add these values to the value in cell B2 to find the distance from Earth’s center.)a. 145,300,000 m b. 65 m c. 25,600 m

1.299615 N 734.5494 N 728.7047 N

d. 300,800,700 m e. 6580 m f. 180,560 m0.316872 N 733.0515 N 694.6873 N

ExercisesExercises

Graphing Calculator InvestigationRegression Equation Lab (Use with Lesson 10-1.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1010


A graphing calculator can be used to determine a regression equation thatbest fits a set of data. This activity requires tiles labeled on one side, and acontainer.

Collect the DataStep 1 Place the tiles on the desktop and count the total number. Record

the total number. Then place the tiles in the container and gentlyshake.

Step 2 Pour the tiles onto the desktop, remove all the tiles with a labelshowing, and set these aside. Count the remaining tiles without thelabels showing and return them to the container.

Step 3 Record the data in atable like this one.

Step 4 Repeat step 2 and 3 until the number of tiles without labels is zeroor the number remains constant.

Step 5 Take the tiles that were set aside in Step 2 and pour them out ofthe container onto the desktop. Remove the tiles without the labelshowing and count the tiles with the label showing. Repeat thisprocess until all the tiles have been removed.

Step 6 Record the data in atable like this one.

Analyze the Data 1-6. Answers will vary.

1. Enter trials in L1 and number of tiles without label showing in L2. Entertrials in L3 and number of tiles with the label showing in L4.

2. Use [STATPLOT] to make a scatter plot. Make a graph on paper for eachplot. Record the window used. Describe the pattern of the points.

3. From the [CALC] menu find the regression equation that best fitsthe data. Record the two closest equations, rounding values to the nearesthundredths. List and discuss the r and/or r2 values. Also include thegraphs in determining the best-fitting regression equation.

4. Sketch your best-fit regression equation choice for each scatter-plot on paper.

5. Describe any problems with the data or the regression equations.

6. Insert (0, total number of tiles) in the tables and the lists. Describe theeffect on the graphs. What happens with [PwrReg] and [ExpReg] whenthis ordered pair is inserted? Explain why this occurs?

STAT

Trials Number of tiles without label showingx y12

Trials Number of tiles with the label showingx y12


Spreadsheet InvestigationNet Present Value (Use after Lesson 10-6.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1010

1. If the NPV is greater than the cost, the investment will pay for itself.Based on the spreadsheet shown above, would it be cost-effective for thecompany to buy the van? Explain. The cost is actually about $75greater than the NPV, so it would not be cost-effective to buythe van.

2. Four times a year, Josey and Drew publish a magazine. They want to buy acolor printer that costs $1750. The cost of capital for this purchase wouldbe 6%. They are planning to raise the price of their magazine from $1 to$2. Create a spreadsheet to determine the NPV for this purchase.a. The last issue of the magazine sold 500 copies. If each issue of the magazine

printed in color sells 100 copies more than the previous issue, is theprinter a good investment after one year? Explain. No, after oneyear the NPV is only about $1682.14.

b. If the sales of the magazine continue to rise at the same rate, is theprinter a good investment after two years? Yes, after two years theNPV is about $5210.28. The NPV is about $3460.28 greaterthan the cost.

3. a. Calculate the NPV for an investment over a period of six years if thecost of capital is 4.5% and the investment will bring a cash flow of $750every year. The NPV would be about $3868.40.

b. Would this be a good investment of $3000? Explain? Yes, the NPV is$1131.60 greater than the cost.

ExercisesExercises

You have learned how to use exponential and logarithmic functions to performa number of financial analyses. Spreadsheets can be used to perform manytypes of analyses, such as calculating the Net Present Value of expendituresor investments. For example, when a businessowner is considering a major purchase, it is agood idea to find out whether the investmentwill be profitable in the future. Consider theexample of a local restaurant-delivery servicethat is debating whether to buy a van for$8000. The owners of the company estimatethat the van will bring in $2500 per year overfour years. They can use the following formulato find the present value of the future cash flowto find the Net Present Value (NPV), that is,how much the profits would be worth in today’s

dollars. NPV � �(1C�Fn

r)n�, where CFn � the cash

flow in period n and r � the cost of capital, which is eitherthe interest that will be paid on a loan or the interest thatthe money would earn if it were invested.

Graphing Calculator InvestigationRecursion and Iteration (Use with Lesson 11-6.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1111


A graphing calculator can be used to perform iterations and recursions.

Find the first 3 iterates of f(x) = 4x +15 if x0 = 5.

Store x0 in X. Then enter the expression on the home screen. Storethe result to X. Repeat the calculation for each iterate.Keystrokes: 5 4 15

.

x1 � 35, x2 � 155, and x3 � 635

ENTERENTERENTER

STO+ENTERSTO

Find the first three iterates of each function.

1. f(x) � 6x � 12 if x0 � 5 2. f(x) � 2x2 � 3 if x0 � �1

x1 � 42, x2 � 264, x3 � 1596 x1 � �1,x2 � �1,x3 � �1

3. f(x) � x2 � 4x � 5 if x0 � 1 4. f(x) � 2x2 � 2x � 1 if x0 � �12�

x1 � 2, x2 � 1, x3 � 2 x1 � �52

�, x2 � �327�, x3 � �

14245�

A bank account has an initial balance of $11,250.00. Interest is paidat the end of each year. Find the account balance under the giveninterest rate after the stated time period.

5. 3.8%, 2 years 6. 4.75%, 5 years 7. 6.05%, 10 years 8. 7.44%, 15 years$12,121.25 $14,188.05 $20,242.27 $33,009.77

Example 1Example 1

A savings account has an initial balance of $3000.00.At the end of each year, the bank pays 6% interest and

charges a $20 annual fee. Find the account balance after 6 years.

Store the initial value and enter an expression to calculate the balance at the end of a year.Keystrokes: 3000 1.06 20

.

At the end of six years, the account has a balance of $4116.05.

ENTERENTERENTERENTERENTERENTER

STO—ENTERSTO

Example 2Example 2

ExercisesExercises


You have learned about the characteristics of numbers in a sequence. Aspreadsheet can calculate a sequence and enable you to find the sum ofterms in the series.

Spreadsheet InvestigationSequences and Series (Use after Lesson 11-2.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1111

1. Create a spreadsheet like the one in the example above. Record the initialsequence as �4, �1, and 2. Repeat the process you followed in the example.What are the next six numbers in the sequence?5, 8, 11, 14, 17, and 20

2. Describe the steps the spreadsheet program completes to find the nextterm in the sequence. First, the program calculates the commondifference by subtracting any term from its succeeding term.Then, it adds the common difference to the last term to findthe next term in the sequence.

3. Use the spreadsheet to find the value for the 16th term in the sequence.41

4. Find the sum of the 3rd through 13th terms in the sequence. 187

ExercisesExercises

Create a spreadsheet like the one below and enter the first three terms of a sequence. Find the first ten

terms of the sequence. Then find the sum of the first ten terms of theseries.

Highlight cells B2 through D2 and move your cursor to any corner of thehighlighted cells until a black cross appears. Drag across the row and releaseit at cell K2. The next values in the sequence will appear in the cells.

To find the sum of the first 10 terms in the series, highlight the cells containing the terms, then click the � symbol on the toolbar. The sum willappear in the next cell. Note that this will work for arithmetic series only.The sum of the first ten terms of this series is 7.5

ExampleExample

Graphing Calculator InvestigationProbabilities (Use with Lesson 12-5.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1212


A graphing calculator can be used to perform calculations involving permu-tations, combinations, and probability.

There are 5 girls and 3 boys on a class committee. Asubcommittee of 3 people is being chosen at random.

What is the probability that the subcommittee will have at least 2 girls?

P(at least 2 girls) � P(2 girls) � P(3 girls). Each probability is theproduct of the combinations of girls and boys divided by the combinations of all the students taken 3 at a time.Keystrokes: 5 3 2 3 3 1 5

3 3 3 3 0 8 3 3

.

The probability that the subcommittee has at least 2 girls is �57�.

ENTER

ENTERMATHMATH�)MATH

MATH+MATH MATH(

Find each probability.

1. There are 5 girls and 4 boys on the school publications committee. A group of 5 membersis being chosen at random to attend a workshop on school newspapers. Find each probability.a. at least 3 girls b. 4 girls or 4 boys c. at least 2 boys

�1201� �

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� �1201�

2. Two cards are drawn from a standard deck of cards. Find each probability.a. both queens or both black b. both kings or both aces c. both face cards or both black

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3. Find the probability that a committee of 6 U.S. Representatives selected at random from

7 Democrats and 7 Republicans will have at least 3 Republicans on the committee. �340029

�

4. Three CDs are randomly selected from a collection of 6 rock and 5 rap CDs. Find the

probability that at least 2 are rock. �1393�

Example 1Example 1

Two cards are randomly selected from a standarddeck of cards. Find the probability that both cards

are kings or that both cards are red.Since these events are mutually inclusive find the combinations of 4kings taken 2 at a time plus 26 red cards taken 2 at a time minus 2red kings taken 2 at a time divided by the combinations of 52 cardstaken 2 at a time.Keystrokes: 4 3 2 26 3 2 2

3 2 52 3 2 .

The probability of choosing 2 kings or two red cards is �25251�

.

ENTERENTERMATHMATH�)

MATH—MATH+MATH(

Example 2Example 2

ExercisesExercises


You have learned the formulas for the number of permutations of n objectstaken r at a time, P(n, r), and the number of combinations of n objects takenr at a time, C(n, r). You are going to set up a spreadsheet like the one shownbelow to perform analyses of these functions.

In the spreadsheet, the values in row 1 represent n, the values in row 2 represent r, and the formulas for P(n, r) and C(n, r) are in rows 3 and 4,respectively.

The formula to calculate P(n, r) is �FACT(B1)/FACT(B1-B2).

FACT is a special function from the function list and should not be enteredfrom the letters on the keyboard. Enter the formula in B3. Then drag thecursor across the row to copy the formula into cells C3 through G3.

The formula for C(n, r) is �FACT(B1)/(FACT(B1�B2)*FACT(B2)) and shouldbe entered in cell B4. Copy the formula into cells C4 through G4.

Spreadsheet InvestigationPermutations and Combinations (Use after Lesson 12-2.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1212

1. Compare the values of P(n, r) and C(n, r) for n � 5 and r � 0 through 5, aswell as for two other choices of n and r. Most of the values of P(n, r)are much larger than the corresponding values of C(n, r). Thevalues of P(n, r) tend to increase, while the values of C(n, r)tend to increase and then decrease.

2. Several identities hold for P(n, r) and C(n, r). Use the spreadsheet to verifythe following identities by finding three examples of each. 2a-2c. Seestudents’ work.a. P(n, n) � P(n, n � 1)

b. C(n � 1, r) � C(n, r � 1) � C(n, r)

c. C(n, 0) � C(n, 1) � C(n, 2) � . . . � C(n, n) � 2n

ExercisesExercises

Graphing Calculator InvestigationLaw of Sines: Ambiguous Case (Use with Lesson 13-4.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1313


A graphing calculator can be used to illustrate the Ambiguous Case for theLaw of Sines. This program constructs a visual representation of given information. From the drawing, the number of solutions can be determined.LAWSINES:Lbl 1 bcos(A→D:bsin(A→EDisp "A�" max(1,int((D�a�.999))→XmaxInput A min(�1,D-int((D�a�.999))→XminDisp "a�" int((E�a�.999)→YmaxInput M min(�1,Ymax�2(Xmax�Xmin)/3) →YminDisp "b�" ZsquareInput B Line(0,0,D,E){0,1,2}→L1:ML1�B→L2 Line(0,0,Xmax,0)LinReg(ax�b) L1,L2 Circle(D,E,a)AxesOff:ClrDraw

In �ABC, A � 35°, a � 34, and b � 45. Determinewhether �ABC has one, two, or no solutions.

Run the program and enter the given information. Examine theresulting figure for intersection points.Keystrokes: to highlight the LAWSINES then press

. Follow the prompts. A � 35 a � 34 b� 45 . Notice that the circle whose radius is a units intersectsthe horizontal segment twice. This indicates there are two solutionsor two triangles are possible.

ENTER


PRGM

Determine whether each triangle has one, two, or no possible solutions.

1. A � 44.3°, a � 22, and b � 20.1 1 2. A � 126°, a � 12, and b � 7 2

3. A � 21°, a � 2, and b � 3 2 4. A � 55°, a � 11, and b � 15 0

5. A � 112°, a � 5, and b � 7 0 6. B � 38.6°, b � 22.9, and c � 33.7 2

7. C � 30°, c � 20.2, and b � 40.4 1 8. B � 50°, b � 13, and c � 15 2

ExampleExample

ExercisesExercises


Spreadsheet InvestigationCofunctions (Use with Lesson 13-1.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1313

1. Use the spreadsheet to make a graph of the sine values for the angles from 0° to 360°. Then make a graph of the cosine values.

2. If f(x) and g(x) are cofunctions, then f(x) � g(90° � x). Compare the shapesof the graphs. How can you tell that sine and cosine are cofunctions by their shapes? The graphs have a similar shape, butthe cosine graph is shifted 90° compared with the sine graphbecause they are cofunctions with sin � � cos (90 � �).

ExercisesExercises

The functions of sine and cosine are cofunctions. Set up a spreadsheet likethe one shown below to investigate the relationships between cofunctions.

In the spreadsheet, the values in row 1 are the angle values in degrees. Thevalues in rows 2 and 3 are the calculated values for the sine and cosine foreach angle, respectively. To use the spreadsheet to find the functions for anyangle, first enter each function into the spreadsheet in the form shown below.

�SIN(B1*PI()/180) �COS(B1*PI()/180)

This form of the formula contains additional information that is necessaryfor the spreadsheet to use degrees to calculate the answer. Without it, thespreadsheet cannot recognize that the angle is measured in degrees and willreturn the wrong answer.

Then, complete the spreadsheet by entering the angles up to 360° whosemeasures are mulitples of 30° and 45°.

