Unit 5 Algebra in Context - Georgia Standards · PDF fileUnit 5 . Algebra in Context . 2nd ... Solve problems that arise in mathematics and in ... equations involving higher degree

Mathematics I Frameworks Student Edition

Unit 5 Algebra in Context

2nd Edition May 19, 2008

Georgia Department of Education

One Stop Shop For Educators

Mathematics I Unit 5 2nd Edition

Georgia Department of Education Kathy Cox, State Superintendent of Schools

May 19, 2008 Copyright 2008 © All Rights Reserved

Unit 5: Page 2 of 41

Table of Contents

INTRODUCTION: ............................................................................................................ 3

Paula’s Peaches Learning Task .......................................................................................... 8

Logo Symmetry Learning Task ....................................................................................... 18

Resistance Learning Task ................................................................................................ 33

Shadows and Shapes Learning Task ................................................................................ 38

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Mathematics I - Unit 5 Algebra in Context

Student Edition

INTRODUCTION: In Units 1 of Mathematics 1 students learn properties of basic quadratic, cubic, absolute value, square root, and rational functions. In Unit 2 students develop skill in adding, subtracting, multiplying, and dividing elementary polynomial, rational, and radical expressions. In this unit students extend the skills and understandings of Units 1 and 2 through further investigation of quadratic, rational, and radical functions. The even and odd symmetry of graphs will be explored, as well as intersections of graphs as solutions to equations. The focus is the development of students’ abilities to solve simple quadratic, rational, and radical equations using a variety of methods. ENDURING UNDERSTANDINGS:

• There is an important distinction between solving an equation and solving an applied problem modeled by an equation. The situation that gave rise to the equation may include restrictions on the solution to the applied problem that eliminate certain solutions to the equation.

• The definitions of even and odd symmetry for functions are stated as algebraic conditions on values of functions but each symmetry has a geometric interpretation related to reflection of the graph of through one or more of the coordinate axes.

• For any graph, rotational symmetry of 180 degrees about the origin is the same as point symmetry of reflection through the origin.

• Techniques for solving rational equations include steps that may introduce extraneous solutions that do not solve the original rational equation and, hence, require an extra step of eliminating extraneous solutions.

• Understand that any equation in can be interpreted as a statement that the values of two functions are equal, and interpret the solutions of the equation domain values for the points of intersection of the graphs of the two functions. In particular, solutions of equations of the form f(x) = 0, where f(x) is an algebraic expression in the variable x, correspond to the x-intercepts of the graph of the equation y = f(x).

KEY STANDARDS ADDRESSED: MM1A1. Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.

c. Graph transformations of basic functions including vertical shifts, stretches, and shrinks, as well as reflections across the x- and y-axes.

d. Investigate and explain characteristics of a function: domain, range, zeros, intercepts, intervals of increase and decrease, maximum and minimum values, and end behavior

h. Determine graphically and algebraically whether a function has symmetry and whether it is even, odd or neither.






i. Understand that any equation in x can be interpreted as the equation f(x) = g(x), and interpret the solutions of the equation as the x-value(s) of the intersection point(s) of the graphs of y = f(x) and y = g(x).

MM1A2. Students will simplify and operate with radical expressions, polynomials, and rational expressions. a. Simplify algebraic and numeric expressions involving square root. b. Perform operations with square roots. c. Add, subtract, multiply, and divide polynomials. d. Add, subtract, multiply, and divide rational expressions.

e. Factor expressions by greatest common factor, grouping, trial and error, and special products limited to the formulas below.

( )( )( )( )( )( ) ( )( )( )

2 2 2

2 2 2

2 2

2

3 3 2 2 3

3 3 2 2 3

2

2

3 3

3 3

x y x xy y

x y x xy y

x y x y x y

x a x b x a b x ab

x y x x y xy y

x y x x y xy y

+ = + +

− = − +

+ − = −

+ + = + + +

+ = + + +

− = − + −

MM1A3. Students will solve simple equations.

a. Solve quadratic equations in the form ax2 + bx + c = 0 where a = 1, by using factorization and finding square roots where applicable.

b. Solve equations involving radicals such as + =x b c , using algebraic techniques. c. Use a variety of techniques, including technology, tables, and graphs to solve equations

resulting from the investigation of 2 0+ + =x bx c . d. Solve simple rational equations that result in linear equations or quadratic equations with

leading coefficient of 1. RELATED STANDARDS ADDRESSED: MM1P1. Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving.

MM1P2. Students will reason and evaluate mathematical arguments.

a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures.






c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof.

MM1P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely.

MM1P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics.

MM1P5. Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena.

UNIT OVERVIEW Prior to this unit students need to have worked extensively with operations on integers, rational numbers, and square roots of nonnegative integers as indicated in the grade 6 – 8 standards for Number and Operations. In the unit students will apply and extend all of standards for algebra listed as key standards addressed in Units 1 and 2 and all of the supporting standards from Grades 6 – 8 referenced in the overviews for Units 1 and 2. In working with symmetry of graphs, students will apply the concepts of similarity and transformations of geometric figures inherent in the Grade 7 standards for geometry. The unit begins with applied problems that can be modeled by quadratic equations. The need to solve such equations provides motivation for the topic of solving quadratic equations by factoring and, hence, for learning to factor. As indicated by the standards for this unit, the quadratic equations to be solved are limited to those which are equivalent to equations of the form x2 + bx + c = 0. Students see application of adding, subtracting, and multiplying polynomials as they take a variety of quadratic equations and put them in this standard form. In work with solving quadratic equations by factoring, it is intended that students learn to factor expressions of the form x2 + bx + c by applying the special product ( )( ) ( )2x a x b x a b x ab+ + = + + + . This method gives a strong foundation for learning to factor other trinomials by the grouping method, a method that allows students who do not yet grasp the “big picture” of factoring to be successful in factoring trinomials that would have a very large number of cases if approached as factoring by trial and error.






