AF U4 L1 SJ...Non–Linear Functions Lesson 1: Introduction to Quadratic Functions Page 1 AIIF...

Non– LinearFunctions

AIIFAlgebra II Foundations

Student Journal

Table of Contents

Lesson Page Lesson 1: Introduction to Quadratic Functions................................................................................................1

Lesson 2: The Quadratic Formula......................................................................................................................8

Lesson 3: Graphing Quadratic Functions and Their Applications .............................................................15

Lesson 4: Power Functions................................................................................................................................27

Lesson 5: Inverse Variation...............................................................................................................................39

Lesson 6: Exponential Functions......................................................................................................................51

Lesson 7: Step Functions ...................................................................................................................................67

Lesson 8: Miscellaneous Non–Linear Functions............................................................................................78

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Non–Linear Functions Lesson 1: Introduction to Quadratic Functions Page 1

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Lesson 1: Introduction to Quadratic Functions Activity 1 For the following exercises, write a matching quadratic equation then solve the equation and answer each question. 1. The square of a number is sixty-four. What are the numbers that make the equation true? 2. The square of the difference of a number and six is one hundred twenty-one. What are the numbers? 3. The area of a square piece of paper is 144 square inches. What are the lengths of the sides of the square

piece of paper?

4. The area of a crop circle is approximately 1257 square feet. Approximately what is the size of the diameter of the circle? Use 3.14 forπ .

5. A company’s cost can be determined by the sum of the variable

costs and the fixed costs. Namely, C(x) = variable costs + fixed costs. It has been determined that the company’s variable costs are the cost of producing one unit times the square of the number of units.

a. If it costs $11.00 to produce a single unit of the product and the company's fixed costs are

$1,000.00, write an equation, in function notation, that represents the total costs of producing x units of the product.

b. If total costs are $2,751,000.00, how many units of the product are produced? 6. A company makes a product. The company has determined the approximate cost to produce a single

unit of the product. The company has fixed costs of $500. The company also knows that it costs $250,500 to produce 100 units of the product. The engineering department’s research shows that the variable portion of the cost function is the cost to produce a single unit times the square of the number of units produced. That is, variable costs = 2cx . Write a cost function, C(x), which represents the total cost of producing x units of the product. Use the information given to determine the cost to produce one unit of the product.

Crop circles are patterns created by the flattening of

crops such as wheat, barley, rapeseed, rye, corn, linseed,

and soy into circles. The term was first used by researcher Colin Andrews to describe

simple circles he was researching.

Page 2 AIIF

Activity 2 In this activity, you will investigate the graphs of various quadratic equations. For the following exercises, find the coordinates of the minimum or maximum value and state the minimum or maximum y–value, all x–intercepts and y–intercepts, and make a sketch of the graph in the grid provided. The exercises are set up in most cases to draw two graphs per grid. You may also want to display the two graphs simultaneously on your graphing calculator as well. NOTE: Displaying table values on your graphing calculator may help you to draw the graph.

1. 212

y x=

2. 24y x=

3. 2 3y x= − + 4. 2 2y x= − −


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5. ( )23y x= +

6. ( )22y x= − 7. ( )22 3y x= − +

8. ( )22 3y x= − −

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9. For the given graph, identify the coordinates of the minimum point, x–intercepts, y–intercept, and equation representing the graph.

10. From Exercises 1 through 9, what conclusions and characteristics can you make about the graphs of

quadratic equations?


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Practice Exercises Solve each of the following. 1. The square of a number is one hundred forty-four. Write a quadratic equation and then solve for the

unknown number(s). 2. The square of the sum of a number and nine is one hundred sixty-nine. Write a quadratic equation and

then solve for the unknown number(s). 3. The Sparkling Diamonds jewelry store sold a diamond studded bracelet and made a profit of $196. The

profit is based on the cost of the necklace to the store. How much did the necklace cost the store if profit

is determined by the equation2

100 100C CP C⎛ ⎞= • =⎜ ⎟

⎝ ⎠, where P is the profit and C is the cost of the item?

4. Graph the quadratic equation 23 12y x= − + on the grid supplied below. Label all intercepts and

determine the maximum or minimum point.

5. Graph the quadratic equation ( )24y x= − on the grid supplied below. Label all intercepts and

determine the maximum or minimum point. 4. 5.

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6. Graph the quadratic equation ( )27 16y x= − − . Label all intercepts and determine the maximum or minimum point.

7. For the given graph, identify the minimum

point, x–intercepts, y–intercept, and equation representing the graph.