Graphing Calculator InvestigationSinusoidal Equations (Use with Lesson 14-2.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1414


A graphing calculator can be used to verify a sinusoidal regression equationin the form y � a sin (bx � c) � d given four data points. The sinusoidalregression is found under [CALC] [C].STAT

As a person rides a Ferris wheel, the person’s distance fromthe ground varies sinusoidally with time. Let t be the

number of seconds that have elapsed since the Ferris wheel started. Therider’s position when the last seat is filled and the Ferris wheel starts iswhen t � 0. Suppose it takes 3 seconds to reach the top of the Ferris wheel,43 feet above the ground. The diameter of the wheel is 40 feet, and it makesa revolution every 8 seconds. Create a table of values and write the sinusoidalequation.

Keystrokes: Enter the data in L1 and L2. Choose an appropriatewindow. Use [STATPLOT] to graph the points. [C]

[L1] [L2] .

a � 20, b � �4π

�, c � 1, and d � 23

h(t) � 20 sin �4π

�(t � 1) � 23

ENTERENTERENTERVARS,2nd,2nd

ALPHASTAT

As the paddlewheel of a steamboat turns, a point on the paddleblade moves so that its distance, h, from the water’s surface is asinusoidal function of time. The wheel’s diameter is 18 feet, and itcompletes a revolution every 10 seconds. The height of the point atvarious times is shown in the table.

1. Why is the height the same after 14 seconds as it is after 4 seconds? The wheel completes a revolution every 10 seconds.

2. What are the values of a, b, c, and d?

a � 9, b � �5π

�, c � �32

�, and d � 7

3. Write a regression equation.

h(t) � 9 sin [�5π

�(t � �32

�)] � 7

ExampleExample

ExercisesExercises

t sec. 1 3 5 7 9 11h(t) ft. 23 43 23 3 23 43

t 1.5 4 6.5 9 11.5 14(seconds)

h(t)7 16 7 �2 7 16(feet)


Spreadsheet InvestigationTrigonometric Identities (Use after Lesson 14-3.)

NAME ______________________________________________ DATE ____________ PERIOD _____

1414

1. Study the values in Columns D and E. What identity seems to be possiblefrom this pattern? tan (�B) � �tan B

2. Enter the formula SIN(B) in Column F. Then enter the formulaCOS(B)*TAN(B) in Column G. What identity do these two new columnssuggest? sin B � cos B tan B

3. Make a column with the formula SIN(PI()-B). What identity do you discover? sin (π� B) � sin B or sin (π� B) � cos B tan B

ExercisesExercises

A trigonometric identity holds for all values of where each expression isdefined. For example, sin � cos � tan . You have learned to provealgbraically that an equation is an identity. You can use a spreadsheet to testequations for specific values to see if an equation might be an identity.

To use the spreadsheet to test the values of expressions for different angles,enter the angle measures in the cells in Column A, and enter the expressionsfrom the equations you want to test in the columns to the right. First, enterthe formula �RADIANS(A) in Column B to convert degrees to radians.(Recall that you can do this by entering the formula RADIANS(A2) in cellB2, copying cell B2, and pasting to fill the rest of Column B.) In the spreadsheet shown, the formula SIN(B)/COS(B) is in the cells in Column C.The cells in Column D contain the formula TAN(B). Column E contains theformula TAN(�B). Notice that the values in Columns C and D agree withthe identity stated above.

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rou

gh o

bser

vati

on o

f C

olu

mn

D,w

hen

cis

neg

ativ

e th

e st

atem

ent

is n

ottr

ue.

Th

e ab

solu

te v

alu

e st

atem

ent,

c|a

�b|

�|c

a�

cb|

is s

omet

imes

tru

e;it

is

tru

e on

ly i

f c

�0.

Example

Example

A


Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Tran

sfo

rmat

ion

s: G

reat

est

Inte

ger

Fu

nct

ion

(Use

wit

h L

esso

n 2

-6.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

22

©G

lenc

oe/M

cGra

w-H

ill29

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

dis

play

tra

nsf

orm

atio

ns

to t

he

grea

test

inte

ger

fun

ctio

n.T

his

is

don

e by

usi

ng

the

int(

com

man

d u

nde

r th

e M

AT

H:

NU

Mm

enu

.Wh

en g

raph

ing

the

grea

test

in

tege

r fu

nct

ion

,it

is i

mpo

rtan

t to

set

the

calc

ula

tor

to D

ot m

ode.

Gra

ph

eac

h f

un

ctio

n.E

valu

ate

it f

or x

�1,

x�

1.3,

and

x�

2.C

omp

are

the

grap

h o

f th

e fu

nct

ion

to

the

grap

h o

f f(

x) �

[[x]

].

1.g(

x)�

[[x]]

�3

2.g(

x) �

[[x�

2]]

3.g(

x) �

[[x+

4]]

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

g(1

)��

2, g

(1.3

) ��

2, g

(2) �

�1

g(1

) ��

1, g

(1.3

) ��

1, g

(2) �

0g

(1) �

5, g

(1.3

) �5,

g(2

) �6

shift

ed d

own

3 un

itssh

ifted

rig

ht 2

uni

tssh

ifted

left

4 un

its

4.g(

x) �

[[�x]

]5.

g(x)

�[[2

x]]

6.g(

x) �

3[[x

]]

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

[�4.

7, 4

.7] s

cl: 1

by

[�3.

1, 3

.1] s

cl: 1

g(1

)��

1, g

(1.3

) ��

2, g

(2) �

�2

g(1

) �2,

g(1

.3) �

2, g

(2) �

4g

(1)�

3, g

(1.3

)�3,

g(2

)�6

refle

cted

acr

oss

the

y-ax

isco

mpr

esse

d by

�1 2�ho

rizo

ntal

lyex

pand

ed b

y 3

vert

ical

ly

Exercises

Exercises

Gra

ph

f(x

)�

[[x]]

and

g(x

) �

[[x]]

�2

on t

he

sam

e ax

es.

Eva

luat

e ea

ch f

un

ctio

n f

or x

�0.

6,x

�1,

x�

1.3,

x�

2,x

��

0.5,

and

x�

�1.

1.C

omp

are

the

grap

hs

of t

he

fun

ctio

ns.

En

ter

f(x)

in

Y1

and

g(x)

in

Y2.

Gra

ph t

he

fun

ctio

ns.

Key

stro

kes:

5 5

2 6

4.

Use

TR

AC

Eto

eva

luat

e ea

ch f

un

ctio

n f

or t

he

give

n v

alu

es.

Key

stro

kes:

0.6

1 1.

3 2

0.5

1.1

0.6

1 1.

3 2

0.5

1.1

.

f(0.

6)�

0,f(

1) �

1,f(

1.3)

�1,

f(2)

�2,

f(�

0.5)

��

1,an

d f(

�1.

1) �

�2

g(0.

6) �

2,g(

1) �

3,g(

1.3)

�3,

g(2)

�4,

g(�

0.5)

��

1,an

d g(

�1.

1) �

0

Th

e gr

aph

of

g(x)

�[[x

�2]

]is

th

e sa

me

as t

he

grap

h o

f f(

x)�

[[x]]

shif

ted

2 u

nit

s u

p.

EN

TER

(–)

EN

TER

(–)

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

(–)

EN

TER

(–)

EN

TER

EN

TER

EN

TER

EN

TER

TR

AC

E

ZO

OM

ZO

OM

EN

TER

+)

MA

TH

EN

TER

)M

AT

HY

=

Example

Example

[�4.

7, 4

.7] s

cl:1

by

[�3.

1, 3

.1] s

cl:1

Answers (Chapter 2)

©G

lenc

oe/M

cGra

w-H

ill30

Gle

ncoe

Alg

ebra

2

Th

e sl

ope

inte

rcep

t fo

rm o

f a

lin

ear

equ

atio

n i

s y

�m

x�

b,w

her

e m

is t

he

slop

e an

d b

is t

he

y-in

terc

ept.

Rec

all

that

th

e fo

rmu

la f

or t

he

slop

e of

a l

ine

thro

ugh

(x 1

,y1)

an

d (x

2,y 2

) is

m�

�y x2 2

� �y x1 1

�.Y

ou c

an u

se t

he

form

ula

for

slo

pe

and

the

slop

e-in

terc

ept

form

to

fin

d th

e va

lue

of b

.

Spre

adsh

eet

Invest

igati

on

Usi

ng

Lin

ear

Eq

uat

ion

s(U

se w

ith L

esso

n 2

-4.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

22

Use

a s

pre

adsh

eet

to f

ind

th

e sl

ope

and

y-i

nte

rcep

t of

th

e li

ne

thro

ugh

eac

h p

air

of p

oin

ts.T

hen

wri

te a

n e

qu

atio

n o

f th

e li

ne

in s

lop

e-in

terc

ept

form

.

1.(0

,�5)

,(2,

5)2.

(4,2

),(�

3,�

5)5,

�5;

y�

�5x

�5

1, �

2; y

��

x�

2

3.(�

1,�

4),(

1,3)

4.(�

4,�

9),(

8,3)

�7 2�, �

�1 2�; y

��

�7 2�x �

�1 2�� 11 21 �

, �5 �

1 3�; y

�� 11 21 �

x�

5 �1 3�

5.(1

2,9)

,(10

,10)

6.(�

1.5,

3.1)

,(0.

9,1.

9)

��1 2�,

15;

y�

��1 2�x

�15

�0.

5, 2

.35;

y�

�0.

5x�

2.35

7.D

oes

the

spre

adsh

eet

wor

k w

hen

tw

o po

ints

hav

e th

e sa

me

x-co

ordi

nat

es?

Exp

lain

.N

o;

Th

e sl

op

e is

un

def

ined

.

Exercises

Exercises

Sta

te t

he

slop

e an

d y

-in

terc

ept

of t

he

grap

h o

f th

e li

ne

thro

ugh

(5,

2) a

nd

(4,

1).T

hen

wri

te a

n e

qu

atio

n o

f th

eli

ne

in s

lop

e-in

terc

ept

form

.

Ste

p1

Use

Col

um

ns

A a

nd

B t

o re

pres

ent

the

firs

t po

int,

and

Col

um

ns

Can

d D

to

repr

esen

t th

e se

con

d po

int

on t

he

lin

e.E

nte

r th

e fo

rmu

lafo

r sl

ope

in C

olu

mn

E.

Ste

p2

Su

bsti

tute

on

e of

th

e gi

ven

poi

nts

in

to t

he

slop

e-in

terc

ept

from

an

dso

lve

for

b.S

ince

we

know

th

e sl

ope

of t

he

lin

e,w

e ca

n s

olve

for

b.

y�

mx

�b

Slo

pe-i

nte

rcep

t fo

rmy 1

�m

x 1�

bS

ubs

titu

te (

x 1,y

1).

y 1�

mx 1

�b

Sol

ve f

or b

.

En

ter

this

for

mu

la i

nto

Col

um

n F

usi

ng

the

nam

es o

f th

e sp

read

shee

t ce

lls.

Th

e sl

ope

of t

he

lin

e th

rou

gh (

5,2)

an

d (4

,1)

is 1

an

dth

e y-

inte

rcep

t is

�3.

Th

us,

the

equ

atio

n o

f th

e li

ne

slop

e-in

terc

ept

form

is

y�

1x�

(�3)

or

y�

x�

3.

Example

Example

Answers (Chapter 3)


A

Gra

ph

th

e sy

stem

x�

3y�

�7,

5x�

y�

13,x

�6y

��

9,3x

�2y

��

7,an

d f

(x,y

)�4x

�3y

.Fin

d t

he

coor

din

ates

of t

he

feas

ible

reg

ion

.Th

en f

ind

th

e m

axim

um

an

d m

inim

um

val

ues

for

the

syst

em.

Sol

ve e

ach

in

equ

alit

y fo

r y.

En

ter

each

bou

nda

ry e

quat

ion

in

th

e Y

�sc

reen

.Fin

d th

e ve

rtic

es o

f th

e fe

asib

le r

egio

n.T

hen

fin

d th

e va

lues

of f

(x,y

) to

det

erm

ine

the

max

imu

m a

nd

min

imu

m v

alu

es.

Key

stro

kes:

1 3

7 3

5 13

1

6 3

2 3

2 7

2 6

[CA

LC

] 5

[QU

IT]

[ {

]

[Y]

4 3

[Y]

[ }

]

[CA

LC

] 5

[QU

IT]

[EN

TR

Y]

[CA

LC

] 5

[QU

IT]

[EN

TR

Y]

[CA

LC

] 5

[QU

IT]

[EN

TR

Y]

.

The

max

imum

val

ue o

f the

sys

tem

is 1

8 an

d th

e m

inim

um v

alue

is�

10.

EN

TER

2nd

2nd

EN

TER

EN

TER

EN

TER

2nd

GR

AP

HE

NTE

R2n

d2n

dE

NTE

RE

NTE

R

EN

TER

2nd

GR

AP

HE

NTE

R2n

d

2nd

EN

TER

EN

TER

EN

TER

2nd

GR

AP

H

EN

TER

2nd

ALP

HA

—,

ALP

HA

,2n

d2n

dE

NTE

RE

NTE

RE

NTE

R2n

d

ZO

OM

)�

(+

)�

(E

NTE

R)

�(

—)

�(

(–)

EN

TER

+(–

)

EN

TER

)�

(+

)�

(Y

=

Example

ExampleG

raph

ing C

alc

ula

tor

Invest

igati

on

Lin

ear

Pro

gra

mm

ing

(Use

wit

h L

esso

n 3

-4.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

33

©G

lenc

oe/M

cGra

w-H

ill31

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

sto

re t

he

x- a

nd

y-co

ordi

nat

es w

hen

usi

ng

the

inte

rsec

tco

mm

and

in t

he

[CA

LC

]m

enu

.Th

is c

an b

e di

spla

yed

on t

he

hom

e sc

reen

an

d u

sed

to e

valu

ate

an e

xpre

ssio

n w

ith

xan

d y

vari

able

s.T

his

proc

ess

is u

sefu

l in

find

ing

the

vert

ices

of

the

feas

ible

reg

ion

and

dete

rmin

ing

the

max

imu

m o

r m

inim

um

val

ue

for

f(x,

y).