The beginning task of the unit also introduces the concept of viewing solutions of equations as first coordinates of points of intersection of the graphs of appropriate functions. This concept is revisited throughout the unit so that students gain understanding of the variety of ways in which the concept can be applied. The unit introduces techniques for solving simple radical equations and motivates the need to solve such equations in application that requires finding intersection points of graphs. Thus, students see the task of finding the intersection points of graphs from two perspectives: (1) as an important question that often requires algebraic techniques for exact solution and (2) as a method for interpreting and verifying algebraic solutions of equations. The treatment of algebraic techniques for radical equations is limited to equations of the form x b c+ = . Solving such equations does introduce the basic techniques for solving radical equations. Solution of more advanced equations involving radicals is addressed Mathematics III. (See standard MM3A3.) The unit also includes the techniques for solving rational equations. The need for solving such equations is introduced by a topic from physical science, the concept of resistance in an electrical circuit, but other applications are included. The presentation focuses on the techniques for solving rational equations and reinforces the topic of solving quadratic equations since rational equations that lead to both linear and quadratic equations are included. For this unit, the denominators that occur in the expressions are limited to rational numbers and first degree polynomials. Solution of rational equations involving higher degree denominators is addressed in Mathematics IV. (See standard MM4A1.) The unit emphasizes the connections among the solution of equations and graphs of functions. This approach gives students many opportunities to consolidate their understandings of the topics from Units 1 and 2. Additionally, the unit includes an in-depth discussion of even and odd symmetry of graphs of functions and transformations of graphs by reflection in the coordinate axes. These topics and the discussion of solving rational equations and quadratic equations of the form x2 – c = 0, c ≥ 0, reinforce topics from geometry, especially the topics of symmetry and transformation of geometric figures, similar triangles, and the Pythagorean Theorem. Students need extensive practice with multiple representations of the same mathematical concept. This unit focuses in integration of algebraic and graphical viewpoints but includes many opportunities to use verbal, tabular, algebraic, and geometric representations to organize, record, and communicate mathematical ideas. Throughout the unit it is important to: • Promote student use of multiple representations of concepts and require students to explain how to

translate information from one representation to another. Such activities especially include requiring students to explain how their equations represent the physical situation they are intended to model and how graphs represent algebraic equalities.

• Regularly use graphing technology to explore graphs of functions and verify calculations. While graphing functions by hand is necessary for developing understanding of many situations, students can deepen their understanding of topics through the use of graphing technology, sometimes by viewing a calculator, or computer, drawn version of a graph that they have already drawn by hand. The unit includes many situations where graphing by hand would so time consuming that it would be a major distraction and hindrance to the focus of the activity. For example, exploration of graphs with technology lead to student conjectures that must be verified algebraically.






• Continue to be extremely careful in the use of language of functions as indicated in the Unit 1 overview.

• In discussing solutions of equations, emphasize the concept of finding equivalent equations, exceptions to this concept, and the concept of solutions set of an equation so that students put particular techniques for solving quadratic, radical, or rational equations in the context of the general theory of solving equations.

TASKS: The remaining content of this framework consists of student tasks or activities presented in a real-world context. Tasks 1 – 4 are designed to allow students to learn by active exploration of the topic in a context and are denoted Learning Tasks. The first task is intended to launch the unit. It introduces the two themes of the unit, techniques for solving non-linear equations and connections among algebraic statements and graphs of functions. The second task explores graph symmetry and odd and even functions. It also includes solving simple radical equations. The other learning tasks focus on techniques for solving rational equations and quadratic equations by finding square roots. The last task is designed to demonstrate the type of assessment activities students should be comfortable with by the end of the unit. Thorough Teacher’s Guides which provide solutions, discuss teaching strategy, and give additional mathematical background are available to accompany each task.






Paula’s Peaches Learning Task Paula is a peach grower in central Georgia and wants to expand her peach orchard. In her current orchard, there are 30 trees per acre and the average yield per tree is 600 peaches. Data from the local agricultural experiment station indicates that if Paula plants more than 30 trees per acre, once the trees are in production, the average yield of 600 peaches per tree will decrease by 12 peaches for each tree over 30. She needs to decide how many trees to plant in the new section of the orchard. Throughout this task assume that, for all peach growers in this area, the average yield is 600 peaches per tree when 30 trees per acre are planted and that this yield will decrease by 12 peaches per tree for each additional tree per acre. 1. Paula believes that algebra can help her determine the best plan for the new section of orchard and

begins by developing a mathematical model of the relationship between the number of trees per acre and the average yield in peaches per tree.

a. Is this relationship linear or nonlinear? Explain your reasoning.

b. If Paula plants 6 more trees per acre, what will be the average yield in peaches per tree?

c. What is the yield in peaches per tree if she plants 42 trees per acre?

d. Let T be the function for which the input x is the number of trees planted on each acre and T(x) is the average yield in peaches per tree. Write a formula for T(x) in terms of x and express it in simplest form. Explain how you know that your formula is correct.

e. Draw a graph of the function T . Given that the information from the agricultural experiment station applies only to increasing the number of trees per acre, what is an appropriate domain for the function T?

