Algebra 1 Notes Quarter 3 2014 – 2015 · 2016-11-26 · ~ 15 ~ Vocabulary Definition Example(s) Monomial A number, a variable, or a product of the two. Polynomial A monomial or

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Algebra 1 Notes

Quarter 3

2014 – 2015

Name: ________________________________________________

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Table of Contents

• Unit 9 – Exponent Rules

• Exponent Rules for Multiplication page 6

• Negative and Zero Exponents page 10

• Exponent Rules Involving Quotients page 12

• Unit 10 – Polynomials

• Naming Polynomials page 15

• Adding and Subtracting Polynomials page 17

• Multiplying and Dividing Polynomials page 19

• Unit 11 – Factoring

• Factoring Binomials page 25

• Factoring Trinomials, Part 1 page 27

• Factoring Trinomials, Part 2 page 29

• Unit 12 – Graphing Quadratics and Best Fit Lines/Curves

• Graphing Quadratic Functions page 32

• Best Fit Lines and Curves page 37

• Graphing Calculator Instructions page 42

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1

-6

-2 3 9

20

4 5 -6 -3

1 -11 7 8

0

-100

8

16

8

-48

-2

-15

20

91

-4

-96

16

28

-10

9

-10

21

5

-66

6

-16

-2

-35

13

30

Diamonds Fill in the blanks using the rules demonstrated in the first two diamonds.

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Property Definition Example

Product of Powers

Power of a Power

Power of a Product

Simplify the expression completely. 1. x² • x³ • x5 2. y-3 • y7 • y-2 3. [(-3)³]² 4. (-2w)2 5. –(6x4y2)3 6. [(a + 1)3]4 Now, put them all together! 7. (4x²y)³ • x5 8. –(3x2y5z3)2 • 5x4y6z

Learning Target: SOL A.2a:

The student will apply the

laws of exponents to

perform operations on

expressions.

Exponent Rules for

Multiplication

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9. (-3xy²)³ • (-2x²y)² 10. (2x6y4)2 + (3x3y2)4 11. 3(x5)2 + (2x3)7

Applying the Rules of Exponents to Scientific Notation Scientific Notation: Write each number in scientific notation. 1. 34,000 2. 0.00000008 Write each number in standard notation. 3. 2.9 x 106 4. 3.5 x 10-5

Simplify the following problems by applying the rules of exponents. 5. (1.4 x 104)(7.6 x 103) 6. (1.5 x 10-3)2

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Try it! Simplify each expression.

1. 2x3y4 • -3x5y8 2. (4x6y10)3 3. -5x11y-4 • (2x3y7)4 4. –(3x5y8)2 • (-4x3y9)2

5. (5.6 x 10-2)(4 x 104) 6. (3.4 x 104)3

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Warm - Up

Fill in the following table.

Power Decimal Whole # or

Fraction

Power Decimal Whole # or

Fraction

104

24

103

23

102

22

101

21

100

20

10-1

2-1

10-2

2-2

10-3

2-3

10-4

2-4

What patterns do you notice?

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Rule Definition Example

Zero Exponents

Negative Exponents

Simplify the expression completely. 1. 80 2. 2x-2y-3 3. (5x)-3 4. (4n-2)-3 5. 8_ 6. –45x3y-8 m-3 z-5

Negative and Zero

Exponents



laws of exponents to perform

operations on expressions.

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7. 8x-2y-6 8. 1 _ 4x-10y14

9. ��(−4��) 10. –16 + 7xy (xy2)-3 x-2y-5

Try it!Try it!Try it!Try it! 1. (x2y4z5)0 2. 4-2 y5 x-6

3. a-5 4. 7a-4b2 b-2 3c5d-3

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Property Definition Example

Quotient of Powers

Power of a Quotient

Simplify the expression completely. 1. x4 2. 1_ • y3 x3 y5

3. 15x5y7z3 4. ��

3x6yz8 Putting it all together!

5. �� ∙ ��

�

�� 6. � ��

�∙ ��

�

� �

Exponent Rules

Involving Quotients



laws of exponents to

perform operations on

expressions.

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7. 1.2 x 104 8. 25a-3b5c-2 1.6 x 10-3 35a5b5c-9 9. 5x-3y2 • (2xy3)-2 10. (-2x4y-5z6)2 • (3x-3y-6z7)-2 x5y-1 xy

Try it!Try it!Try it!Try it!

1. ��

��

2. (��)�(��)� 3. ��

��

�

4. m4 • m2 5. 4r3s4t-9 • -12r-12st6 6. 8.4 x 109 m7 3r6s2t 2r-8s-5t11 3.2 x 104

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Unit 9 Scratch Paper

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Vocabulary Definition Example(s) Monomial A number, a variable, or a

product of the two.