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl:1

Gra

ph

eac

h s

yste

m.F

ind

th

e co

ord

inat

es o

f th

e ve

rtic

es o

f th

e fe

asib

lere

gion

.Th

en f

ind

th

e m

axim

um

an

d m

inim

um

val

ues

for

th

e sy

stem

.

1.2x

�3y

�6

2.y

�4x

�6

3.y

�16

�x

3x�

2y�

�4

x�

4y�

70

�2y

�17

5x�

y�

152x

�y

�7

2x�

3y�

11f(

x,y)

�x

�3y

x�

6y�

10y

�3x

�1

f(x,

y) �

2x�

yy

�2x

�13

y�

7 �

2xf(

x,y)

�5x

�6y

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl: 1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl: 1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl: 1

(0, 2

), (

3, 0

), (

2, 5

);(�

1, 2

), (

�2,

�2)

, (3,

1),

(4,

�1)

;(5

.5, 0

), (

6.5,

0),

(7.

5, 8

.5),

m

in.�

3, m

ax. �

17m

in. �

�4,

max

. �9

(1.2

, 4.6

), (

2.5,

2),

(9.

66, 6

.33)

;m

in. �

24.5

, max

. �88

.5

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill32

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Bre

ak-E

ven

Po

int

(Use

wit

h L

esso

n 3

-1.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

33

1.If

Car

ly c

ould

dec

reas

e h

er a

nn

ual

ove

rhea

d to

$14

,000

,wh

at w

ould

th

ebr

eak-

even

poi

nt

be?

bet

wee

n 1

400

and

150

0 ca

nd

les

2.S

upp

ose

Car

ly d

ecre

ases

her

an

nu

al o

verh

ead

to $

14,0

00 a

nd

incr

ease

sth

e pr

ice

of a

can

dle

to $

14.0

0.W

hat

is

the

new

bre

ak-e

ven

poi

nt?

bet

wee

n 1

200

and

130

0 ca

nd

les

Exercises

Exercises

Car

ly E

rics

on i

s co

nsi

der

ing

open

ing

a ca

nd

le

bu

sin

ess.

Sh

e es

tim

ates

th

at s

he

wil

l h

ave

an a

nn

ual

over

hea

d o

f $1

5,00

0.It

cos

ts C

arly

$3.

00 t

o m

ake

a ja

r ca

nd

le,w

hic

hsh

e se

lls

for

$12.

50.W

hat

is

Car

ly’s

bre

ak-e

ven

poi

nt?

Use

Col

um

n A

for

th

e n

um

ber

of c

andl

es.C

olu

mn

s B

and

C a

re t

he

cost

an

d th

e in

com

e,re

spec

tive

ly.

Ext

end

the

row

s of

th

e sp

read

shee

t to

fin

d th

e po

int

atw

hich

the

inco

me

firs

t ex

ceed

s th

e co

st.T

he b

reak

-eve

npo

int

occu

rs b

etw

een

th

is p

oin

t an

d th

e pr

evio

us

poin

t.In

th

is c

ase,

the

brea

k ev

en p

oin

t oc

curs

bet

wee

n 1

500

and

1600

can

dles

.

Th

e ch

art

tool

of

the

spre

adsh

eet

allo

ws

you

to

grap

hth

e da

ta.T

he

grap

h v

erif

ies

the

solu

tion

.

Example

Example

You

hav

e le

arn

ed t

hat

th

e br

eak-

even

poi

nt

is t

he

poin

t at

wh

ich

th

e in

com

eeq

ual

s th

e co

st.Y

ou c

an u

se t

he

form

ula

s an

d ch

arts

in

a s

prea

dsh

eet

to f

ind

a br

eak-

even

poi

nt.


Answers (Chapter 4)

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Mat

rice

s fo

r 30

°, 4

5°, 6

0°R

ota

tio

ns

(Use

wit

h L

esso

n 4

-4.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

44

©G

lenc

oe/M

cGra

w-H

ill33

Gle

ncoe

Alg

ebra

2

Th

e ro

tati

on m

atri

x fo

r 90

°co

un

terc

lock

wis

e ab

out

the

orig

in i

s [0

�1

10

].

Th

e ge

ner

al r

otat

ion

mat

rix

for

any

angl

e

cou

nte

rclo

ckw

ise

abou

t th

e

orig

in i

s [co

s�

sin

].

Qu

adri

late

ral

AB

CD

has

ver

tice

s A

(0,0

),B

(4,0

),C

(4,6

),an

d D

(0,6

).F

ind

th

e co

ord

inat

es o

f th

e ve

rtic

es o

f th

e im

age

afte

r ea

ch

cou

nte

rclo

ckw

ise

rota

tion

.Rou

nd

to

the

nea

rest

ten

th.

1.45

°2.

30°

3.60

°A

(0,

0),

B(

2.8,

2.8

),A

(0,

0),

B(

3.5,

2),

A(

0, 0

), B

(2,

3.5

),C

(�

1.4,

7.1

), D

(�

4.2,

4.2

)C

(0.

5, 7

.2),

D(

�3,

5.2

)C

(�

3.2,

6.5

), D

(�

5.2,

3)

4.12

0°5.

75°

6.22

5°A

(0,

0),

B(

�2,

3.5

),A

(0,

0),

B(

1, 3

.9),

A(

0, 0

), B

(�

2.8,

2.8

),C

(�

7.2,

0.5

),D(

-5.2

, �3)

C(

�4.

8, 5

.4),

D(

�5.

8, 1

.6)

C(

�7.

1, �

1.4)

, D(

�4.

2, �

4.2)

Exercises

Exercises

Fin

d t

he

coor

din

ates

of

the

imag

e of

�A

BC

wit

h

vert

ices

A(0

,0),

B(6

,0)

and

C(3

,4)

afte

r a

cou

nte

rclo

ckw

ise

rota

tion

of

30°

abou

t th

e or

igin

.E

nte

r th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f th

e in

ver

tex

mat

rix

[A]

and

the

rota

tion

mat

rix

in m

atri

x [B

].B

e su

re t

he

calc

ula

tor

is s

et i

nD

egre

e m

ode.

Key

stro

kes:

[MA

TR

X]

2 3

0 6

3 0

0 4

[MA

TR

X]

2 2

2 30

30

30

30

[Q

UIT

] [M

AT

RX

] 2

[MA

TR

X]

1 .H

old

to

scro

ll a

cros

s to

see

th

e ot

her

coo

rdin

ates

.

Th

e co

ordi

nat

es o

f th

e im

age

of �

AB

Car

e A

(0,

0),B

(5.

2,3)

,an

dC

(0.

6,5.

0).

EN

TER

2nd

2nd

2nd

EN

TER

)C

OS

)S

INE

NTE

R)

SIN

(–)

EN

TER

)C

OS

EN

TER

EN

TER

2nd

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

2nd

Example

1Example

1

Fin

d t

he

coor

din

ates

of

the

imag

e of

�A

BC

wit

h

vert

ices

A(0

,0),

B(6

,0)

and

C(3

,4)

afte

r a

two

rota

tion

s of

45°

cou

nte

rclo

ckw

ise

abou

t th

e or

igin

.In

ord

er t

o ro

tate

th

e im

age

twic

e,st

ore

the

vert

ex m

atri

x of

th

e fi

rst

imag

e.K

eyst

roke

s:[M

AT

RX

] 2

45

45

45

45

[QU

IT]

[MA

TR

X]

2 [M

AT

RX

] 1

[EN

TR

Y]

[MA

TR

X]

6 [E

NT

RY

]

[EN

TR

Y]

[MA

TR

X]

6 .

Th

e ve

rtic

es o

f �

AB

C

are

A(

0,0)

,B(

0,6)

,an

d C

(�

4,3)

.

EN

TER

2nd

2nd

2nd

EN

TER

2nd

ST

O

2nd

EN

TER

2nd

2nd

2nd

EN

TER

)C

OS

)S

INE

NTE

R)

SIN

(–)

EN

TER

)C

OS

EN

TER

EN

TER

2nd

Example

2Example

2

sin

co

s

©G

lenc

oe/M

cGra

w-H

ill34

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Cra

mer

’s R

ule

(Use

wit

h L

esso

n 4

-5.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

44

You

hav

e le

arn

ed t

o so

lve

syst

ems

of l

inea

r eq

uat

ion

s by

usi

ng

mat

rix

equ

atio

ns

and

the

inve

rse

mat

rix.

An

oth

er w

ay t

o so

lve

syst

ems

is t

o u

seC

ram

er’s

Ru

le.S

tudy

th

e sp

read

shee

t be

low

to

disc

over

Cra

mer

’s R

ule

.

To

use

th

e sp

read

shee

t to

sol

ve a

syst

em o

f eq

uat

ion

s,w

rite

eac

heq

uat

ion

in

th

e fo

rm b

elow

.

ax�

by�

c

Th

e va

lues

for

th

e sy

stem

6x

�3y

��

12 a

nd

5x�

y�

8 ar

e sh

own

.In

th

e sp

read

shee

t,th

e va

lues

of

a,b,

and

cfo

r th

e fi

rst

equ

atio

n a

reen

tere

d in

cel

ls A

1,B

1,an

d C

1,re

spec

tive

ly.T

he

valu

es o

f a,

b,an

dc

for

the

seco

nd

equ

atio

n a

reen

tere

d in

cel

ls A

2,B

2,an

d C

2,re

spec

tive

ly.

Th

e va

lues

in

cel

ls B

10 a

nd

B11

repr

esen

t th

e so

luti

on f

or t

he

syst

em.

1.S

tudy

the

for

mul

a in

cel

l A4.

Wri

te a

mat

rix

who

se d

eter

min

ant

is f

ound

usi

ng t

his

form

ula.

[A1

B1 ]

A2

B2

2.W

rite

mat

rice

s w

hos

e de

term

inan

ts a

re f

oun

d u

sin

g th

e fo

rmu

las

in c

ells

A6

and

A8.

[C1

B1 ] ; [A

1C

1 ]C

2B

2A

2C

23.

Exp

lain

how

th

e va

lues

of

xan

d y

are

fou

nd

usi

ng

Cra

mer

’s r

ule

.

|C1

B1

||A

1C

1|

x�

C2

B2

; y

�A

2C

2

|A1

B1

||A

1B

1|

A2

B2

A2

B2

Use

th

e sp

read

shee

t to

sol

ve e

ach

sys

tem

of

equ

atio

ns.

4.6x

�3y

��

125.

5x�

3y�

196.

8x�

3y�

115x

�y

�8

7x�

2y�

86x

�9y

�15

(4,�

12)

(2, �

3)(1

.6, 0

.6)

7.0.

3x�

1.6y

�0.

448.

3y�

4x+

289.

y�

�0.

5x�

40.

4x�

2.5y

�0.

665x

�7y

�8

y

�4x

�5

(0.4

, 0.2

)(�

4, 4

)(2

, 3)

Exercises

Exercises

Answers (Chapter 5)


A

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Usi

ng

Tab

les

to F

acto

r b

y G

rou

pin

g(U

se w

ith L

esso

n 5

-4.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

55

©G

lenc

oe/M

cGra

w-H

ill35

Gle

ncoe

Alg

ebra

2

Th

e T

AB

LE

feat

ure

of

a gr

aph

ing

calc

ula

tor

can

be

use

d to

hel

p fa

ctor

apo

lyn

omia

l of

th

e fo

rm a

x2�

bx�

c.

Fac

tor

10x2

�43

x�

28 b

y gr

oup

ing.

Mak

e a

tabl

e of

th

e n

egat

ive

fact

ors

of 1

0�

28 o

r 28

0.L

ook

for

a pa

irof

fac

tors

wh

ose

sum

is

�43

.

En

ter

the

equ

atio

n y

��28 x0 �

in Y

1to

fin

d th

e fa

ctor

s of

280

.Th

en,

fin

d th

e su

m o

f th

e fa

ctor

s u

sin

g y

��28 x0 �

�x

in Y

2.S

et u

p th

e ta

ble

to d

ispl

ay t

he

neg

ativ

e fa

ctor

s of

280

by

sett

ing

�T

bl

= to

�1.

Exa

min

e th

e re

sult

s.

Key

stro

kes:

280

[TB

LS

ET

] 1

1 [T

AB

LE

].

Th

e la

st l

ine

of t

he

tabl

e sh

ows

that

�43

xm

ay b

e re

plac

ed w

ith

�

8x+(

�35

x).

10x2

�43

x�

28�

10x2

�8x

�(�

35x)

�28

�2x

(5x

�4)

�(�

7)(5

x�

4)�

(5x

�4)

(2x

�7)

Th

us,

10x2

�43

x�

28 �

(5x

�4)

(2x

�7)

.

2nd

EN

TER

(–)

EN

TER

(–)

2nd

EN

TER

+E

NTE

RE

NTE

RV

AR

SE

NTE

R�

Y=

Fac

tor

each

pol

ynom

ial.

1.y2

�20

y�

962.

4z2

�33

z�

353.

4y2

�y

�18

4.6a

2�

2a�

15(y

�4)

(y�

24)

(4z

�5)

(z�

7)(4

y �

9)(y

�2)

prim

e

5.6m

2�

17m

�12

6.24

z2�

46z

�15

7.36

y2�

84y

�49

8.4b

2�

36b

�40

3(2

m�

3)(3

m�

4)(1

2z�

5)(2

z�

3)(6

y�

7)2

(2b

�31

)(2b

�13

)

Example

1Example

1

Fac

tor

12x2

�7x

�12

.

Loo

k at

th

e fa

ctor

s of

12

��

12 o

r�

144

for

a pa

ir w

hos

e su

m i

s�

7.E

nte

r an

equ

atio

n t

o de

term

ine

the

fact

ors

in Y

1an

d an

equ

atio

n t

ofi

nd

the

sum

of

fact

ors

in Y

2.E

xam

ine

the

tabl

e to

fin

d a

sum

of�

7.K

eyst

roke

s:14

4

[TB

LS

ET

] 1

1 [T

AB

LE

].