2. Since her income from peaches depends on the total number of peaches she produces, Paula

realized that she needed to take a next step and consider the total number of peaches that she can produce per acre.






a. With the current 30 trees per acre, what is the yield in total peaches per acre?

b. If Paula plants 36 trees per acre, what will be the yield in total peaches per acre?

c. What is the yield in total peaches per acre if she plants 42 trees per acre?

d. Find the average rate of change of peaches per acre with respect to number of trees per acre when the number of trees per acre increases from 30 to 36. Write a sentence to explain what this number means.

e. Find the average rate of change of peaches per acre with respect to the number of trees per acre when the number of trees per acre increases from 36 to 42. Write a sentence to explain the meaning of this number.

f. Is the relationship between number of trees per acre and yield in peaches per acre linear? Explain your reasoning.

g. Let Y be the function that expresses this relationship, that is, the function for which the input x is the number of trees planted on each acre and the output Y(x) is the total yield in peaches per acre. Write a formula for Y(x) in terms of x and express your answer in expanded form.

h. Calculate Y(30), Y(36), and Y(42). What is the meaning of these values? How are they related to your answers to parts a through c?

i. What is the relationship between the domain for the function T and the domain for the function Y? Explain.






3. Paula wants to know whether there is a different number of trees per acre that will give the same yield per acre as the yield when she plants 30 trees per acre. a. Write an equation that expresses the requirement that x trees per acre yields the same total

number of peaches per acre as planting 30 trees per acre.

b. Use the algebraic rules for creating equivalent equations to obtain an equivalent equation with an expression in x on one side of the equation and 0 on the other.

c. Multiply this equation by an appropriate rational number so that the new equation is of the form . Explain why this new equation has the same solution set as the equations

from parts a and b.

2 0x bx c+ + =

d. When the equation is in the form 2 0x bx c+ + = , what are the values of b and c

e. Find integers m and n such that m⋅n = c and m + n = b.

f. Using the values of m and n found in part e, form the algebraic expression ( )( )x m x n+ + and simplify it.

g. Combining parts d through f, rewrite the equation from part c in the form . ( )( ) 0x m x n+ + =

h. This equation expresses the idea that the product of two numbers, x m+ and x n+ , is equal to 0. We know from the discussion in Unit 2 that, when the product of two numbers is 0, one of the numbers has to be 0. This property is called the Zero Factor Property. For these particular values of m and n, what value of x makes 0x m+ = and what value of x makes 0+ = ? x n






i. Verify that the answers to part h are solutions to the equation written in part a. It is appropriate to use a calculator for the arithmetic.

j. Write a sentence to explain the meaning of your solutions to the equation in relation to planting peach trees.

4. Paula saw another peach grower, Sam, from a neighboring county at a farm equipment auction and

began talking to him about the possibilities for the new section of her orchard. Sam was surprised to learn about the agricultural research and said that it probably explained the drop in yield for a orchard near him. This peach farm has more than 30 trees per acre and is getting an average total yield of 14,400 peaches per acre. a. Write an equation that expresses the situation that x trees per acre results in a total yield per

acre of 14,400 peaches per acre.

b. Use the algebraic rules for creating equivalent equations to obtain an equivalent equation with an expression in x on one side of the equation and 0 on the other.

c. Multiply this equation by an appropriate rational number so that the new equation is of the form . Explain why this new equation has the same solution set as the equations

from parts a and b.

2 0x bx c+ + =

d. When the equation is in the form 2 0x bx c+ + = , what is value of b and what is the value of c?

e. Find integers m and n such that m⋅n = c and m + n = b .






f. Using the values of m and n found in part e, form the algebraic expression ( )( )x m x n+ + and

simplify )( )(x m x n+ + .

g. Combining parts d through f, rewrite the equation from part d in the form . ( )( ) 0x m x n+ + =

h. This equation expresses the idea that the product of two numbers, x m+ and x n+ , is equal to 0. We know from the discussion in Unit 2 that, when the product of two numbers is 0, one of the numbers has to be 0. What value of x makes 0x m+ = ? What value of x makes

0 ? x n+ =

i. Verify that the answers to part h are solutions to the equation written in part a. It is appropriate to use a calculator for the arithmetic.

j. Which of the solutions verified in part i is (are) in the domain of the function Y? How many peach trees per acre are planted at the peach orchard getting 14400 peaches per acre?

The steps in items 3 and 4 outline a method of solving equations of the form . These

equations are called quadratic equations and an expression of the form

2 0x bx c+ + =2x bx c+ + is called a

quadratic expression. In general, quadratic expressions may have any nonzero coefficient on the 2x term, but in Mathematics I we focus on quadratic expressions with coefficient 1 on the 2x term. An

important part of this method for solving quadratic equations is the process of rewriting an expression of the form 2x bx c+ + in the form ( )( )x m x n+ + . The rewriting step is an application of Identity 1from Unit 2. The identity tells us that the product of the numbers m and n must equal c and that the sum of m and n must equal b. In Mathematics I, we will apply Identity 1 in this way only when the values of b, c, m, and n are integers.