8. How does the process of squaring relate to quadratic functions?


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Outcome Sentences To solve a quadratic equation I know a quadratic equation will have a minimum when I know a quadratic equation will have a maximum when The minimum or maximum of a quadratic equation can be determined by I can use the graphing calculator to I would like to find out more about I now understand I still have a question about

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Lesson 2: The Quadratic Formula Activity 1 In this activity, you will be solving quadratic equations using the quadratic formula to find the values of x when y=0. Make sure the equations are written in general form before determining the coefficients a, b, and c. For the following exercises: a. Write the quadratic equation in general form. b. Identify the values of a, b, and c. c. State the nature of the roots by calculating the

discriminant. d. Find all solutions, if any, for the quadratic equation. 1. 2 3 4y x x= + − 2. 22 5 2y x x= + + 3. 23 8 3y x x= − − 4. 2 4 5y x x− + = 5. 26 9y x x+ − =

Standard Form: 2y ax bx c= + + General Form: 2 0ax bx c+ + =

Quadratic Formula: 2 4

2b b acx

a− ± −=

Non–Linear Functions Lesson 2: The Quadratic Formula Page 9

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6. For the given quadratic formula, identify the values a, b, and c and write the matching quadratic

equation in standard form. Note: use 2y ax bx c= + + .

a. − ± −

=25 5 4(1)(3)

2(1)x

b. ± − − −

=27 ( 7) 4(3)( 4)

2(3)x

c. ± − − −

=−

28 ( 8) 4( 9)(1)2( 9)

x

Page 10 AIIF

Activity 2 In this activity, your teacher will guide you through writing a program for the quadratic formula on the classroom graphing calculator. Test your program on the first two activity. Things you will need to pay attention to in your program are:

• The discriminant • Programming logic • Data input • Data output • Calculations using the quadratic formula

Use the supplemental exercises below to further test your program by solving for x when y=0. Round your answers to 3 decimal places. 1. 23 6 9y x x= − − 2. 212 5y x x= + − 3. 213 6 1y x x= + + 4. 20.25 6 4y x x= + + 5. 2 12 36y x x= − + 6. Can you think of any improvements in the program you wrote? .


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Activity 3 In this activity, you will continue to use the quadratic formula to solve quadratic equations for real-world applications. Use the same process from Activity 1 to find your solutions (write the equation in general form; identify the coefficients a, b, and c.) Make sure your answers make sense for the real-world application problem.

1. You have a part–time job working for a local machine shop. The owner plans to make a certain product

to sell. The product's costs are related by the function 2( ) 6250 50C x x x= + + and the owner knows he can sell the product for $325 each, giving him a total revenue of ( ) 325R x x= , where x represents the number of items produced. The owner would like you to find the break-even points so he can determine the number of product items he should produce each week.

2. A ball is thrown downward from the top of a building into a river. The height of the ball from the river

can be modeled by 2( ) 16 15 600H t t t= − − + , where t is the time, in seconds, after the ball was thrown. How long after the ball is thrown is it 75 feet above the river? How long, to the nearest tenth of a second, does it take the ball to land in the river?

3. It takes a 2004 Corvette 4.3 seconds to accelerate from 0 to 60 miles per hour. The same car can do the

quarter mile, 1320 feet, in 12.7 seconds. The displacement function can be described by the equation 2( ) 4.09 51.99s t t t= + .

a. How far has the Corvette traveled after 4.3 seconds, to the nearest foot?

b. How long does it take the Corvette to travel half a mile? Round your answer to the nearest tenth of a second. Note: A mile is 5,280 feet.

4. The Coast Guard is testing two rescue flares from two competing companies. The Coast Guard plans to

sign a contract with the company whose rescue flare travels the farthest. The Coast Guard fires the two flares into the air over the ocean. The paths of the flares are given by:

Company A: 20.000253 15y x x= − + +

Company B:2 56 15

243 3xy x−= + +

where y is the height and x is the horizontal distance traveled. Determine which flare the Coast Guard

should purchase by substituting y = 0 into each equation and finding x. What does the constant 15 represent in each equation?

Break Even Point – The point where the revenue, R(x), equals

the cost, C(x). Symbolically, R(x) = C(x).

Page 12 AIIF

Practice Exercises For Exercises 1 through 3: a. Write the quadratic equation in general form. b. Identify the values of a, b, and c. c. State the nature of the roots by calculating the discriminant. d. Find all solutions, if any, for x when y=0 for the quadratic equation. Round all answers to the nearest tenth. 1. 211 10 1y x x= − − 2. 23 5 12y x x= − + + 3. 22 8 8y x x= − − − 4. Cox’s formula for measuring velocity of water draining

from a reservoir through a horizontal pipe is 21200 4 5 2HD v v

L= + − , where v represents the velocity

of the water in feet per second, D represents the diameter of the pipe in inches, H represents the height of the reservoir in feet, and L represents the length of pipe in feet. How fast is water flowing through a 30 foot long pipe with diameter of 24 inches that is draining from a pond with a depth of 30 feet? Round your answer to the nearest tenth of a foot per second.

H

L

D


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SJ Page # 5. A ball is thrown upward with an initial velocity of 146 feet per second from a height of 7 feet. How long

does it take the ball to hit the ground? The equation for projectile motion is s(t) = –16t2 +v0t + h0, where s is the height of the projectile in feet, t is the time in seconds, v0 is the initial velocity, and h0 is the initial height. Round your answer to the nearest tenth of a second.