12x2

�7x

�12

� 1

2x2

�9x

�(�

16x)

�12

�3x

(4x

�3)

�4(

4x�

3)�

(4x

�3)

(3x

�4)

T

hu

s,12

x2�

7x�

12 �

(4x

�3)

(3x

�4)

.

2nd

EN

TER

EN

TER

2nd

EN

TER

+

EN

TER

EN

TER

VA

RS

EN

TER

�(–

)Y

=

Exercises

Exercises

Example

2Example

2

©G

lenc

oe/M

cGra

w-H

ill36

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Ap

pre

ciat

ion

an

d D

epre

ciat

ion

(Use

wit

h L

esso

n 5

-7.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

55

1.If

Mr.

Bla

ckst

ock

choo

ses

anot

her

pro

pert

y in

th

e n

eigh

borh

ood

that

cos

ts$9

9,90

0,w

hat

are

the

expe

cted

val

ues

of t

hat

hom

e in

the

sam

e pe

riod

s of

tim

e?$1

03,8

96.0

0, $

105,

953.

55, $

116,

868.

87, $

130,

178.

88

2.W

hat

wou

ld M

r.B

lack

stoc

k’s

prof

it b

e on

th

e $9

9,90

0 h

ome

if h

e so

ld i

taf

ter

9 ye

ars

and

3 m

onth

s?$1

43,5

89.8

9

3.If

an

an

tiqu

e ch

air

wor

th $

165.

00 i

ncr

ease

s in

val

ue

an a

vera

ge o

f 3 �

1 2�%ev

ery

year

,how

mu

ch w

ill

it b

e w

orth

nex

t ye

ar?

$170

.78

4.O

ften

ass

ets

like

car

s de

crea

se i

n v

alu

e ov

er t

ime.

Th

is a

sset

is

said

to

dep

reci

ate.

If t

he

valu

e de

crea

ses

by a

fix

ed p

erce

nt

each

yea

r,or

oth

erpe

riod

of

tim

e,th

e am

oun

t y

of t

hat

qu

anti

ty a

fter

tye

ars

is g

iven

by

y�

a(1

�r)

t ,w

her

e a

is t

he

init

ial

amou

nt

and

ris

th

e pe

rcen

t of

dec

reas

eex

pres

sed

as a

dec

imal

.Use

a s

prea

dsh

eet

to f

ind

the

valu

e of

a c

ar

purc

has

ed f

or $

18,5

00 a

fter

2 y

ears

,2 y

ears

an

d 6

mon

ths,

and

4 ye

ars

and

3 m

onth

s if

th

e ca

r de

prec

iate

s at

a r

ate

of 1

2% p

er y

ear.

$14,

326.

40, $

13,4

39.3

5, $

10,7

45.4

1

Exercises

Exercises

Mic

hae

l B

lack

stoc

k i

s co

nsi

der

ing

bu

yin

g a

pie

ce o

fin

vest

men

t p

rop

erty

for

$95

,000

.Th

e h

omes

in

th

ear

ea a

re a

pp

reci

atin

g at

an

ave

rage

rat

e of

4%

per

yea

r.F

ind

th

eex

pec

ted

val

ue

of t

he

hom

e in

1 y

ear,

1 ye

ar a

nd

6 m

onth

s,4

year

s,an

d 6

yea

rs a

nd

9 m

onth

s.U

se r

ows

1 an

d 2

to e

nte

r th

e in

itia

l am

oun

t an

d th

e ra

te o

fin

crea

se.T

hen

use

Col

um

n A

to

ente

r th

e am

oun

ts o

f ti

me.

En

ter

the

nu

mbe

rs o

f m

onth

s as

a f

ract

ion

of

a ye

ar s

ince

tis

mea

sure

d in

yea

rs.C

olu

mn

B c

onta

ins

the

form

ula

s fo

rth

e va

lue

of t

he

hom

e.

For

mat

th

e ce

lls

con

tain

ing

the

valu

es a

s cu

rren

cy s

o th

atth

ey a

re d

ispl

ayed

as

doll

ars

and

cen

ts.T

he

expe

cted

val

ue

of t

he

hom

e af

ter

each

am

oun

t of

tim

e is

sh

own

in

th

esp

read

shee

t.

Example

Example

Wh

en a

n a

sset

su

ch a

s a

hou

se i

ncr

ease

s in

val

ue

over

tim

e,it

is

said

to

appr

ecia

te.I

f th

e va

lue

incr

ease

s by

a f

ixed

per

cen

t ea

ch y

ear,

or o

ther

pe

riod

of

tim

e,th

e am

oun

t y

of t

hat

qu

anti

ty a

fter

tye

ars

is g

iven

by

y�

a(1

�r)

t ,

wh

ere

ais

th

e in

itia

l am

oun

t an

d r

is t

he

perc

ent

of i

ncr

ease

exp

ress

ed a

s a

deci

mal

.You

can

use

a s

prea

dsh

eet

to i

nve

stig

ate

futu

re v

alu

es o

f an

ass

et.


Answers (Chapter 6)

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Qu

adra

tic

Ineq

ual

itie

s an

d t

he

Test

Men

u(U

se w

ith L

esso

n 6

-7.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

66

©G

lenc

oe/M

cGra

w-H

ill37

Gle

ncoe

Alg

ebra

2

The

ine

qual

ity

sym

bols

,cal

led

rela

tion

al o

pera

tors

,in

the

TE

ST

men

u ca

n be

used

to

disp

lay

the

solu

tion

of

a qu

adra

tic

ineq

uali

ty.A

noth

er m

etho

d th

at c

anbe

use

d to

fin

d th

e so

luti

on s

et o

f a

quad

rati

c in

equ

alit

y is

to

grap

h e

ach

sid

eof

an

ineq

ualit

y se

para

tely

.Exa

min

e th

e gr

aphs

and

use

the

in

ters

ect

func

tion

to d

eter

min

e th

e ra

nge

of

valu

es f

or w

hic

h t

he

ineq

ual

ity

is t

rue.

Sol

ve e

ach

in

equ

alit

y.

1.�

x2�

10x

�21

�0

2.x2

�9

�0

3.x2

�10

x�

25�

0{x

| x

�7

or

x�

�3}

{x| �

3

x

3}{x

| x�

�5}

4.x2

�3x

�28

5.2x

2�

x�

36.

4x2

�12

x�

9

0{x

| �7

�x

��

4 }

{x| x

��

1.5

or

x�

1}{x

| x

�1.

5 o

r x

��

1.5}

7.23

�

x2�

10x

8.x2

�4x

�13

�0

9.(x

�1)

(x�

3)

0{x

| x�

3.58

or

x�

6.41

}{x

| �2.

12 �

x�

6.12

}{x

| x

�1

or

x�

3}

Exercises

Exercises

Sol

ve x

2�

x�

6.

Pla

ce t

he

calc

ula

tor

in D

ot m

ode.

En

ter

the

ineq

ual

ity

into

Y1.

Th

en t

race

th

e gr

aph

an

d de

scri

be t

he

solu

tion

as

an i

neq

ual

ity.

Key

stro

kes:

[TE

ST] 4

6

4.

Use

TR

AC

Eto

det

erm

ine

the

endp

oin

ts o

f th

e se

gmen

ts.

Th

eses

val

ues

are

use

d to

exp

ress

th

e so

luti

on o

f th

e in

equ

alit

y,{

x|

x�

�3

or x

�2

}.

ZO

OM

2nd

+x

2Y

=

Example

1Example

1

Sol

ve 2

x2�

4x�

5�

3.

Pla

ce t

he

left

sid

e of

th

e in

equ

alit

y in

Y1

and

the

righ

t si

de i

n Y

2.D

eter

min

e th

e po

ints

of

inte

rsec

tion

.Use

th

e in

ters

ecti

on p

oin

ts

to e

xpre

ss t

he

solu

tion

set

of

the

ineq

ual

ity.

Be

sure

to

set

the

calc

ula

tor

to C

onn

ecte

dm

ode.

Key

stro

kes:

2 4

5 3

6.

Pre

ss

[CA

LC

] 5

and

use

th

e ke

y to

mov

e th

e cu

rsor

to

th

e le

ft o

f th

e fi

rst

inte

rsec

tion

poi

nt.

Pre

ss

.Th

en m

ove

the

curs

or t

o th

e ri

ght

of t

he

inte

rsec

tion

poi

nt

and

pres

s .O

ne

of t

he

valu

es u

sed

in t

he

solu

tion

set

is

disp

laye

d.R

epea

t th

e pr

oced

ure

on

th

e ot

her

in

ters

ecti

on p

oin

t.

Th

e so

luti

on i

s {

x|

�3.

24�

x�

1.24

}.

EN

TER

EN

TER

EN

TER

2nd

ZO

OM

EN

TER

EN

TER

—+

x2

Y=

Example

2Example

2

[�4.

7, 4

.7] s

cl:1

by

[�3.

1, 3

.1] s

cl:1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl:1

[�10

, 10]

scl

:1 b

y [�

10, 1

0] s

cl:1

©G

lenc

oe/M

cGra

w-H

ill38

Gle

ncoe

Alg

ebra

2

You

hav

e le

arn

ed t

he

Loc

atio

n P

rin

cipl

e,w

hic

h c

an b

e u

sed

to a

ppro

xim

ate

the

real

zer

os o

f a

poly

nom

ial.

In t

he

spre

adsh

eet

abov

e,th

e po

siti

ve r

eal

zero

of

ƒ(x)

�x2

�2

can

be

appr

oxim

ated

in

th

e fo

llow

ing

way

.Set

th

e sp

read

shee

t pr

efer

ence

to

man

ual

calc

ula

tion

.Th

e va

lues

in

A2

and

B2

are

the

endp

oin

ts o

f a

ran

ge o

f va

lues

.T

he

valu

es i

n D

2 th

rou

gh J

2 ar

e va

lues

equ

ally

in

th

e in

terv

al f

rom

A2

toB

2.T

he fo

rmul

as fo

r th

ese

valu

es a

re A

2,A

2�

(B2

�A

2)�6

,A2

� 2

*(B

2�

A2)

/6,

A2

� 3

*(B

2�

A2)

/6,A

2 �

4*(

B2

�A

2)/6

,A2

� 5

*(B

2�

A2)

/6,a

nd

B2,

resp

ecti

vely

.

Row

3 g

ives

th

e fu

nct

ion

val

ues

at

thes

e po

ints

.Th

e fu

nct

ion

ƒ(x

) �

x2�

2 is

ente

red

into

th

e sp

read

shee

t in

Cel

l D

3 as

D2^

2 �

2.T

his

fu

nct

ion

is

then

copi

ed t

o th

e re

mai

nin

g ce

lls

in t

he

row

.

You

can

use

th

is s

prea

dsh

eet

to s

tudy

th

e fu

nct

ion

val

ues

at

the

poin

ts i

nce

lls

D2

thro

ugh

J2.

Th

e va

lue

in c

ell

F3

is p

osit

ive

and

the

valu

e in

cel

l G

3is

neg

ativ

e,so

th

ere

mu

st b

e a

zero

bet

wee

n �

1.66

67 a

nd

0.E

nte

r th

ese

valu

es i

n c

ells

A2

and

B2,

resp

ecti

vely

,an

d re

calc

ula

te t

he

spre

adsh

eet.

(You

wil

l h

ave

to r

ecal

cula

te a

nu

mbe

r of

tim

es.)

Th

e re

sult

is

a n

ew t

able

fro

mw

hic

h y

ou c

an s

ee t

hat

th

ere

is a

zer

o be

twee

n 1

.414

14 a

nd

1.41

4306

.B

ecau

se t

hes

e va

lues

agr

ee t

o th

ree

deci

mal

pla

ces,

the

zero

is

abou

t 1.

414.

Th

is c

an b

e ve

rifi

ed b

y u

sin

g al

gebr

a.

By

solv

ing

x2�

2 �

0,w

e ob

tain

x�

��

2�.T

he

posi

tive

roo

t is

x

��

�2�

�1.

4142

13..

.,w

hic

h v

erif

ies

the

resu

lt.

Spre

adsh

eet

Invest

igati

on

Ap

pro

xim

atin

g t

he

Rea

l Zer

os

of

Po

lyn

om

ials

(Use

wit

h L

esso

n 6

-5.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

66

1.U

se a

spr

eads

hee

t li

ke t

he

one

abov

e to

app

roxi

mat

e th

e ze

ro o

f ƒ(

x) �

3x�

2 to

th

ree

deci

mal

pla

ces.

Th

en v

erif

y yo

ur

answ

er b

y u

sin

g al

gebr

a to

fin

d th

e ex

act

valu

e of

th

ero

ot.

Th

e sp

read

shee

t g

ives

x�

0.66

7. B

y so

lvin

g f

or

xal

geb

raic

ally

, x

��2 3�.

So

, th

e ap

pro

xim

atio

n is

co

rrec

t.

2.U

se a

spr

eads

heet

like

the

one

abo

ve t

o ap

prox

imat

e th

e re

al z

eros

of

f(x)

�x2

�2x

�0.

5.R

oun

d yo

ur

answ

er t

o fo

ur

deci

mal

pla

ces.

Th

en,v

erif

y yo

ur

answ

er b

y u

sin

g th

e qu

adra

tic

form

ula

.T

he

pro

cess

giv

es�

1.70

71 a

nd

�0.

2929

to

th

e n

eare

st

ten

-th

ou

san

dth

. Th

e q

uad

rati

c fo

rmu

la g

ives

x�

�1

�� 22 � �

. �

1�

�� 22 � ��

�1.

7071

an

d �

1�

�� 22 � ��

�0.

2929

.3.

Use

a s

prea

dshe

et li

ke t

he o

ne a

bove

to

appr

oxim

ate

the

real

zer

o of

ƒ(x

) �x3

��3 2�x

2�

6x�

2be

twee

n �

0.4

and

�0.