5. Since the whole expression ( )( )x m x n+ + is a product, we call the expressions x m+ and x n+

the factors of this product. For the following expressions in the form 2x bx c+ + , rewrite the expression as a product of factors of the form x + m and x + n. Verify each answer by drawing a rectangle with sides of length x m+ and x n+ , respectively, and showing geometrically that the

area of the rectangle is 2x bx c+ + .

a. 2 3 2x x+ +e. 22 8 1x x+ +

b. 2 6 5x x+ +

f. 2 13 36x x+ +

c. 2 5 6x x+ + g. 2 13 12x x+ +

d. 2 2 7 1x x+ +

6. In item 5, the values of b and c were positive. Now use Identity 1 in reverse to factor each of the

following quadratic expressions of the form 2x bx c+ + where c is positive but b is negative. Verify each answer by multiplying the factored form to obtain the original expression. a. 2 8 7x x− +

e. 2 11 24x x− + b. 8 2 9 1x x− +

f. 2 11 18x x− +

c. 2 4 4x x− + g. 2 12 27x x− +

d. 5 2 8 1x x− +






7. Use Identity 1 in reverse to factor each of the following quadratic expressions of the form 2x bx c+ + where c is negative. Verify each answer by multiplying the factored form to obtain the

original expression. a. 2 6 7x x+ −

b. 2 6 7x x− −

c. 2 42x x+ −

d. 2 42x x− −

e. 2 10 24x x+ −

f. 2 10 24x x− −

8. In items 3 and 4, we used factoring as part of a process to solve equations that are equivalent to

equations of the form where b and c are integers. Look back at the steps you did in

items 3 and 4, and describe the process for solving an equation of the form . Use this process to solve each of the following equations, that is, to find all of the numbers that satisfy the original equation. Verify your work by checking each solution in the original equation.

2 0x bx c+ + =2 0x bx c+ + =

a. 0 2 6 8x x− + =

b. 2 15 36 0x x− + =

c. 2 28 27 0x x+ + = d. 0 2 3 10x x− − =

e. 02 2 15x x+ − = f. 02 4 21x x− − =

g. 2 7 0x x− =

h. 2 13 0x x+ =






9. The process you used in item 8 works whenever you have an equation in the form . There are many equations, like those in items 3 and 4, that look somewhat different from this form but are, in fact, equivalent to an equation in this form. Remember that the Addition Property of Equality allows us to get an equivalent equation by adding the same expression to both sides of the equation and the Multiplicative Property of Equality allows us to get an equivalent equation by multiplying both sides of the equation by the same number as long as the number we use is not 0. For each equation below, find an equivalent equation in the form

2 0x bx c+ + =

2 0x bx c+ + = . a. 0 26 12 48x x+ − =

b. 2 8 9x x− =

c. 0 23 21 3x x= −

d. 24 24 20x x+ = e. ( )11 30 0x x − + =

f. ( )1 8 102

x x + =

g. ( )( )1 5 3x x 0+ + + =

h. 9( )25 4x + =

i. ( )( )2 3 4 2x x x 4+ + = +

j. ( )5 3 200x x + =

10. Now we return to the peach growers in central Georgia. How many peach trees per acre would result in only 8400 peaches per acre?

11. If there are no peach trees on a property, then the yield is zero peaches per acre. Write an equation to express the idea that the yield is zero peaches per acre with x trees planted per acre, where x is number greater than 30. Is there a solution to this equation, that is, is there a number of trees per acre that is more than 30 and yet results in a yield of zero peaches per acre? Explain.




12. At the same auction where Paula heard about the peach grower who was getting a low yield, she talked to the owner of a major farm supply store in the area. Paula began telling the store owner about her plans to expand her orchard, and the store owner responded by telling her about a local grower that gets 19,200 peaches per acre. Is this number of peaches per acre possible? If so, how many trees were planted?

13. Using graph paper, explore the graph of Y as a function of x. a. What points on the graph correspond to the answers for part j from items 3and 4?

b. What points on the graph correspond to the answers to items 10, 11, and 12?

c. What is the relationship of the graph of the function Y to the graph of the function f , where the formula for f(x) is the same as the formula for Y(x) but the domain for f is all real numbers?

d. Items 4, 10, and 11 give information about points that are on the graph of f but not on the

graph of Y. What points are these?

e. Graph the functions f and Y on the same axes. How does your graph show that the domain of f is all real numbers? How is the domain of Y shown on your graph?

14. In answering parts a, b, and d of item 13, you gave one geometric interpretation of the

solutions of the equations solved in items 3, 4, 10, 11, and 12. We now explore a slightly different viewpoint. a. Draw the line y = 18000 on the graph drawn for item 13, part e. This line is the graph of

the function with constant value 18000. Where does this line intersect the graph of the function Y? Based on the graph, how many trees per acre give a yield of more than 18000 peaches per acre?






b. Draw the line y = 8400 on your graph. Where does this line intersect the graph of the

function Y? Based on the graph, how many trees per acre give a yield of fewer than 8400 peaches per acre?

c. Use a graphing utility and this intersection method to find the number of trees per acre that give a total yield closest to the following numbers of peaches per acre: (i) 10000 (ii) 15000 (iii) 20000

d. Find the value of the function Y for the number of trees given in answering (i) – (iii) in part c above.

15. For each of the equations solved in item 8, do the following.

a. Use technology to graph a function whose formula is given by the left-hand side of the

equation.

b. Find the points on the graph which correspond to the solutions found in item 8.

c. How is each of these results an example of the intersection method explored in item 14?






Logo Symmetry Learning Task In middle school you learned about line and rotational symmetry. Remember that a figure has line symmetry if there is a line that divides the figure into two parts that are mirror images of each other. A figure has rotational symmetry if, when rotated by an angle of 180 degrees or less about its center, the figure aligns with itself. A figure can also have point symmetry. A figure is symmetric about a single point if when rotated about that point 180 degrees it aligns with itself. So, rotational symmetry of 180 degrees is also symmetry about the center. 1. We all see many company logos everyday. These logos often have symmetry. For each logo

shown below, identify and explain any symmetries you see.