Polynomial A monomial or the sum or difference of many monomials.

Degree of a Monomial

The sum of the exponents of the variables in the monomial.

Term In a polynomial, each individual monomial is called a term.

Degree of a Polynomial

The greatest degree of the polynomial’s terms.

Standard Form of a Polynomial

A polynomial is written in standard form when the terms are listed in descending order by degree (highest degree to lowest degree).

What is a Polynomial?

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Classifying Polynomials by Degree:

Degree Example Name 0

1

2

3

4

5 or more

Classifying Polynomials by Terms: # of Terms Example Name

1

2

3

4 or more

Try it! Classify each polynomial by degree AND term. 1. 6x3 – 9x 2. 5x2 – 19x + 7 3. 14x4 4. 8 5. 6x – 2 6. -9x6 + 17x5 – 2x4 + x3

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• You can add and subtract polynomials using two different formats: vertical and

horizontal.

• When you add or subtract, you are really combining like terms so you DO NOT

CHANGE THE EXPONENT!

1. (4x2 + 6x – 9) + (5x2 – 7x + 2) 2. (9x3 – 4x2 + 2x – 6) + (4x3 – 3x + 1)

3. (2x2 + x – 5) + (6x2 – 3x + 7) 4. (6x3 + 2x2 – 5x + 9) – (10x3 – 5x2 + 3x – 8)

5. (7x2 – 5x) – (2x2 + 6x – 1) 6. (-4x3 + 9x2 – 8) – (5x2 + 7x – 3)

Adding and Subtracting

Polynomials

Learning Target: SOL A.2b:

The student will add, subtract,

multiply and divide polynomials.

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Try it!Try it!Try it!Try it!

1. (9w – 4) + (w + 5) 2. (6x2 – 3x + 1) + (x2 + x – 1)

3. (5x3 – 6x2 + 4x – 1) – (2x3 + 8x2 – 9x – 5) 4. (8b2 + 6) – (3b2 + 6b + 1)

5. (x2 + xy + 2y2) + (6x2 – y2) 6. (4x2 + y2) – (-3xy + y2)

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It’s time to use the Distributive Property!

Remember, when you MULTIPLY with variables, you ADD exponents!

A monomial times a polynomial

1. 3x²(4x² – 6x + 7) 2. -5xy(8x³y² + 3x²y – 10x)

3. 4x(8x² + 9x – 1) + 3x(7x² – 6x + 2) 4. 8x(5x² + 3x – 3) – 3x(9x² + x – 2)

A binomial times a binomial

Using a Table Using FOIL (Horizontal Method)__________

5. (5x – 2)(6x + 1) 6. (x –y)(x – 5y)

Multiplying and Dividing

Polynomials

Learning Target: SOL A.2b:

The student will add, subtract,

multiply and divide polynomials.

F

O

I

L

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A binomial times a trinomial

Using a Table __________ Horizontal Method ________

7. (3x + 2)(4x² – 2x – 7) 8. (x – 4)(x² + 6x + 3)

Dividing a polynomial by a monomial

Remember, when you DIVIDE powers, you SUBTRACT the exponents!

9. 18x3y4 + 27x2y5 – 36xy6 10. -80x4 + 24x3 – 8x2 -3xy2 8x2

Application

11. Tara is putting a pool in her backyard. The length of the pool is four more than twice the width. The depth of the pool is half of the width.

Write an expression in simplest form that represents the – a. Perimeter of the pool

b. Area of the pool

c. Volume of the pool d. If the width is 8 ft, how much water can fit in the pool?

Formulas to

Know

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Special Cases for Multiplying Binomials

Directions: Use a table or the horizontal method to multiply the following binomials.

(x + 4)(x + 4) (y – 2)2 (3z + 13)(3z + 13)


Directions: Use a table or the horizontal method to multiply the following binomials.

(x + 9)(x - 9) (x – 15)(x + 15) (6x3 + 7)(6x3 - 7)


Directions: Find each product using mental math.

(2x + 11)(2x + 11) (9z + 1)(9z - 1) (4y – x2)(4y – x2)

(6z - 14)(6z + 14) (a + 5b)2 (3x5 - 4y)2

The Special Cases

The Square of a Binomial (Difference of Squares)

The Sum and Difference Patter

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You Try It!

DIRECTIONS: Simplify the following expressions using any method.