3.�

0.37

81 t

o t

he

nea

rest

ten

-th

ou

san

dth

Exercises

Exercises

Answers (Chapter 7)


A

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Rat

ion

al R

oo

t T

heo

rem

(Use

wit

h L

esso

n 7

-6.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

77

©G

lenc

oe/M

cGra

w-H

ill39

Gle

ncoe

Alg

ebra

2

Th

e fo

llow

ing

prog

ram

per

form

s sy

nth

etic

div

isio

n a

nd

disp

lays

th

ede

pres

sed

poly

nom

ial

coef

fici

ents

in

rat

ion

al f

orm

.Th

e pr

ogra

m w

ill

allo

wth

e te

stin

g of

pos

sibl

e ra

tion

al z

eros

of

a po

lyn

omia

l fu

nct

ion

.

PR

OG

RA

M:S

YN

TH

DIV

Dis

p "

DE

GR

EE

OF

DIV

IDE

ND

"P

�1→

PQ

→L

2(P

)In

pu

t M

Dis

p "

CO

EF

FIC

IEN

T"

P+1

→P

Dis

p "

CO

EF

FIC

IEN

TS

?"In

pu

t A

If P

�M

+1D

isp

"0�

SA

ME

"A

→L

1(P

)G

oto

3D

isp

"1�

QU

OT

IEN

T"

If P

M

�1

Sto

pD

isp

"2�

NE

W"

Got

o 1

Lb

l 4

Inp

ut

UL

bl

20→

PD

isp

"P

OS

SIB

LE

RO

OT

"1→

PL

bl

5In

pu

t R

0→S

1�P

→P

If U

�0

Lb

l 3

L2(

P)→

L1(

P)

Got

o 2

L1(

P)

→F

If P

M

�1

If U

�1

F�

S→

QG

oto

5G

oto

4D

isp

Q �

Fra

cG

oto

20→

PP

ause

L

bl

1R

Q→

S

Fin

d a

ll o

f th

e ra

tion

al z

eros

of

f(x)

�2x

3�

11x2

�12

x�

9.

Use

th

e pr

ogra

m t

o te

st p

ossi

ble

zero

s.K

eyst

roke

s:[S

YN

TH

DIV

] 3

2 1

2 11

12

9

.P

ress

u

nti

l th

e sc

reen

dis

play

s D

one.

The

col

umn

of n

umbe

rs a

re t

he c

oeff

icie

nts

of t

he d

epre

ssed

pol

ynom

ial.

Sin

ce t

he

last

nu

mbe

r is

not

zer

o,pr

ess

3 .C

hoo

se 0

for

th

e sa

me

coef

fici

ents

.Pre

ss

1 th

en

un

til

fin

ish

ed.R

epea

t th

is u

nti

l a

zero

is

fou

nd.

Th

en p

ress

2

for

the

degr

ee o

f th

e de

pres

sed

poly

nom

ial

and

1 fo

r th

e qu

otie

nt.

Th

e ze

ros

are

3,3,

and

�1 2�.

EN

TER

EN

TER

EN

TER

(–)

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

(–)

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

PR

GM

Fin

d a

ll t

he

zero

s of

eac

h f

un

ctio

n.

1.f(x

)�x3

�8x

2�

23x

�30

1,�

3, 1

02.

f(x) �

x3�

7x2

�2x

�40

�2,

4, 5

3.f(x

) �2x

3�

x2�

32x

�16

4, �

4, �1 2�

4.f(x

) �x4

�x3

�11

x2�

9x�

181,

�2,

3, �

3

5.p(

x) �

3x4

�11

x3�

11x2

�x

�2

�1,

�2,

�1 3�6.

p(x)

�x4

�2x

3�

x2�

8x�

12�

1, 3

,�2i

7.p(

x) �

3x5

�x4

�24

3x�

813,

�3,

��1 3�

8.p(

x) �

3x4

�13

x3�

15x2

�4

�2,

��1

�6�

13 ��

Example

Example

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill40

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Op

erat

ion

s o

n F

un

ctio

ns

(Use

with L

esso

n 7

-7.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

77

Stu

dy

and

use

th

e sp

read

shee

t ab

ove.

1.F

ind

k(x)

�(3

x�

2) �

(x2

�2x

).H

ow d

oes

it c

ompa

re t

o h

(x)?

k(x)

�x

2�

x�

2 �

h(x

)

2.C

hang

e th

e fu

ncti

ons

in t

he s

prea

dshe

et t

o f(

x) �

� 2x �,g(

x) �

1 �

x2,a

nd

h(x)

�1

�� 2x �

�x2

.How

are

th

ese

fun

ctio

ns

rela

ted?

Is

it t

rue

that

f(x)

�g(

x) �

h(x

)?(f

�g

)(x)

�h

(x);

yes

3.M

ake

a co

nje

ctu

re a

bou

t (f

�g)

(x)

for

any

fun

ctio

ns

f(x)

an

d g(

x).

(f�

g)(

x) �

f(x)

�g

(x)

4.M

ake

a co

nje

ctu

re a

bou

t (f

�g)

(x)

for

any

fun

ctio

ns

f(x)

an

d g(

x).U

se t

he

spre

adsh

eet

to t

est

you

r co

nje

ctu

re.D

oes

it a

ppea

r to

be

tru

e? E

xpla

inyo

ur

answ

er.

(f�

g)(

x) �

f(x)

�g

(x);

See

stu

den

ts’w

ork

.

Fin

d (

f�

g)(x

),(f

�g)

(x),

for

each

f(x

) an

d g

(x).

Use

th

e sp

read

shee

tto

fin

d f

un

ctio

n v

alu

es t

o ve

rify

you

r so

luti

ons.

5-7.

See

stu

den

ts’

spre

adsh

eets

.

5.f(

x) �

6x�

86.

f(x)

�x2

�1

7.f(

x) �

10x2

g(x)

�9

�x

g(x)

�3x

�4

g(x)

�6

�x2

7x�

17;

5x�

1x

2�

3x�

3; x

2�

3x�

59x

2�

6; 1

1x2

�6

Exercises

Exercises

It i

s po

ssib

le t

o pe

rfor

m o

pera

tion

s on

fu

nct

ion

s su

ch a

s ad

diti

on,s

ubt

rac-

tion

,mu

ltip

lica

tion

an

d di

visi

on.Y

ou c

an u

se a

spr

eads

hee

t to

in

vest

igat

eth

e re

lati

onsh

ips

amon

g fu

nct

ion

s.

Con

sid

er t

he

fun

ctio

ns

f(x)

�3x

�2,

g(x)

�x2

�2x

,an

d h

(x)

�x2

�x

+ 2.

Fin

d t

he

fun

ctio

n v

alu

es o

f ea

ch f

un

ctio

n f

or s

ever

al v

alu

es o

f x.

Doe

s it

ap

pea

r th

at f

(x)

�g(

x) �

h(x

)?

Use

Col

um

n A

for

th

e ch

osen

val

ues

of

x.C

olu

mn

s B

,C,a

nd

E a

re f

(x),

g(x)

,an

d h

(x)

resp

ecti

vely

.Use

Col

um

n D

for

f(x

)�g(

x).

For

eve

ry v

alu

e of

x,f

(x)�

g(x)

�h

(x).


Answers (Chapter 8)

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Mat

rice

s an

d E

qu

atio

ns

of

Cir

cles

(Use

wit

h L

esso

n 8

-3.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

88

©G

lenc

oe/M

cGra

w-H

ill41

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

wri

te t

he

equ

atio

n o

f a

circ

le i

n t

he

form

x2

�y2

�D

x�

Ey

�F

�0

give

n a

ny

thre

e po

ints

on

th

e ci

rcle

.

Wri

te t

he

equ

atio

n o

f th

e ci

rcle

th

at p

asse

s th

rou

gh t

he

give

n p

oin

ts.I

den

tify

th

e ce

nte

r an

dra

diu

s of

eac

h c

ircl

e.

a.A

(5,3

),B

(�2,

2),a

nd

C(�

1,�

5)

Su

bsti

tute

eac

h o

rder

ed p

air

for

(x,y

) in

x2

�y2

�D

x�

Ey

�F

�0

to f

orm

the

a sy

stem

of

equ

atio

ns.

5D �

3E �

F �

�34

�2D

�2E

�F

��

8�

D �

5E �

F �

�26

Sol

ve t

he

syst

em u

sin

g a

mat

rix

equ

atio

n t

o fi

nd

D,E

,an

d F

.Rep

lace

th

eco

effi

cien

ts in

th

e ex

pan

ded

form

.Th

en,c

ompl

ete

the

squ

are

to w

rite

th

eeq

uat

ion

in s

tan

dard

for

m t

o id

enti

fy t

he

cen

ter

and

radi

us.

Key

stro

kes:

[MA

TR

X]

3 3

5 3

1 2

2 1

1 5

1 [M

AT

RX

] [E

DIT

] 2

3 1

34

8 26

[QU

IT]

[MA

TR

X]

[MA

TR

X] 2

.T

hu

s,D

��

4,E

�2,

and

F�

�20

.T

he

expa

nde

d fo

rm is

x2

�y2

�4x

�2y

�20

�0.

Aft

er c

ompl

etin

g th

e sq

uar

e,th

e st

anda

rd f

orm

is (

x�

2)2

�(y

�1)

2�

25.

Th

e ce

nte

r is

( 2

,�1)

,an

d th

e ra

diu

s is

5.

b.A

(�2,

3),B

(6,�

5),a

nd

C(0

,7)

Fin

d a

syst

em o

f eq

uat

ion

s.T

hen

en

ter

the

equ

atio

ns

into

an

au

gmen

ted

mat

rix.

Red

uce

th

e m

atri

x to

row

red

uce

d e

chel

on fo

rmu

sin

g th

e rr

ef(

com

man

d.T

he

row

red

uce

d ec

hel

on f

orm

of

an a

ugm

ente

d m

atri

x w

ill

disp

lay

the

solu

tion

to

the

syst

em.

�2D

�3E

+ F

��

136D

�5E

�F

��

617E

�F

��

49K

eyst

roke

s:E

nter

the

sys

tem

of e

quat

ions

as

[A],

a 3

�4

augm

ente

d m

atri

x.T

hen

use

th

e re

duce

d ro

w e

chel

on f

orm

by

pres

sin

g [M

AT

RX

] [B

] [M

AT

RX

] .

Th

e so

luti

on is

D=

�10

,E=

�4,

and

F=

�21

.Th

e ex

pan

ded

form

is x

2�

y2�

10x

�4y

�21

= 0

,sta

nda

rd f

orm

is (

x�

5)2

�(y

�2)

2

�50

.Th

e ce

nte

r is

(5,

2) a

nd

the

radi

us

is 5

�2�.

EN

TER

)E

NTE

R2n

dA

LPH

A

2nd

EN

TER

2nd

x

–1E

NTE

R2n

d2n

d

EN

TER

(–)

EN

TER

(–)

EN

TER

(–)

EN

TER

EN

TER

EN

TER

2nd

EN

TER

EN

TER

(–)

EN

TER

(–)

EN

TER

EN

TER

EN

TER

(–)

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

2nd

Example

Example

Wri

te t

he

equ

atio

n o

f th

e ci

rcle

th

at p

asse

s th

rou

gh t

he

give

np

oin

ts.I

den

tify

th

e ce

nte

r an

d r

adiu

s of

eac

h c

ircl

e.

1.(0

,�1)

,(�

3,�

2),a

nd (�

6,�

1)2.

(7,�

1),(

11,�

5),a

nd (3

,�5)

3.(�

2,7)

,(�

9,0)

,and

(�10

,�5)

x2�

y2�

6x�

6y�

7�

0;x2

�y2

�14

x�

10y

�58

�0;

x2�

y2�

6x�

10y

�13

5�

0;C

(�3,

3),

R�

5C

(7, �

5), R

�4

C(3

, �5)

, R�

13

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill42

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Par

abo

las

(Use

wit

h L

esso

n 8

-2.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

88

Th

e sp

read

shee

t b

elow

use

s th

e eq

uat

ion

of

a p

arab

ola

in t

he

form

y

�a

(x�

h)2

�k

or x

�a

(y�

k)2

�h

to f

ind

in

form

atio

n a

bou

t th

ep

arab

ola.

xor

yis

en

tere

d i

n C

olu

mn

D a

nd

th

e va

lues

of

a,h

,an

d k

are

ente

red

in

to C

olu

mn

s A

,B,a

nd

C r

esp

ecti

vely

.

1.W

hic

h r

ow r

epre

sen

ts t

he

equ

atio

n y

�3x

2�

24x

�50

?ro

w 3

2.W

rite

th

e st

anda

rd f

orm

of

the

equ

atio

n r

epre

sen

ted

by r

ow 2

.

x�

�1 4�(y

�1)

2�

33.

Wh

at f

orm

ula

sh

ould

be

use

d in

cel

l F

2?1/

AB

S(A

2)

4.F

ind

the

vert

ex,l

engt

h o

f la

tus

rect

um

,axi

s of

sym

met

ry,f

ocu

s,di

rect

rix,

and

dire

ctio

n o

f op

enin

g of

a p

arab

ola

wit

h e

quat

ion

(y

�8)

2�

�4(

x�

4).

(8, 4

); 4

; y

�4;

(7,

4);

x�

9; le

ft

Exercises

Exercises

You

hav

e le

arn

ed m

any

of t

he

char

acte

rist

ics

of p

arab

olas

wit

h v

erti

cal

and

hor

izon

tal

axes

of

sym

met

ry.T

he

info

rmat

ion

is

sum

mar

ized

in

th

e ta

ble

at t

he

righ

t.Yo

uca

n u

se w

hat

you

kn

ow t

o cr

eate

a s

prea

dsh

eet

to

anal

yze

give

n e

quat

ion

s of

para

bola

s.

form

of e

quat

ion

y�

a(x

�h)

2�

kx

�a(

y�

k)2

�h

verte

x(h

, k)

(h, k

)ax

is o

f sym

met

ryx

�h

y�

kfo

cus

(h, k

� � 41 a�

)(h

� � 41 a�

, k )

dire

ctrix

y�

k�

� 41 a�x

�h

�� 41 a�

dire

ctio

n of

ope

ning

upw

ard

if a

0,

right

if a

0,

left

dow

nwar

d if

a�

0if

a�

0

leng

th o

f lat

us

|�1 a� |un

its|�1 a� |

units

rect

um

Answers (Chapter 9)


A

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Ho

rizo

nta

l Asy

mp

tote

s an

d T

able

s(U

se w

ith L

esso

n 9

-3.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

99

©G

lenc

oe/M

cGra

w-H

ill43

Gle

ncoe

Alg

ebra

2

Th

e li

ne

y�

bis

a h

oriz

onta

l as

ympt

ote

for

the

rati

onal

fu

nct

ion

f(x

) if

f(

x)→

bas

x→

�or

as

x→

��

.Th

e h

oriz

onta

l as

ympt

ote

can

be

fou

nd

byu

sin

g th

e T

AB

LE

feat

ure

of

the

grap

hin

g ca

lcu

lato

r.