2. A textile company called “Uniform Universe” has been hired to manufacture some military uniforms. To complete the order, they need embroidered patches with the military insignia for a sergeant in the United States Army. To save on costs, Uniform Universe subcontracted a portion of their work to a foreign company. The machines that embroider the insignia design require a mathematical description. The foreign company incorrectly used the design at the right, which is the insignia for a British Sergeant. They sent the following description for the portion of the design to be embroidered in black.

Function Domain 1y x= − – 7 x 7 ≤ ≤

3y x= − – 7 x 7 ≤ ≤

5y x= − – 7 x 7 ≤ ≤

7y x= − – 7 x 7 ≤ ≤

Black embroidery instructions:

Vertical line Restriction x = – 7 0 y 6 ≤ ≤x = 7 0 y 6 ≤ ≤

a. Match the lines in the design to the functions indicated in the table.

b. When the design is stitched on a machine, wide black stitching is centered along the lines given by the equations above. The British Sergeant’s insignia has a light-colored embroidery between the lines of black. Write a description for the lines on which the light-colored stitching will be centered. Use a table format similar to that shown above.

3. Jessica, a manager at Uniform Universe, immediately noticed the design error when she saw some of the prototype uniforms. The sergeant’s insignia was upside down from the correct insignia for a U.S. sergeant, which is shown at the right. Jessica checked the description that had been sent by the foreign contractor. She immediately realized how to fix






the insignia. So, she emailed the foreign supplier to point out the mistake and to inform the company that the error could be corrected the by reflecting each of the functions in the x-axis.

Ankit, an employee at the foreign textile company, e-mailed Jessica back and included the graph at the right to verify that Uniform Universe would be satisfied with the new formulas. a. What type of symmetry does the incorrect insignia

have?

b. If it is symmetric about a point, line, or lines, write the associated coordinates of the point or equation(s) for the lines of symmetry.

c. Does the corrected insignia have the same symmetry?

d. Write the mathematical description of the design for the U.S. sergeant insignia, as shown

in the graph above. Verify that your mathematical description yields the graphs shown.

e. Let f denote any one of the functions graphed in the British sergeant’s insigniz or the U.S.

sergeant’s insignia. Compare f(1) and f(–1), f(2.5) and f(–2.5), f(3.7) and f(–3.7). If x is a number such that 0 ≤ x ≤7, how do f(x) and f(– x) compare?

f. Let a be a constant other than the number 0 and let g denote the function whose formula is given by g(x) = ax2. You studied the shapes of these graphs in Unit 1. Look back at some examples for particular choices of a. What type of symmetry do these graphs have? If x is a positive number, how do g(x) and g(– x) compare?

4. We call a function f an even function if, for any number x in the domain of f, – x is also in the domain and f(– x) = f(x). a. Suppose f is an even function and the point (3, 5) is on the graph of f. What other point

do you know must be on the graph of f? Explain.






b. Suppose f is an even function and the point (– 2, 4) is on the graph of f. What other point do you know must be on the graph of f? Explain.

c. If (a, b) is a point on the graph of an even function f, what other point is also on the graph of f?

d. What symmetry does the graph of an even function have? Explain why.

e. Consider the function k, which is an even function. Part of the graph of k is shown to the right. Using the information that k is an even function, complete the graph for the rest of the domain.

5. When Jessica’s supervisor, Malcom, saw the revised design, he told Jessica that he did not think that the U.S. Army would be satisfied. He pointed out that, while the revision did turn the design right-side up, it did not account for the slight curve in the lines in the real U.S. sergeant’s insignia. He suggested that a square root function might be a better choice than an absolute value function and told Jessica to work with the foreign contractor to get a more accurate design. Jessica emailed Ankit to let him know that the design needed to be revised again to have lines with a curve similar to the picture from the U.S. Army website, as shown above, and suggested that he try the square root function.






a. Ankit graphed the square root function, y = x , and decided to limit the domain to 4x≤ ≤ . Set up a grid using a scale of ½ -inch for each unit, and graph the square root

function on this limited domain. For accuracy, plot points for the following domain values: 0,

0

14 , 1, 25

16 , 94 , 49

16 , 4.

b. Ankit saw that his graph of the square root function (on the domain 0 4x≤ ≤ ) looked like the curve that forms the lower right edge of the British sergeant’s insignia. He knew that he could reflect the graph through the x-axis to turn the curve over for the U.S. sergeant’s insignia, but first he needed to reflect the graph through the y-axis in order to get the full curve to form the lower edge of the British sergeant’s insignia. He knew that he needed to input a negative number and then get the square root of the corresponding positive number, so he tried graphing the function y = x− on the domain 4 0x . Reproduce Ankit’s graph using the scale of ½-inch for each unit and using the opposites of the domain values in part a as some accurate points on the graph: 0,

− ≤ ≤

14− , – 1, 25

16− , 9

4− , 4916− ,– 4.

c. To see an additional example of reflecting a graph through the axes, graph each of the

following and compare the graphs.