1. 3x(5x² - 9x + 2) 2. 3a(4a² + 7a – 8) – 5a(7a² - 3a + 4) 3. (7b – 4)(2b + 8) 4. (3x – 6)(x – 2) 5. (4x + 5)² 6. (x – 8)(x + 8) 7. (8x - 3)(4x3 – 2x2 + 6x – 7) 8. -30y8 + 18y5 – 24y2 6y2

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Perfect Squares Warm Up Fill in the chart below. Simplify.

1. (5x – 12)(5x + 12)

2. (3x + 14y)(3x – 14y)

3. (11x2 – 6y)(11x2 + 6y)

4. (15x + 7)(15x – 7)

5. (8xy – 1)(8xy + 1)

Number (n) Number Squared (n2)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

What is a “Perfect Square”?

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Factor out the Greatest Common Factor (GCF)

1. 3x + 6y 2. ab + ac 3. 12x3y4 – 6x2y7

Factor the Difference of Two Squares 4. 4x2 – 25 5. x2y2 – 1 6. 81x4 – 16y4

Factoring Binomials

Learning Target: SOL A.2c:

The student will factor completely first-

and second-degree binomials and

trinomials in one or two variables.

Steps for Factoring Binomials 1. 2.

a. b. c.

3. 4.

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Factor out the GCF and THEN the difference of two squares

7. 18x2 – 8 8. 27x6 – 3y2

Prime

9. x2 + 16 10. 4y2 + 11

Try it: Factor Completely!

1. 8 – 18x² 2. 225x² – 144y² 3. -45x² + 20y²

4. x² – y² 5. 5x2y3 – 15x3y4 6. y² – 36

7. 4x² + 16 8. x4 – y4 9. 9x² – 25

10. 9x² + 25 11. 9x² – 24 12. 16x² – 64

13. 12x + 42y 14. 4x4 + 24x3 15. 2x2 + 7x

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ALWAYS LOOK FOR A GCF 1st!

Do you remember the diamonds activity? Factor completely. 1. x² + 7x + 12 2. 2x² – 14x – 16 3. x² – 6x + 9 4. -2x³ – 14x² – 20x Simplify completely. 5. Find the quotient of x2 + 9x – 36 and x – 3.





Factoring Trinomials, Part 1

Helpful Hints

• If the last term is negative, one of the factors

will have a subtraction sign and one of the

factors will have an addition sign. Look at the

middle term to determine which one is which.

• If the last term is positive, look at the middle

term. If the middle term is positive, both

factors will be addition. If the middle term is

negative, both factors will be subtraction.

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Try it! 1. x² – 10x + 16 2. x² + x – 20 3. x² + 5x – 24 4. 8x² + 80x + 72 5. 3x² – 21x – 90 6. x² – 12x + 36 7. x² + 6x – 5 8. x3 – 2x2 – 48x Simplify completely. 9. x² – x – 56 10. x² – 9x – 52 (x – 8) (x + 4)

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ALWAYS look for a GCF first!

1. 2x2 + 15x + 7 2. 6x2 – x – 15 3. 15x2 – 24x – 12 4. 25x3 – 30x2 + 9x Simplify completely. 5. 12x2 + 17x – 5 6. 24x2 + 54x – 15 (3x + 5) (4x - 1) 7. The area of a rectangle is 6x2 + 19x – 36. Find the dimensions of the rectangle. Then, find the actual dimensions if x = 2 inches.





Factoring Trinomials, Part 2

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Try it! 1. 4x2 + 8x + 3 2. 3x3 – 13x2 – 10x 3. 4x2 + 18x – 10 4. 6x2 – 25x + 14 5. 25x2 + 20x + 4 6. 4x2 –13x – 12 Simplify completely. 7. 6x2 + 31x + 5 8. 30x2 - 5x - 10 (x + 5) (3x - 2)

9. The area of a rectangle is 15x2 + 38x + 24.

a. Write two expressions that represent the dimensions of the rectangle. b. Find the actual dimensions if x = 3 centimeters. 10. The volume of a rectangular prism is 10x3 + 13x2 – 30x. If the height of the prism is x units, write an expression for the length and width of the prism.

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The graph of a quadratic function Standard Form: is a PARABOLA.

Finding the Vertex:

Axis of Symmetry Equation: Find the vertex and axis of symmetry for each quadratic function. 1. � � 3�� − 8 2. � � −�� ! 4� ! 5 3. � � 2�� ! 2� � 4

Trends in Quadratic Graphs Reflect (Flip) Graph

$ � %&' ! (& ! ) If ________ is positive, the parabola

will open _________ and the vertex is the

____________ point.

Ex. $ � &'

If ________ is negative, the parabola will

open _________ and the vertex is the

____________ point.

Ex. $ � �&'

Width of Graph

$ � %&' ! (& ! )

If |+| = 1, the graph has a __________width.