Exercises

Exercises

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

a.f(

x)�� x2

�41 x

�5

�

En

ter

the

fun

ctio

n in

to Y

1.P

lace

[T

blS

et]

in t

he

Ask

mod

e.E

nte

r th

en

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

inth

e x-

list

.K

eyst

roke

s:1

4 5

[TB

LS

ET

] [T

AB

LE

].T

hen

en

ter

the

valu

es f

or x

.

Not

ice

that

as

xin

crea

ses,

yap

proa

ches

0.T

hu

s,w

hen

y�

0 is

th

eh

oriz

onta

l asy

mpt

ote.

b.f

(x)

�� 2x

2�3x

52 x�

6�

En

ter

the

equ

atio

n in

to Y

1.E

nte

r th

e n

um

bers

10,

000,

100,

000,

1,00

0,00

0,an

d 5,

000,

000

and

thei

r op

posi

tes

in t

he

x-li

st.N

ote

the

patt

ern

.As

xin

crea

ses,

yap

proa

ches

1.5

.Th

us,

y�

1.5

is t

he

hor

izon

tal a

sym

ptot

e.

2nd

EN

TER

2nd

)—

+x

2(

�Y

=

Example

Example

Fin

d t

he

hor

izon

tal

asym

pto

te f

or e

ach

fu

nct

ion

.

1.f(

x)�

� x2 �x

1�

y�

22.

f(x)

�� 2x

2x �2

7� x1 �

12�

y�

�1 2�3.

f(x)

�� 2x

3�

6 2x x3

2�

2�

y�

3

4.f(

x)�� 3x

2�

2 5x x�

1�

y�

05.

f(x)

��15

x2� x33x

�7

�y

�0

6.f(

x)�

y�

0

7.f(

x)�

�5 xx2 ��23

�n

on

e8.

f(x)

�� 2x

2�6x

33 x�

6�

no

ne

9.f(

x)�

�2x2�

4�

no

ne

x3�

8x2

�4x

�11

��

�x4

�3x

3 �4x

�6

You

have

lear

ned

to s

olve

pro

blem

s in

volv

ing

dire

ct,i

nver

se,a

nd jo

int

vari

atio

n.M

any

phys

ical

sit

uat

ion

s in

volv

e at

lea

st o

ne

of t

hes

e ty

pes

of v

aria

tion

.For

exam

ple,

acco

rdin

g to

New

ton

’s l

aw o

f u

niv

ersa

l gr

avit

atio

n,t

he

wei

ght

of a

mas

s n

ear

Ear

th d

epen

ds o

n t

he

dist

ance

bet

wee

n t

he

mas

s an

d th

e ce

nte

rof

Ear

th.S

tudy

th

e sp

read

shee

t be

low

to

dete

rmin

e th

e ty

pe o

f va

riat

ion

that

exi

sts

betw

een

th

e qu

anti

ty o

f an

ast

ron

aut’s

wei

ght

and

the

dist

ance

of

th

e as

tron

aut

from

th

e ce

nte

r of

Ear

th.

In t

he

spre

adsh

eet,

the

valu

es f

or t

he

astr

onau

t’s w

eigh

t in

new

ton

s ar

een

tere

d in

th

e ce

lls

in c

olu

mn

A,a

nd

the

valu

es f

or t

he

astr

onau

t’s d

ista

nce

in m

eter

s fr

om t

he c

ente

r of

Ear

th a

re e

nter

ed in

cel

ls in

col

umn

B.C

olum

n C

con

tain

s th

e as

tron

aut’s

dis

tan

ce f

rom

Ear

th’s

su

rfac

e.

©G

lenc

oe/M

cGra

w-H

ill44

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Var

iati

on

(Use

wit

h L

esso

n 9

-4.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

99

1.U

se t

he

valu

es i

n t

he

spre

adsh

eet

to m

ake

a gr

aph

of

the

astr

onau

t’s w

eigh

t pl

otte

d ag

ain

st t

he

astr

onau

t’s

dist

ance

fro

m E

arth

’s c

ente

r.

2.B

ased

on

you

r gr

aph

,is

this

an

in

vers

e or

dir

ect

vari

atio

n?

inve

rse

3.W

rite

an

equ

atio

n t

hat

rep

rese

nts

th

is s

itua

tion

.Let

Wre

pres

ent

the

astr

onau

t’sw

eigh

t,k

the

con

stan

t of

vari

atio

n,a

nd

R t

he

dist

ance

fro

m E

arth

’s c

ente

r.

W�

� RK2�

4.U

se t

he

equ

atio

n t

o fi

nd

the

wei

ght

of t

he

astr

onau

t at

th

ese

dist

ance

s fr

om E

arth

’s s

urf

ace.

(Hin

t:R

emem

ber

to a

dd t

hes

e va

lues

to

the

valu

e in

cel

l B

2 to

fin

d th

e di

stan

ce f

rom

Ear

th’s

cen

ter.

)a.

145,

300,

000

mb

.65

mc.

25,6

00 m

1.29

9615

N73

4.54

94 N

728.

7047

N

d.3

00,8

00,7

00 m

e.65

80 m

f.18

0,56

0 m

0.31

6872

N73

3.05

15 N

694.

6873

N

Exercises

Exercises

Weight (N)

200

300

100 0

400

500

600

700

800

Dis

tan

ce (

mill

ion

s o

f m

eter

s)20

4060

100


Answers (Chapter 10)

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Reg

ress

ion

Eq

uat

ion

L

ab(U

se w

ith L

esso

n 1

0-1

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1010

©G

lenc

oe/M

cGra

w-H

ill45

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

det

erm

ine

a re

gres

sion

equ

atio

n t

hat

best

fit

s a

set

of d

ata.

Th

is a

ctiv

ity

requ

ires

til

es l

abel

ed o

n o

ne

side

,an

d a

con

tain

er.

Col

lect

th

e D

ata

Ste

p1

Pla

ce t

he

tile

s on

th

e de

skto

p an

d co

un

t th

e to

tal

nu

mbe

r.R

ecor

dth

e to

tal

nu

mbe

r.T

hen

pla

ce t

he

tile

s in

th

e co

nta

iner

an

d ge

ntl

ysh

ake.

Ste

p2

Pou

r th

e ti

les

onto

th

e de

skto

p,re

mov

e al

l th

e ti

les

wit

h a

lab

elsh

owin

g,an

d se

t th

ese

asid

e.C

oun

t th

e re

mai

nin

g ti

les

wit

hou

t th

ela

bels

sh

owin

g an

d re

turn

th

em t

o th

e co

nta

iner

.S

tep

3R

ecor

d th

e da

ta i

n a

tabl

e li

ke t

his

on

e.

Ste

p4

Rep

eat

step

2 a

nd

3 u

nti

l th

e n

um

ber

of t

iles

wit

hou

t la

bels

is

zero

or t

he

nu

mbe

r re

mai

ns

con

stan

t.S

tep

5T

ake

the

tile

s th

at w

ere

set

asid

e in

Ste

p 2

and

pou

r th

em o

ut

ofth

e co

nta

iner

on

to t

he

desk

top.

Rem

ove

the

tile

s w

ith

out

the

labe

lsh

owin

g an

d co

un

t th

e ti

les

wit

h t

he

labe

l sh

owin

g.R

epea

t th

ispr

oces

s u

nti

l al

l th

e ti

les

hav

e be

en r

emov

ed.

Ste

p6

Rec

ord

the

data

in

ata

ble

like

th

is o

ne.

An

alyz

e th

e D

ata

1-6

. An

swer

s w

ill v

ary.

1.E

nte

r tr

ials

in

L1

and

nu

mbe

r of

til

es w

ith

out

labe

l sh

owin

g in

L2.

En

ter

tria

ls i

n L

3an

d n

um

ber

of t

iles

wit

h t

he

labe

l sh

owin

g in

L4.

2.U

se [

ST

AT

PL

OT

] to

mak

e a

scat

ter

plot

.Mak

e a

grap

h o

n p

aper

for

eac

hpl

ot.R

ecor

d th

e w

indo

w u

sed.

Des

crib

e th

e pa

tter

n o

f th

e po

ints

.

3.F

rom

th

e [C

AL

C]

men

u f

ind

the

regr

essi

on e

quat

ion

th

at b

est

fits

the

data

.Rec

ord

the

two

clos

est

equ

atio

ns,

rou

ndi

ng

valu

es t

o th

e n

eare

sth

un

dred

ths.

Lis

t an

d di

scu

ss t

he

ran

d/or

r2

valu

es.A

lso

incl

ude

th

egr

aph

s in

det

erm

inin

g th

e be

st-f

itti

ng

regr

essi

on e

quat

ion

.

4.S

ketc

h yo

ur b

est-

fit

regr

essi

on e

quat

ion

choi

ce f

or e

ach

scat

ter-

plot

on

pape

r.

5.D

escr

ibe

any

prob

lem

s w

ith

th

e da

ta o

r th

e re

gres

sion

equ

atio

ns.

6.In

sert

(0,

tota

l n

um

ber

of t

iles

) in

th

e ta

bles

an

d th

e li

sts.

Des

crib

e th

eef

fect

on

th

e gr

aph

s.W

hat

hap

pen

s w

ith

[P

wrR

eg]

and

[Exp

Reg

]w

hen

this

ord

ered

pai

r is

in

sert

ed?

Exp

lain

wh

y th

is o

ccu

rs?

ST

AT

Tria

lsN

umbe

r of

tile

s w

ithou

t lab

el s

how

ing

xy

1 2

Tria

lsN

umbe

r of

tile

s w

ith th

e la

bel s

how

ing

xy

1 2

©G

lenc

oe/M

cGra

w-H

ill46

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Net

Pre

sen

t V

alu

e(U

se a

fter

Les

son 1

0-6

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1010

1.If

th

e N

PV

is

grea

ter

than

th

e co

st,t

he

inve

stm

ent

wil

l pa

y fo

r it

self

.B

ased

on

th

e sp

read

shee

t sh

own

abo

ve,w

ould

it

be c

ost-

effe

ctiv

e fo

r th

eco

mpa

ny

to b

uy

the

van

? E

xpla

in.

Th

e co

st is

act

ual

ly a

bo

ut

$75

gre

ater

th

an t

he

NP

V, s

o it

wo

uld

no

t b

e co

st-e

ffec

tive

to

bu

yth

e va

n.

2.F

our

tim

es a

yea

r,Jo

sey

and

Dre

w p

ubl

ish

a m

agaz

ine.

Th

ey w

ant

to b

uy

aco

lor

prin

ter

that

cos

ts $

1750

.Th

e co

st o

f ca

pita

l fo

r th

is p

urc

has

e w

ould

be 6

%.T

hey

are

pla

nn

ing

to r

aise

th

e pr

ice

of t

hei

r m

agaz

ine

from

$1

to$2

.Cre

ate

a sp

read

shee

t to

det

erm

ine

the

NP

V f

or t

his

pu

rch

ase.

a.T

he la

st is

sue

of t

he m

agaz

ine

sold

500

cop

ies.

If e

ach

issu

e of

the

mag

azin

epr

inte

d in

col

or s

ells

100

cop

ies

mor

e th

an t

he

prev

iou

s is

sue,

is t

he

prin

ter

a go

od i

nve

stm

ent

afte

r on

e ye

ar?

Exp

lain

.N

o, a

fter

on

eye

ar t

he

NP

V is

on

ly a

bo

ut

$168

2.14

.b

.If

the

sale

s of

th

e m

agaz

ine

con

tin

ue

to r

ise

at t

he

sam

e ra

te,i

s th

epr

inte

r a

good

in

vest

men

t af

ter

two

year

s?Y

es, a

fter

tw

o y

ears

th

eN

PV

is a

bo

ut

$521

0.28

. Th

e N

PV

is a

bo

ut

$346

0.28

gre

ater

than

th

e co

st.

3.a.

Cal

cula

te t

he

NP

V f

or a

n i

nve

stm

ent

over

a p

erio

d of

six

yea

rs i

f th

eco

st o

f ca

pita

l is

4.5

% a

nd

the

inve

stm

ent

wil

l br

ing

a ca

sh f

low

of

$750

ever

y ye

ar.

Th

e N

PV

wo

uld

be

abo

ut

$386

8.40

.

b.W

ould

th

is b

e a

good

in

vest

men

t of

$30

00?

Exp

lain

?Y

es, t

he

NP

V is

$113

1.60

gre

ater

th

an t

he

cost

.

Exercises

Exercises

You

have

lear

ned

how

to

use

expo

nent

ial a

nd lo

gari

thm

ic f

unct

ions

to

perf

orm

a nu

mbe

r of

fin

anci

al a

naly

ses.

Spr

eads

heet

s ca

n be

use

d to

per

form

man

yty

pes

of a

naly

ses,

such

as

calc

ulat

ing

the

Net

Pre

sent

Val

ue o

f ex

pend

itur

esor

in

vest

men

ts.F

or e

xam

ple,

wh

en a

bu

sin

ess

own

er i

s co

nsi

deri

ng

a m

ajor

pu

rch

ase,

it i

s a

good

ide

a to

fin

d ou

t w

het

her

th

e in

vest

men

tw

ill

be p

rofi

tabl

e in

th

e fu

ture

.Con

side

r th

eex

ampl

e of

a l

ocal

res

tau

ran

t-de

live

ry s

ervi

ceth

at i

s de

bati

ng

wh

eth

er t

o bu

y a

van

for

$800

0.T

he

own

ers

of t

he

com

pan

y es

tim

ate

that

th

e va

n w

ill

brin

g in

$25

00 p

er y

ear

over

fou

r ye

ars.