(i) (ii) 2 3y x= +( )2 32 3

y xx

= − +

= − − (iii)

( )2 32 3

y xx

= − +

= − +

d. Given an equation of the form y = f(x), there is a specific process for creating a second equation whose graph is obtained by reflecting the graph of y = f(x) through the x-axis, and there is different process for creating an equation whose graph is obtained by reflecting the graph of of y = f(x) through the y-axis. Explain these processes in your own words.

e. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part a in the x-axis and then shifting it up 8 units. Call this function f1, read “f sub one.”

f. Write a formula for the function whose graph coincides with the graph obtained by reflecting the graph from part b in the x-axis and then shifting it up 8 units. Call this function f2, read “f sub two.” When graphed on the same axes, the graphs of the functions f1 and f2 give the curve shown at the right.






g. Ankit completed his mathematical definition for the U.S. Army sergeant’s insignia and sent it to Jessica along with the graph shown at the right. Based on the graph and your work above, write Ankit’s specifications for black embroidery of the insignia as shown in the graph.

Function Domain

y = y = y = y = y = y = y = y =

Vertical line Restriction x =

x =






6. A few days after Ankit sent the completed specifications to Uniform Universe, he looked back at the picture of the insignia and his graphs and thought that he could improve upon the proportions of the design and the efficiency of his mathematical formula. He sent the revised instructions below. Use graphing technology to examine Ankit’s final mathematical definition for the insignia. Explain the effect of the changes from the functions you wrote in answer to item 4.

Vertical line Restriction x = – 4 0.4 ≤ y ≤ 3.5 x = 4 0.4 ≤ y ≤ 3.5

Function Domain

7.5 2.0y = − x – 4 x ≤ ≤ 4

6.0 1.8y x= − – 4 x ≤ ≤ 4

4.7 1.65y x= − – 4 x ≤ ≤ 4

3.5 1.55y x= − ≤ – 4 x ≤ 4

7. Now, we will use some functions involving square root and some linear functions to create another logo. The functions are listed in the table below. The logo is the shape completely enclosed by the graphs of the functions. Thus, in order to draw the logo, you will need to find the points of intersections among the graphs. Once you have the points of intersections, you can determine how to limit the domain of each function to specify the boundary of the logo. You are also asked to specify the relationship of the other graphs to the graph of

2y x= and to find the range for each function after you have restricted the domain.

Function Relation of the graph to graph of (i) Domain

What is the range of the function with limited domain?

(i) (ii) Reflection through (iii) Reflection through (iv) Rotation of (v) Intersects at ( __, __) (vi) No intersection






To find the x-coordinates of the points of intersection algebraically, solve the following equations. In solving the equations remember the definition of square root:

• a b= if and only if a and b are nonnegative real numbers with 2a b= .

Verify your solutions geometrically by checking that each solution is the x-coordinate of the point where the graph of the function given by the expression on the left side of the equation intersects either the line or the line 4y = 4y = − .

Fill in the table and graph the logo with the limited domain. Be creative and color the logo after you are finished graphing it.

a. This logo is symmetric about a point. What is the point of symmetry?

b. This logo is symmetric about two different lines, write the associated equations for the lines of symmetry. Where do these lines intersect?

c. If the figure has rotational symmetry, determine the angle of rotation for which it is symmetric.

The last part of this task explores another logo, but first we explore the coordinate geometry of reflections and rotations a little further.






8. Start with a point ( , )a b . a. Assuming that ( , )a b is in Quadrant I, reflect the point in the x-axis. Which coordinate

stays the same? Which coordinate changes? What are the coordinates of the new point obtained by reflecting in the x-axis?

b. Repeat part a assuming that ( , )a b is in Quadrant II.

c. Repeat part a assuming that ( , )a b is in Quadrant III.

d. Repeat part a assuming that ( , )a b is in Quadrant IV.

e. Repeat part a assuming that you have a point on the y-axis.

f. Summarize: For any point ( , )a b , reflecting the point through the x-axis results in a point whose coordinates are ( ),a b− . Explain why this rule applies to points on the x-axis even though reflecting such points in the x-axis results in the same point as the original one.






9. Start with a point ( , )a b . a. Assuming that ( , )a b is in Quadrant I, reflect the point in the y-axis. Which coordinate

stays the same? Which coordinate changes? What are the coordinates of the new point obtained by reflecting in the y-axis?

b. Repeat part a assuming that ( , )a b is in Quadrant II.

c. Repeat part a assuming that ( , )a b is in Quadrant III.

d. Repeat part a assuming that ( , )a b is in Quadrant IV.

e. Repeat part a assuming that you have a point on the x-axis.

f. Summarize: For any point ( , )a b , reflecting the point through the y-axis results in a point whose coordinates are ),a b− . Explain why this rule applies to points on the y-axis even though reflecting such points in the y-axis results in the same point as the original one.

(






10. Start with a point ( , )a b . a. Reflect the point through the x-axis and then reflect that point through the y-axis. What

are the coordinates of the twice-reflected point in terms of a and b? Does the location of the point (Quadrant I, II, III, IV, or on one of the axes) affect your answer?

b. Start with point (a, b) again. This time reflect through the y-axis first and then the x-axis. What are the coordinates of this twice-reflected point in terms of a and b? Does the location of the point (Quadrant I, II, III, IV or on one of the axes) affect your answer?

c. Compare the results from parts a and b. Does the order in which you perform the two different reflections have an effect on the final answer?

d. Using the concept of slope, explain why the points (a, b) and (–a, –b) are on the same line through the origin. What happens if a = 0 and slope is not defined for the line through the origin and the point (a, b)?

e. What are the coordinates of the point obtained by rotating the point (a, b) through 180° about the origin? Explain how to use reflection through the axes to rotate a point 180° about the origin. Note: the two points are symmetric with respect to the point (0, 0), so find this second point is also called a reflection of the point (a, b) through the origin.