Ex. $ � &' If |+| > 1, the graph is __________. Ex. $ � ,&' If 0 < |+| < 1, the graph is __________.

Ex. $ � -

.&'

Graphing Quadratic

Functions

Learning Target: SOL A.4c

The student will solve quadratic equations

in two variables including solving

graphically.

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Graph each quadratic equation. Answer the questions provided.

1. $ � &' � /& ! 0 Does the parabola open up or down? Is the vertex the minimum point or the maximum point? a = b = c = Vertex: Axis of Symmetry: Table of Values: Roots:

2. $ � �-

'&' ! '& ! .

Does the parabola open up or down? Is the vertex the minimum point or the maximum point? Compared to the parent function (y = x2), is it skinnier, wider or the same width? a = b = c = Vertex: Axis of Symmetry: Table of Values: Roots:

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Use your calculator to graph the quadratic. (Calculator instructions are on page 42).

3. $ � '&' ! '& � , Is the vertex the minimum point or the maximum point?

Vertex: Axis of Symmetry: Table of Values: Roots:

Try it!

Match each equation on the left to its correct graph on the right. ____ 1. � � �� ! 3 ____ 2. � � 4�� − 1

____3. � � �� − 1

____4. � � �� ! 3

A B

C D

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Try it, Continued Directions: Graph each quadratic. Answer the questions provided.

5. $ � �&' ! /& � 2 a = b = c = a. Graph opens: up or down b. Vertex is a: min or max

c. Coordinates of vertex: ( , ) d. Axis of symmetry: ________

e. Table of values: x y f. Roots: _____________

Directions: Use the calculator to graph each quadratic.

6. $ � -

'&' ! '& � .

a = b = c = a. Graph opens: up or down b. Vertex is a: min or max

c. Coordinates of vertex: ( , ) d. Axis of symmetry: ________

e. Table of values: x y f. Roots: _____________

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Scatter Plot Warm Up Plot the data in each table as a scatter plot on the coordinate grid provided. 1. Gold’s Gym followed 3 members for a week. Each person recorded the calories they burned during their treadmill workout. The results are in the table below.

(Hint: Count by 5’s on the x-axis and by 20’s on the y-axis!)

2. A football was kicked from an initial height of 2.5 feet. The height was recorded at different intervals and recorded in the chart below.

(Hint: Count by 0.3 on the x-axis and by 5’s on the y-axis!)

Minutes on

Treadmill

Calories Burned

5 20 5 60 10 70 15 80 15 110 20 160 25 150 30 200 35 260

Seconds in the air

Distance from the

ground (ft) 0 2.5

0.5 21 1 31.5

1.3 33.96 1.8 31.66 2 28.5

2.3 21.36 2.7 7.36

Minutes on Treadmill

Ca

lori

es

Bu

rne

d

Seconds in the Air

Dis

tan

ce f

rom

th

e G

rou

nd

(ft

)

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What is Linear Regression? A linear regression line is used to represent data that continually increases (positive correlation) or

continually decreases (negative correlation) at a fairly constant rate.

Given a Graph (Scatter Plot)

1. Draw a best fit line for the scatter plot. 2. Pick 2 points on the line and find slope. 3. Estimate the y-intercept. 4. Then write the equation of the line in slope-intercept form.

Write the equation of the best fit line. 1. 2.

3. Consider the data in the table below. Write the equation of the line of best fit. Then, determine the approximate value of f(25).

x 1 4 4 8 9 10 13 y 1 3 7 6 10 13 10

Best Fit Lines &

Best Fit Curves

Learning Target: SOL A.11:

The student will determine the equation

of the curve of best fit in order to make predictions.

Calculator Steps

1. Using the calculator, open a new spreadsheet page. Name each list and enter the x values and the y values.

2. Return to a calculator page and press Menu, 6, 1, 3. 3. Select the appropriate x list and y list from the drop down menu. 4. Write the equation of the line in slope-intercept (y = mx + b) form.

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What is Quadratic Regression? Quadratic regression is used to represent data that will increase, reach a peak, and then decrease or

vice versa.

Example 1

Larry made a scatter plot showing the apparent height of a football at one-second intervals during the

time period the ball was in the air.

Which is most likely the equation for the curve of best fit for the relationship?