Th

ey c

an u

se t

he

foll

owin

g fo

rmu

lato

fin

d th

e pr

esen

t va

lue

of t

he

futu

re c

ash

flo

wto

fin

d th

e N

et P

rese

nt

Val

ue

(NP

V),

that

is,

how

mu

ch t

he

prof

its

wou

ld b

e w

orth

in

tod

ay’s

doll

ars.

NP

V�

� (1C �F

n r)n

�,w

her

e C

Fn

�th

e ca

sh

flow

in

per

iod

nan

d r

�th

e co

st o

f ca

pita

l,w

hic

h i

s ei

ther

the

inte

rest

tha

t w

ill

be p

aid

on a

loan

or

the

inte

rest

tha

tth

e m

oney

wou

ld e

arn

if it

wer

e in

vest

ed.



A

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Rec

urs

ion

an

d It

erat

ion

(Use

wit

h L

esso

n 1

1-6

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1111

©G

lenc

oe/M

cGra

w-H

ill47

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

per

form

ite

rati

ons

and

recu

rsio

ns.

Fin

d t

he

firs

t 3

iter

ates

of

f(x)

= 4

x+1

5 if

x0

= 5.

Sto

re x

0in

X.T

hen

en

ter

the

expr

essi

on o

n t

he

hom

e sc

reen

.Sto

reth

e re

sult

to

X.R

epea

t th

e ca

lcu

lati

on f

or e

ach

ite

rate

.K

eyst

roke

s:5

4 15

.

x 1�

35,x

2�

155,

and

x 3�

635

EN

TER

EN

TER

EN

TER

ST

O+

EN

TER

ST

O

Fin

d t

he

firs

t th

ree

iter

ates

of

each

fu

nct

ion

.

1.f(

x)�

6x�

12 i

f x 0

�5

2.f(

x)�

2x2

�3

if

x 0�

�1

x 1�

42, x

2�

264,

x3

�15

96x 1

��

1,x 2

��

1,x 3

��

1

3.f(

x)�

x2�

4x�

5 if

x0

�1

4.f(

x)�

2x2

�2x

�1

if x

0�

�1 2�

x 1�

2, x

2�

1, x

3�

2x 1

��5 2�,

x2

��3 27 �

, x3

��14

245 �

A b

ank

acc

oun

t h

as a

n i

nit

ial

bal

ance

of

$11,

250.

00.I

nte

rest

is

pai

dat

th

e en

d o

f ea

ch y

ear.

Fin

d t

he

acco

un

t b

alan

ce u

nd

er t

he

give

nin

tere

st r

ate

afte

r th

e st

ated

tim

e p

erio

d.

5.3.

8%,2

yea

rs6.

4.75

%,5

yea

rs7.

6.05

%,1

0 ye

ars

8.7.

44%

,15

year

s$1

2,12

1.25

$14,

188.

05$2

0,24

2.27

$33,

009.

77

Example

1Example

1

A s

avin

gs a

ccou

nt

has

an

in

itia

l b

alan

ce o

f $3

000.

00.

At

the

end

of

each

yea

r,th

e b

ank

pay

s 6%

in

tere

st a

nd

char

ges

a $2

0 an

nu

al f

ee.F

ind

th

e ac

cou

nt

bal

ance

aft

er 6

yea

rs.

Sto

re t

he

init

ial

valu

e an

d en

ter

an e

xpre

ssio

n t

o ca

lcu

late

th

e ba

lan

ce a

t th

e en

d of

a y

ear.

Key

stro

kes:

3000

1.

06

20

.

At

the

end

of s

ix y

ears

,th

e ac

cou

nt

has

a b

alan

ce o

f $4

116.

05.

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

ST

O—

EN

TER

ST

O

Example

2Example

2

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill48

Gle

ncoe

Alg

ebra

2

You

hav

e le

arn

ed a

bou

t th

e ch

arac

teri

stic

s of

nu

mbe

rs i

n a

seq

uen

ce.A

spre

adsh

eet

can

cal

cula

te a

seq

uen

ce a

nd

enab

le y

ou t

o fi

nd

the

sum

of

term

s in

th

e se

ries

.

Spre

adsh

eet

Invest

igati

on

Seq

uen

ces

and

Ser

ies

(Use

aft

er L

esso

n 1

1-2

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1111

1.C

reat

e a

spre

adsh

eet

like

th

e on

e in

th

e ex

ampl

e ab

ove.

Rec

ord

the

init

ial

sequ

ence

as

�4,

�1,

and

2.R

epea

t th

e pr

oces

s yo

u fo

llow

ed in

the

exa

mpl

e.W

hat

are

th

e n

ext

six

nu

mbe

rs i

n t

he

sequ

ence

?5,

8, 1

1, 1

4, 1

7, a

nd

20

2.D

escr

ibe

the

step

s th

e sp

read

shee

t pr

ogra

m c

ompl

etes

to

fin

d th

e n

ext

term

in

th

e se

quen

ce.

Fir

st, t

he

pro

gra

m c

alcu

late

s th

e co

mm

on

dif

fere

nce

by

sub

trac

tin

g a

ny

term

fro

m it

s su

ccee

din

g t

erm

.T

hen

, it

add

s th

e co

mm

on

dif

fere

nce

to

th

e la

st t

erm

to

fin

dth

e n

ext

term

in t

he

seq

uen

ce.

3.U

se t

he

spre

adsh

eet

to f

ind

the

valu

e fo

r th

e 16

th t

erm

in

th

e se

quen

ce.

41

4.F

ind

the

sum

of

the

3rd

thro

ugh

13t

h t

erm

s in

th

e se

quen

ce.

187

Exercises

Exercises

Cre

ate

a sp

read

shee

t li

ke

the

one

bel

ow a

nd

en

ter

the

firs

t th

ree

term

s of

a s

equ

ence

.Fin

d t

he

firs

t te

nte

rms

of t

he

seq

uen

ce.T

hen

fin

d t

he

sum

of

the

firs

t te

n t

erm

s of

th

ese

ries

.

Hig

hli

ght

cell

s B

2 th

rou

gh D

2 an

d m

ove

you

r cu

rsor

to

any

corn

er o

f th

eh

igh

ligh

ted

cell

s u

nti

l a

blac

k cr

oss

appe

ars.

Dra

g ac

ross

th

e ro

w a

nd

rele

ase

it a

t ce

ll K

2.T

he

nex

t va

lues

in

th

e se

quen

ce w

ill

appe

ar i

n t

he

cell

s.

To

fin

d th

e su

m o

f th

e fi

rst

10 t

erm

s in

th

e se

ries

,hig

hli

ght

the

cell

s co

nta

inin

g th

e te

rms,

then

cli

ck t

he

�sy

mbo

l on

th

e to

olba

r.T

he

sum

wil

lap

pear

in

th

e n

ext

cell

.Not

e th

at t

his

wil

l w

ork

for

arit

hm

etic

ser

ies

only

.T

he

sum

of

the

firs

t te

n t

erm

s of

th

is s

erie

s is

7.5

Example

Example



Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Pro

bab

iliti

es(U

se w

ith L

esso

n 1

2-5

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1212

©G

lenc

oe/M

cGra

w-H

ill49

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

per

form

cal

cula

tion

s in

volv

ing

perm

u-

tati

ons,

com

bin

atio

ns,

and

prob

abil

ity.

Th

ere

are

5 gi

rls

and

3 b

oys

on a

cla

ss c

omm

itte

e.A

sub

com

mit

tee

of 3

peo

ple

is

bei

ng

chos

en a

t ra

nd

om.

Wh

at i

s th

e p

rob

abil

ity

that

th

e su

bco

mm

itte

e w

ill

hav

e at

lea

st 2

gir

ls?

P(a

t le

ast

2 gi

rls)

�P

(2 g

irls

)�P

(3 g

irls

).E

ach

pro

babi

lity

is

the

prod

uct

of

the

com

bin

atio

ns

of g

irls

an

d bo

ys d

ivid

ed b

y th

e co

mbi

nat

ion

s of

all

th

e st

ude

nts

tak

en 3

at

a ti

me.

Key

stro

kes:

5 3

2 3

3 1

5

3 3

3 3

0 8

3 3

.

Th

e pr

obab

ilit

y th

at t

he

subc

omm

itte

e h

as a

t le

ast

2 gi

rls

is �5 7�.

EN

TER

EN

TER

MA

TH

MA

TH

�)

MA

TH

MA

TH

+M

AT

H

MA

TH

(

Fin

d e

ach

pro

bab

ilit

y.

1.T

her

e ar

e 5

girl

s an

d 4

boys

on

th

e sc

hoo

l pu

blic

atio

ns

com

mit

tee.

A g

rou

p of

5 m

embe

rsis

bei

ng c

hose

n at

ran

dom

to

atte

nd a

wor

ksho

p on

sch

ool n

ewsp

aper

s.F

ind

each

pro

babi

lity.

a.at

leas

t 3

girl

sb

.4 g

irls

or

4 bo

ysc.

at le

ast

2 bo

ys

�1 20 1�� 12 25 6�

�1 20 1�

2.T

wo

card

s ar

e dr

awn

fro

m a

sta

nda

rd d

eck

of c

ards

.Fin

d ea

ch p

roba

bili

ty.

a.bo

th q

uee

ns

or b

oth

bla

ckb

.bot

h k

ings

or

both

ace

sc.

both

fac

e ca

rds

or b

oth

bla

ck

� 25 25 1�� 22 21�

�1 68 68 3�

3.F

ind

the

prob

abil

ity

that

a c

omm

itte

e of

6 U

.S.R

epre

sen

tati

ves

sele

cted

at

ran

dom

fro

m

7 D

emoc

rats

an

d 7

Rep

ubl

ican

s w

ill

hav

e at

lea

st 3

Rep

ubl

ican

s on

th

e co

mm

itte

e.�3 40 02 9�

4.T

hre

e C

Ds

are

ran

dom

ly s

elec

ted

from

a c

olle

ctio

n o

f 6

rock

an

d 5

rap

CD

s.F

ind

the

prob

abil

ity

that

at

leas

t 2

are

rock

.�1 39 3�

Example

1Example

1

Tw

o ca

rds

are

ran

dom

ly s

elec

ted

fro

m a

sta

nd

ard

dec

k o

f ca

rds.

Fin

d t

he

pro

bab

ilit

y th

at b

oth

car

ds

are

kin

gs o

r th

at b

oth

car

ds

are

red

.S

ince

th

ese

even

ts a

re m

utu

ally

in

clu

sive

fin

d th

e co

mbi

nat

ion

s of

4ki

ngs

tak

en 2

at

a ti

me

plu

s 26

red

car

ds t

aken

2 a

t a

tim

e m

inu

s 2

red

kin

gs t

aken

2 a

t a

tim

e di

vide

d by

th

e co

mbi

nat

ion

s of

52

card

sta

ken

2 a

t a

tim

e.K

eyst

roke

s:4

3 2

26

3 2

2 3

2 52

3

2 .

Th

e pr

obab

ilit

y of

ch

oosi

ng

2 ki

ngs

or

two

red

card

s is

� 25 25 1�.

EN

TER

EN

TER

MA

TH

MA

TH

�)

MA

TH

—M

AT

H+

MA

TH

(

Example

2Example

2

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill50

Gle

ncoe

Alg

ebra

2

You

hav

e le

arn

ed t

he

form

ula

s fo

r th

e n

um

ber

of p

erm

uta

tion

s of

nob

ject

sta

ken

rat

a t

ime,

P(n

,r),

and

the

nu

mbe

r of

com

bin

atio

ns

of n

obje

cts

take

nr

at a

tim

e,C

(n,r

).Yo

u a

re g

oin

g to

set

up

a sp

read

shee

t li

ke t

he

one

show

nbe

low

to

perf

orm

an

alys

es o

f th

ese

fun

ctio

ns.

In t

he

spre

adsh

eet,

the

valu

es i

n r

ow 1

rep

rese

nt

n,t

he

valu

es i

n r

ow 2

re

pres

ent

r,an

d th

e fo

rmu

las

for

P(n

,r)

and

C(n

,r)

are

in r

ows

3 an

d 4,

resp

ecti

vely

.

Th

e fo

rmu

la t

o ca

lcu

late

P(n

,r)

is�

FA

CT

(B1)

/FA

CT

(B1-

B2)

.

FA

CT

is a

spe

cial

fu

nct

ion

fro

m t

he

fun

ctio

n l

ist

and

shou

ld n

ot b

e en

tere

dfr

om t

he

lett

ers

on t

he

keyb

oard

.En

ter

the

form

ula

in

B3.

Th

en d

rag

the

curs

or a

cros

s th

e ro

w t

o co

py t

he

form

ula

in

to c

ells

C3

thro

ugh

G3.

Th

e fo

rmu

la f

or C

(n,r

) is

�F

AC

T(B

1)/(

FA

CT

(B1�

B2)

*FA

CT

(B2)

) an

d sh

ould

be e

nte

red

in c

ell

B4.

Cop

y th

e fo

rmu

la i

nto

cel

ls C

4 th

rou

gh G

4.

Spre

adsh

eet

Invest

igati

on

Per

mu

tati

on

s an

d C

om

bin

atio

ns

(Use

aft

er L

esso

n 1

2-2

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1212

1.C

ompa

re t

he

valu

es o

f P

(n,r

) an

d C

(n,r

) fo

r n

�5

and

r�

0 th

rou

gh 5

,as

wel

l as

for

tw

o ot

her

ch

oice

s of

nan

d r.