11. We call a function f an odd function if, for any number x in the domain of f, – x is also in the domain and f(– x) = – f(x). a. Suppose f is an odd function and the point (3, 5) is on the graph of f. What other point do

you know must be on the graph of f?

b. Suppose f is an odd function and the point (– 2, 4) is on the graph of f. What other point do you know must be on the graph of f?

c. If (a, b) is a point on the graph of an odd function f, what is f(a)? What other point is also on the graph of f?

d. What symmetry does the graph of an odd function have? Explain why.

e. Consider the function k, which is an odd function. The part of the graph of k which has nonnegative numbers for the domain is shown at the right below. Using the information that k is an odd function, complete the graph for the rest of the domain.






f. In Mathematics I, you have studied six basic functions:

f(x) = x, f(x) = x2, f(x) = x3, f(x) = x , f(x) = |x|, and f(x) = x1

Classify each of these basic functions as even, odd, or neither.

g. For each basic function you classified as even, let g be the function obtained by shifting

the graph down five units, and determine whether g is even, odd, or neither.

h. For each basic function you classified as odd, let h be the function obtained by shifting

the graph up three units, and determine whether h is even, odd, or neither.






12. Next we explore a logo based on transformations of the basic function 1yx

= . The graph of

the logo is shown at the right. The curved parts of the logo lie on the graphs of the functions listed in the table below.

13.

a. Complete the mathematical definition for this logo by completing the table. Determine the remaining information by applying what you know about the graphs of the functions listed to obtain equations in x and y for each curved or straight line shown at the right.

Function

Domain

Vertical line

Restriction

4yx

= 1 4x≤ ≤ or ____________

1x = 4 y≤ ≤ 9

9yx

= 1 9x≤ ≤ or ____________

12

x = 182

y− ≤ ≤ −

2yx−

= 1 42

x≤ ≤ or

____________

4yx−

= 1 82

x≤ ≤ or

____________






b. The completed logo, without grid lines, is shown at the right. What symmetry does this logo have?

14. Create a logo using any combination of vertical shifts, vertical stretches or shrinks, reflection through the x-axis, or reflection through the y-axis of the basic functions listed below as well as horizontal and vertical lines.

f(x) = x, f(x) = x2, f(x) = x3, f(x) = x , f(x) = |x|, and f(x) = 1x .

Your logo should be aesthetically appealing and must include the following: • at least one of the functions from the list of basic functions • at least four different equations • at least two examples of vertical shifts, vertical stretches and/or shrinks • at least one reflection • at least one type of symmetry

o Explain how your logo meets each of the requirements listed above. o Identify any important points, lines, and or angles associated with your logo’s symmetry.






Resistance Learning Task

We use electricity every day to do everything from brushing our teeth to powering our cars. Electricity results from the presence and flow of electric charges. Electrons with a negative charge are attracted to those with a positive charge. Electrons cannot travel through the air. They need a path to move from one charge to the other. This path is called a circuit. A simple circuit can be seen in the connection of the negative and positive ends of a battery.

When a circuit is created, electrons begin moving from the one charge to the other. In the circuit below, a bulb is added to the circuit. The electrons pass through the filament in the bulb heating it and causing it to glow and give off light.

Electrons try to move as quickly as possible. If a circuit is not set up carefully, too many electrons can move across at one time causing the circuit to break. We can limit the number of electrons crossing over a circuit to protect it. Adding objects that use electricity, such as the bulb in the above circuit, is one way to limit the flow of electrons. This limiting of the flow of electrons is called resistance. It is often necessary to add objects called resistors to protect the circuit and the objects using the electricity passing through the circuit. In the circuit below, R1 represents a resistor.

More than one resistor can be placed on a circuit. The placement of the resistors determines the total effect on the circuit. The resistors in the diagram below are placed in parallel (this refers to the fact that there are no resistors directly between two resistors, not to the geometric definition of parallel). Parallel resistors allow multiple paths for the electricity to flow. Two examples of parallel resistors are shown below.






The resistors in the next circuit below are not parallel. These resistors are placed in series because the electricity must travel through all three resistors as it travels through the circuit. Resistance is measured in units called ohms and must always be a positive number. The omega symbol, Ω, is used to represent ohms. For n resistors in parallel, R1, R2, R3, etc. the total resistance, RT, across a circuit can be found using the equation:

1 2 3

1 1 1 1 1

T nR R R R R= + + + +L

For n resistors is series, R1, R2, R3, etc. the total resistance, RT, across a circuit can be found using the equation: 1 2 3T nR R R R R= + + + +L 1. What is the total resistance for a circuit with three resistors in series if the resistances are 2

ohms, 5 ohms, and 4 ohms, respectively?

2. What is the total resistance for a circuit with two parallel resistors, one with a resistance of 3 ohms and the other with a resistance of 7 ohms?






3. What is the total resistance for a circuit with four resistors in parallel if the resistances are 1

ohm, 3 ohms, 52

ohms, and 35

ohms, respectively?

4. What is the total resistance for the circuit to the right?

5. A circuit with a total resistance of 2811

has

two parallel resistors. One of the resistors has a resistance of 4 ohms. a. Let x represent the resistance of the other of the other resistor, and write an equation for

the total resistance of the circuit.

b. The equation in part a contains rational expressions. If you have any complex fractions,

simplify them. In your equation containing no complex fractions, what is the least common denominator of the rational expressions?

c. Use the Multiplication Principle of Equality to obtain a new equation that has the same

solutions as the equation in part a but does not contain any rational expressions. Why do you know that x ≠ 0? How does knowing that x ≠ 0 allow you to conclude that this new equation has the same solutions as, or is equivalent to, the equation from part a.

d. Solve the new equation to find the resistance in the second resistor. Check your answer.