A. y = - 0.4x + 9.0

B. y = 9.0x + 0.4

C. y = 5.3x2 – 0.9x + 4.9

D. y = - 0.9x2 + 5.3x + 4.9

Example 2

a. Look at the table. What do you notice about the y-values? Does this make the equation linear

or quadratic?

b. Use the graphing calculator to calculate a curve of best fit. Round to the nearest 100

th if

necessary.

x 1 3 4 6 7 10 12 14 y 1 3 7 11 12 11 6 3

Calculator Steps

1. Using the calculator, return to the spreadsheet page. Name each list and enter the values found in the table.

2. Return to a calculator page and press Menu, 6, 1, 6 (Quadratic Regression). 3. Select the appropriate x list and y list from the drop down menu. 4. Write the equation of the parabola in standard quadratic form (y = ax2 + bx + c.)

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Practice: The table below shows the cost, y, for a specified number of phone lines.

Number of lines, x 4 5 6 7 8 9 Total cost, y $10.40 $12.25 $14.10 $15.95 $17.80 $19.65

a. How do you know this data can be represented using a liner regression line?

b. Use your graphing calculator to find a line of best fit.

c. Determine the approximate total cost for 15 lines.

d. Determine the number of lines if the cost is $6.70.

Practice: Consider the table below.

a. How do you know this data can be represented using a quadratic regression line?

b. Use your graphing calculator to find the curve of best fit. Round to the nearest 100th if necessary.

c. Determine the approximate weight at a depth of 8 inches.

Depth (in) 6 7.5 9 10.5 12 13.5 Weight (lbs) 68 137 242 389 340 279

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Try It! Determine if the given data represents a linear function or a quadratic function. Then use your

graphing calculator to find the line or curve of best fit. Round to the nearest 100th

if necessary.

1.

2.

3. The table below shows the body temperature B (in degrees Celsius) of a desert spiny

lizard at various air temperatures A (in degrees Celsius).

a. Is this data best modeled by a linear or a quadratic function? Write an equation for the best fit

line/curve.

b. Use your equation to predict the body temperature of the spiny lizard when the air

temperature is 35° C.

4. The table below shows the number of students who attended Mrs. Knipper’s math club in the

first seven weeks of school.

a. Is this data best modeled by a linear or a quadratic function? Write an equation for the best fit

line/curve.

b. Use your equation to predict the number of students who attended Math Club during the 10th

week of school.

x -3 -2 -1 0 1 y -7 -5 -3 -1 1

Time

(sec)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

Height

(ft)

6 6.3 9 10.2 12 12.4 13.1 12 12.1 10.9 9 6.1

Air temperature, A (°C) 26 27 28 29 30

Body temperature, B (°C) 33.44 33.82 34.08 34.52 34.91

Week 1 3 5 7 9 11 13

Number of Students 4 6 7 9 6 3 2

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Review Day Warm Up 1. Consider the equation � � �� 2� − 8.

a. Find the vertex:

b. Identify the axis of symmetry:

c. Create a table of values and graph the equation. x y_

d. What are the roots of this equation? 2. Identify the following using the graph shown. Coordinate of the Vertex: Axis of Symmetry: Roots: 3. Determine if the given data represents a linear function or a quadratic function. Then use your graphing calculator to find the line or curve of best fit. Round to the nearest 100th if necessary. Then find f(8).

x 1 3 4 6 7 10 12 14 y 1 3 7 11 12 11 6 3

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Graphing Calculator Instructions

Always begin by clearing the memory on your calculator! Graph an Equation Steps: 1. Add a graphing page. This will open up a blank graph. At the bottom of the screen you will see “f1(x)=.” This is where you will type in your equation. To create a second graph: 1. Press the “tab” key. The bottom of the screen should now read “ f2(x)= ” Note: Lines must be in slope-intercept form before using the calculator to graph! Finding the Vertex Steps 1. Graph the equation. Decide if the vertex is the minimum or maximum point. 2. Press Menu 3. Select 6: Analyze Graph 4. Depending on the graph, select 2: minimum or 3: maximum 5. A pointing hand will appear. Use the cursor or the arrow keys to move the hand. You will need to select a lower bound by pressing enter. Then, select an upper bound by pressing enter. Displaying a Table of Values Steps 1. Graph the equation. 2. Press Ctrl T to display the table. Find the Roots Using a Graph Steps 1. Open a graphing page and enter the equation. 2. Press Menu 3. Select 6: Analyze Graph 4. Select 1: Zero 5. A pointing hand will appear. Use the cursor or the arrow keys to move the hand. You will need to select a lower bound by pressing enter. Then, select an upper bound by pressing enter.

• s • 5: Delete All • Select “Yes”

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Documents

Algebra 1 Notes Quarter 3 2014 – 2015 · 2016-11-26 · ~ 15 ~ Vocabulary Definition Example(s) Monomial A number, a variable, or a product of the two. Polynomial A monomial or