Mo

st o

f th

e va

lues

of

P(n

, r)

are

mu

ch la

rger

th

an t

he

corr

esp

on

din

g v

alu

es o

f C

(n, r

). T

he

valu

es o

f P

(n, r

) te

nd

to

incr

ease

, wh

ile t

he

valu

es o

f C

(n, r

)te

nd

to

incr

ease

an

d t

hen

dec

reas

e.

2.S

ever

al i

den

titi

es h

old

for

P(n

,r)

and

C(n

,r).

Use

th

e sp

read

shee

t to

ver

ify

the

foll

owin

g id

enti

ties

by

fin

din

g th

ree

exam

ples

of

each

.2a

-2c.

See

stu

den

ts’w

ork

.a.

P(n

,n)�

P(n

,n�

1)

b.C

(n�

1,r)

�C

(n,r

�1)

�C

(n,r

)

c.C

(n,0

)�C

(n,1

)�C

(n,2

)�..

.�C

(n,n

)�2n

Exercises

Exercises



A

Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Law

of

Sin

es:

Am

big

uo

us

Cas

e(U

se w

ith L

esso

n 1

3-4

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1313

©G

lenc

oe/M

cGra

w-H

ill51

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

ill

ust

rate

th

e A

mbi

guou

s C

ase

for

the

Law

of

Sin

es.T

his

pro

gram

con

stru

cts

a vi

sual

rep

rese

nta

tion

of

give

n

info

rmat

ion

.Fro

m t

he

draw

ing,

the

nu

mbe

r of

sol

uti

ons

can

be

dete

rmin

ed.

LA

WS

INE

S:

Lb

l 1

bco

s(A

→D

:bsi

n(A

→E

Dis

p "

A�

" m

ax(1

,in

t((D

�a�

.999

))→

Xm

axIn

pu

t A

min

(�1,

D-i

nt(

(D�

a�.9

99))

→X

min

Dis

p "

a�"

int(

(E�

a�.9

99)→

Ym

axIn

pu

t M

min

(�1,

Ym

ax�

2(X

max

�X

min

)/3)

→Y

min

Dis

p "

b�

"Z

squ

are

Inp

ut

BL

ine(

0,0,

D,E

){0

,1,2

}→L

1:M

L1�

B→

L2

Lin

e(0,

0,X

max

,0)

Lin

Reg

(ax�

b)

L1,

L2

Cir

cle(

D,E

,a)

Axe

sOff

:Clr

Dra

w In �

AB

C,A

�35

°,a

�34

,an

d b

�45

.Det

erm

ine

wh

eth

er �

AB

Ch

as o

ne,

two,

or n

o so

luti

ons.

Ru

n t

he

prog

ram

an

d en

ter

the

give

n i

nfo

rmat

ion

.Exa

min

e th

ere

sult

ing

figu

re f

or i

nte

rsec

tion

poi

nts

.K

eyst

roke

s:to

hig

hli

ght

the

LA

WS

INE

Sth

en p

ress

.F

ollo

w t

he

prom

pts.

A�

35

a�

34

b�

45

.Not

ice

that

th

e ci

rcle

wh

ose

radi

us

is a

un

its

inte

rsec

tsth

e h

oriz

onta

l se

gmen

t tw

ice.

Th

is i

ndi

cate

s th

ere

are

two

solu

tion

sor

tw

o tr

ian

gles

are

pos

sibl

e.

EN

TER

EN

TER

EN

TER

EN

TER

EN

TER

PR

GM

Det

erm

ine

wh

eth

er e

ach

tri

angl

e h

as o

ne,

two,

or n

o p

ossi

ble

sol

uti

ons.

1.A

�44

.3°,

a�

22,a

nd

b�

20.1

12.

A�

126°

,a�

12,a

nd

b�

72

3.A

�21

°,a

�2,

and

b�

32

4.A

�55

°,a

�11

,an

d b

�15

0

5.A

�11

2°,a

�5,

and

b�

70

6.B

�38

.6°,

b�

22.9

,an

d c

�33

.72

7.C

�30

°,c

�20

.2,a

nd

b�

40.4

18.

B�

50°,

b�

13,a

nd

c�

152

Example

Example

Exercises

Exercises

©G

lenc

oe/M

cGra

w-H

ill52

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Co

fun

ctio

ns

(Use

wit

h L

esso

n 1

3-1

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1313

1.U

se t

he

spre

adsh

eet

to m

ake

a gr

aph

of

the

sin

e va

lues

for

th

e an

gles

fro

m 0

°to

360

°.T

hen

mak

e a

grap

h o

f th

e co

sin

e va

lues

.

2.If

f(x

) an

d g(

x) a

re c

ofu

nct

ion

s,th

en f

(x)�

g(90

°�

x).C

ompa

re t

he

shap

esof

th

e gr

aph

s.H

ow c

an y

ou t

ell

that

sin

e an

d co

sin

e ar

e co

fun

ctio

ns

by t

hei

r sh

apes

?T

he

gra

ph

s h

ave

a si

mila

r sh

ape,

bu

tth

e co

sin

e g

rap

h is

sh

ifte

d 9

0°co

mp

ared

wit

h t

he

sin

e g

rap

hb

ecau

se t

hey

are

co

fun

ctio

ns

wit

h s

in �

�co

s (9

0 �

�).

Exercises

Exercises

Th

e fu

nct

ion

s of

sin

e an

d co

sin

e ar

e co

fun

ctio

ns.

Set

up

a sp

read

shee

t li

keth

e on

e sh

own

bel

ow t

o in

vest

igat

e th

e re

lati

onsh

ips

betw

een

cof

un

ctio

ns.

In t

he

spre

adsh

eet,

the

valu

es i

n r

ow 1

are

th

e an

gle

valu

es i

n d

egre

es.T

he

valu

es i

n r

ows

2 an

d 3

are

the

calc

ula

ted

valu

es f

or t

he

sin

e an

d co

sin

e fo

rea

ch a

ngl

e,re

spec

tive

ly.T

o u

se t

he

spre

adsh

eet

to f

ind

the

fun

ctio

ns

for

any

angl

e,fi

rst

ente

r ea

ch f

un

ctio

n i

nto

th

e sp

read

shee

t in

th

e fo

rm s

how

n b

elow

.

�S

IN(B

1*P

I()/

180)

�C

OS

(B1*

PI(

)/18

0)

Th

is f

orm

of

the

form

ula

con

tain

s ad

diti

onal

in

form

atio

n t

hat

is

nec

essa

ryfo

r th

e sp

read

shee

t to

use

deg

rees

to

calc

ula

te t

he

answ

er.W

ith

out

it,t

he

spre

adsh

eet

can

not

rec

ogn

ize

that

th

e an

gle

is m

easu

red

in d

egre

es a

nd

wil

lre

turn

th

e w

ron

g an

swer

.

Th

en,c

ompl

ete

the

spre

adsh

eet

by e

nte

rin

g th

e an

gles

up

to 3

60°

wh

ose

mea

sure

s ar

e m

uli

tple

s of

30°

and

45°.

0

1.50

1.00

0.50

0.00

�0.

50

�1.

00

�1.

50

6012

018

024

030

036

060

120

180

240

300

360

0

1.50

1.00

0.50

0.00

�0.

50

�1.

00

�1.

50


Gra

ph

ing C

alc

ula

tor

Invest

igati

on

Sin

uso

idal

Eq

uat

ion

s(U

se w

ith L

esso

n 1

4-2

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1414

©G

lenc

oe/M

cGra

w-H

ill53

Gle

ncoe

Alg

ebra

2

A g

raph

ing

calc

ula

tor

can

be

use

d to

ver

ify

a si

nu

soid

al r

egre

ssio

n e

quat

ion

in t

he

form

y�

asi

n (

bx�

c)�

dgi

ven

fou

r da

ta p

oin

ts.T

he

sin

uso

idal

regr

essi

on i

s fo

un

d u

nde

r [C

AL

C]

[C].

ST

AT

As

a pe

rson

rid

es a

Fer

ris

wh

eel,

the

pers

on’s

dis

tan

ce f

rom

the

grou

nd

vari

es s

inu

soid

ally

wit

h t

ime.

Let

tbe

th

e n

um

ber

of s

econ

ds t

hat

hav

e el

apse

d si

nce

th

e F

erri

s w

hee

l st

arte

d.T

he

ride

r’s

posi

tion

wh

en t

he

last

sea

t is

fil

led

and

the

Fer

ris

wh

eel

star

ts i

sw

hen

t�

0.S

upp

ose

it t

akes

3 s

econ

ds t

o re

ach

th

e to

p of

th

e F

erri

s w

hee

l,43

fee

t ab

ove

the

grou

nd.

Th

e di

amet

er o

f th

e w

hee

l is

40

feet

,an

d it

mak

esa

revo

luti

on e

very

8 s

econ

ds.C

reat

e a

tabl

e of

val

ues

and

wri

te t

he s

inus

oida

leq

uat

ion

.

Key

stro

kes:

En

ter

the

data

in

L1

and

L2.

Ch

oose

an

app

ropr

iate

win

dow

.Use

[S

TA

TP

LO

T]

to g

raph

th

e po

ints

.[C

] [L

1]

[L2]

.

a�

20,b

�� 4π �,

c�

1,an

d d

�23

h(t

)�20

sin

� 4π �(t�

1) �

23

EN

TER

EN

TER

EN

TER

VA

RS

,2n

d,

2nd

ALP

HA

ST

AT

As

the

pad

dle

wh

eel

of a

ste

amb

oat

turn

s,a

poi

nt

on t

he

pad

dle

bla

de

mov

es s

o th

at i

ts d

ista

nce

,h,f

rom

th

e w

ater

’s s

urf

ace

is a

sin

uso

idal

fu

nct

ion

of

tim

e.T

he

wh

eel’s

dia

met

er i

s 18

fee

t,an

d i

tco

mp

lete

s a

revo

luti

on e

very

10

seco

nd

s.T

he

hei

ght

of t

he

poi

nt

atva

riou

s ti

mes

is

show

n i

n t

he

tab

le.

1.W

hy

is t

he

hei

ght

the

sam

e af

ter

14 s

econ

ds a

s it

is

afte

r 4

seco

nds

? T

he

wh

eel c

om

ple

tes

a re

volu

tio

n e

very

10

seco

nd

s.

2.W

hat

are

th

e va

lues

of

a,b,

c,an

d d

?

a�

9, b

�� 5π �,

c�

�3 2�, a

nd

d�

7

3.W

rite

a r

egre

ssio

n e

quat

ion

.

h(t

) �

9 si

n [� 5π � (t�

�3 2� )]�7

Example

Example

Exercises

Exercises

t sec

.1

35

79

11h(

t) ft.

2343

233

2343

t 1.

54

6.5

911

.514

(sec

onds

)h

(t)

716

7�

27

16(f

eet)


©G

lenc

oe/M

cGra

w-H

ill54

Gle

ncoe

Alg

ebra

2

Spre

adsh

eet

Invest

igati

on

Trig

on

om

etri

c Id

enti

ties

(Use

aft

er L

esso

n 1

4-3

.)

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1414

1.S

tudy

th

e va

lues

in

Col

um

ns

D a

nd

E.W

hat

ide

nti

ty s

eem

s to

be

poss

ible

from

th

is p

atte

rn?

tan

(�

B)

��

tan

B

2.E

nte

r th

e fo

rmu

la S

IN(B

) in

Col

um

n F

.Th

en e

nte

r th

e fo

rmu

laC

OS

(B)*

TA

N(B

) in

Col

um

n G

.Wh

at i

den

tity

do

thes

e tw

o n

ew c

olu

mn

ssu

gges

t?si

n B

� c

os

Bta

n B

3.M

ake

a co

lum

n w

ith

th

e fo

rmu

la S

IN(P

I()-

B).

Wh

at i

den

tity

do

you

di

scov

er?

sin

(π

�B

)�si

n B

or

sin

(π

�B

)�co

s B

tan

B

Exercises

Exercises

A t

rigo

nom

etri

c id

enti

ty h

olds

for

all

val

ues

of

w

her

e ea

ch e

xpre

ssio

n i

sde

fin

ed.F

or e

xam

ple,

sin

�

cos

�

tan

.Y

ou h

ave

lear

ned

to

prov

eal

gbra

ical

ly t

hat

an

equ

atio

n i

s an

ide

nti

ty.Y

ou c

an u

se a

spr

eads

hee

t to

tes

teq

uat

ion

s fo

r sp

ecif

ic v

alu

es t

o se

e if

an

equ

atio

n m

igh

t be

an

ide

nti

ty.

To

use

th

e sp

read

shee

t to

tes

t th

e va

lues

of

expr

essi

ons

for

diff

eren

t an

gles

,en

ter

the

angl

e m

easu

res

in t

he

cell

s in

Col

um

n A

,an

d en

ter

the

expr

essi

ons

from

th

e eq

uat

ion

s yo

u w

ant

to t

est

in t

he

colu

mn

s to

th

e ri

ght.

Fir

st,e

nte

rth

e fo

rmu

la�

RA

DIA

NS

(A)

in C

olu

mn

B t

o co

nve

rt d

egre

es t

o ra

dian

s.(R

ecal

l th

at y

ou c

an d

o th

is b

y en

teri

ng

the

form

ula

RA

DIA

NS

(A2)

in

cel

lB

2,co

pyin

g ce

ll B

2,an

d pa

stin

g to

fil

l th

e re

st o

f C

olu

mn

B.)

In

th

e sp

read

shee

t sh

own

,th

e fo

rmu

laS

IN(B

)/C

OS

(B)

is i

n t

he

cell

s in

Col

um

n C

.T

he

cell

s in

Col

um

n D

con

tain

th

e fo

rmu

laT

AN

(B).

Col

um

n E

con

tain

s th

efo

rmu

laT

AN

(�B

).N

otic

e th

at t

he

valu

es i

n C

olu

mn

s C

an

d D

agr

ee w

ith

the

iden

tity

sta

ted

abov

e.

Documents

Graphing Calculator and Spreadsheet Masters · 2017. 6. 15. · and be used solely in conjunction with Glencoe Algebra 2. Any other reproduction, for use or sale, is prohibited without