6. A circuit has been built using two parallel resistors. a. One resistor has twice the resistance of the other. If the total resistance of the circuit is

34

ohms, what is the resistance of each of the two resistors?






b. One resistor has a resistance of 4 ohms. If the total resistance is one-third of that of the other parallel resistor, what is the total resistance?

7. A circuit has been built using two paths for the flow of the current; one of the paths has a single resistor and the other has two resistors in series as shown in the diagram at the right. a. Assume that, for the two resistors in series, the second

has a resistance that is three times the resistance of the first one in the series. The single resistor has a resistance that is 6 ohms more than the resistance of the first resistor in series, and the total resistance of the circuit is 4 Ω. Write an equation to model this situation, and solve this equation. What is the solution set of the equation? What is the resistance of the each of the resistors?

b. Assume that, for the two resistors in series, the second has a resistance that is 3 ohms more than twice the resistance of the first one in the series. The single resistor has a resistance that is 1 ohm more than the resistance of the first resistor in series, and the total resistance of the circuit is 3 Ω. Write an equation to model this situation, and solve this equation. What is the solution set of the equation? What is the resistance of the each of the resistors?

c. Assume that, for the two resistors in series, the second has a resistance that is 2 ohms more than the first one in the series. The single resistor has a resistance that is 3 ohms more than the resistance of the first resistor in series, and the total resistance of the circuit is 2 Ω. Write an equation to model this situation, and solve this equation. What is the solution set of the equation? What is the resistance of the each of the resistors?






d. Assume that, for the two resistors in series, the second has a resistance that is 4 ohms more than the first one in the series. The single resistor has a resistance that is 3 ohms less than the resistance of the first resistor in series, and the total resistance of the circuit is 4 Ω. . Write an equation to model this situation, and solve this equation. What is the solution set of the equation? What is the resistance of the each of the resistors?

8. A circuit has three resistors in parallel. The second resistor has a resistance that is 4 ohms more than the first. The third resistor has a resistance of 8 ohms. The total resistance is one-half the resistance of the first resistor. Find each of the unknown resistances.






Shadows and Shapes Learning Task 1. At a particular time one spring day in a park area with level ground, a pine tree casts a 60-

foot shadow while a nearby post that is 5 feet high casts an 8-foot shadow. a. Using the figure below, draw appropriate lines to indicate the rays of sunlight, which are

always parallel.

b. Let h represent the height of the pine tree. Use relationships about similar triangles to write an equation involving h.

c. Find the height of the pine tree, to the nearest foot.






2. On bright sunny day, two men are standing in an open plaza. The shorter man is wearing a hat that makes him appear to be the same height as the other man, who is not wearing a hat. The hat is designed so that the top of the hat is 4 inches above the top of the wearer’s head. When the shorter man takes off his hat, he casts a 51-inch shadow while the taller man casts a 54-inch shadow. How tall is each man?

3. The plaza floor has a geometric design. The design includes

similar right triangles with the relationships shown in the figure at the right. a. If the hypotenuse of the larger right triangle is 15 inches,

what is the length of the other hypotenuse? Explain why.

b. Assuming the hypotenuse of the larger right triangle is 15 inches, for each triangle, write an equation expressing a relationship among the lengths of the sides of the triangle.

c. Solve at least one of the equations from part b, and find the lengths of the legs of both triangles.

4. Another shape in the plaza floor design is a right triangle with hypotenuse 26-inches long and one leg 24 inches long.








a. Let x denote the length of the other leg of this triangle. Write a quadratic equation expressing a relationship among the lengths of the sides of the triangle.

b. Put this equation in the standard form 2 0x bx c+ + = . What are the values of b and c?

c. What is the length of the other leg of the triangle?

5. When you put the equation you solved in item 4 in standard form, you were working with an

equation in the form . In item 4, the value of c is a perfect square, so you could use the Difference of Square identity to solve the equation by factoring. Use this identity and factoring to solve each of the equations below, each of which has the form .

2 0x c− =

2 0x c− = a. 2 25 0x − =

d. 2 81 0x − =

b. 2 49 0x − =

e. 2 121 0x − =

f. 2 64 0x − = c. 2 4 0x − =

5. Make a quick sketch of the graphs of each of the functions listed below, and discuss how the

solutions from item 5 can be seen in the graphs. a. ( ) 2 25f x x= − d. ( ) 2 81f x x= −

b. ( ) 2 49f x x= − e. ( ) 2 121f x x= −

c. ( ) 2 4f x x= − f. ( ) 2 64f x x= −




6. Use the graphs of appropriate functions to solve each of the following equations.

a. 2 5 0x − =

d. 2 8 0x − =

b. 2 11 0x − =

e. 2 2 0x − =

c. 2 5 0x − =

f. 2 12 0x − =

g. Explain why each equation has two solutions.

h. For a quadratic equation of the form 2 0x c− = with 0c > , how many solutions does the equation have? In terms of c, what are the solutions?

7. In the figure at the right, a line has been added to the part of the design from item 3. As

shown, the shorter leg of the smaller right triangle and the longer leg of the larger right triangle form the legs of this new right triangle. a. Let z represent the length of the hypotenuse of this new right

triangle, as indicated. Use the Pythagorean Theorem, and the lengths you found in item 3 to find z.

b. If a line segment were drawn from point A to point B in the figure, what would be the length of the segment?



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