6. For the given quadratic formula, identify the values a, b, and c and write the quadratic equation from

these values.

a. − ± −

=211 (11) 4(5)(6)

2(5)x

b. ± − − − −

=−

212 ( 12) 4( 2)( 19)2( 2)

x

c. For part b. above, will the quadratic equation have any real solutions? Explain.

7. Find the mistake below and correct it.

− =

− − ± − −=

± −=

±=

±≈

+≈ =

−≈ =

2

2

13 7

( 13) ( 13) 4(1)(7)2(1)

13 169 282

13 1412

13 11.92

13 11.9 12.452

13 11.9 0.552

x x

and

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Outcome Sentences I know that the discriminant portion of the quadratic formula is used to I know that the quadratic equation must be in ______________________________form to be When solving real-world applications using the quadratic formula The part of the quadratic formula I don’t understand is ________________________________________ because

Non–Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications Page 15

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Lesson 3: Graphing Quadratic Functions and Their Applications Activity 1 In this activity, you will use your knowledge of the vertical line of symmetry to plot points, draw a graph, and find the equation for the vertical line of symmetry. 1. Using the given dashed vertical line of symmetry, plot and draw the missing half of the graph. Write the

equation for the vertical line of symmetry. State whether the graphs have a minimum or a maximum value and explain why.

a. b. 2. Complete the tables below using the values in the table along with the equation for the vertical line of

symmetry. Plot the points in the table, draw the graph, and draw the vertical line of symmetry. State the equation of the vertical line of symmetry and whether the data tables have a minimum or a maximum value and explain why.

a. Vertical line of symmetry:

b. Vertical line of symmetry:

x y –6 –73 –3 –28 0 –1 3 8

x y

–1/2 –5 1 –1/2

5/2 13 4 71/2

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3. Write the equation for the vertical line of symmetry for the given graphs. State whether the graphs have a minimum or a maximum value and explain why.

a. b. 4. Write the equation for the vertical line of symmetry for the data tables below. State whether the data

tables have a minimum or a maximum value and explain why. Also state the minimum or maximum value.

a. b.

x y –18 10 –14 0 –10 –6 –6 –8 –2 –6 2 0 6 10

x y 0 –7 2 –2 4 1 6 2 8 1

10 –2 12 –7


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Activity 2 In this activity, you will be determining specific characteristics of quadratic functions and real–world problems involving quadratic functions and then graphing the quadratic functions from the characteristics. In the following exercises you will need to: a. Identify the values of a, b, and c b. Vertex coordinates c. All intercepts d. Line of symmetry e. Several points on either side of the vertex 1. = − + −2 4 5y x x 2. 23 8 3y x x= − −

3. 22 5 2y x x= − +

Vertex – x coordinate: −=2

bxa

Vertex – y coordinate: −⎛ ⎞= ⎜ ⎟⎝ ⎠2

by fa

Line of Symmetry: −=2

bxa

Quadratic Formula: 2 4

2b b acx

a− ± −=

Discriminant: 2 4b ac−

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4. Photosynthesis is the process in which plants use the energy from the sun's rays to convert carbon

dioxide to oxygen. The intensity of light is measured in lumens. Let R be the rate that a certain plant uses to convert the sun's light energy. Let x be the intensity of the light. The plant converts the carbon dioxide at a rate according to the equation = − 2240 80R x x . Sketch the graph of this equation and determine the intensity that gives the maximum rate of photosynthesis. State the domain which makes sense for the application.

5. The cost function to make a certain product is = − +2( ) 0.2 10 360C x x x . The revenue function for the

same product is given by = − +2( ) 0.2 50R x x x .

a. Graph the cost and revenue cost functions on the same set of axes. b. What level of production will produce the maximum revenue? What is the maximum revenue? c. What level of production will produce the minimum cost? What is the minimum cost? d. Graph the profit function (profit = revenue minus cost) on a separate set of axes. e. What level of production will produce the maximum profit? What is the maximum profit?

4000

2000

30 60 90 120 150

1600

800

30 60 90 120 150


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6. The graph below represents the profit function for a company that produces widgets. Find the

equation of profit function P(x). Note: You should be able to determine the value of c from the graph. Also, use the coordinates of the vertex to find a and b. Use –b/2a for the x coordinate and solve for b in terms of a and substitute this value into y = ax2 + bx + c to find a and then b.

7. A town is having a parade and celebration for its high school marching band. The school’s marching

band recently marched in Macy’s® Annual Thanksgiving Day Parade. This was the first time the marching band is being honored for its hard work and achievement in the state competition. The town wants to hang a banner on a steel cable between its two tallest buildings -- each 100 feet tall. The distance between the two buildings is 50 feet. The weight of the banner caused the bottom of the banner to be 20 feet lower than the top of the building. Assume the bottom of the banner is parabolic in shape. What is the quadratic function that represents the lower portion of the banner?

–200

200

10

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Activity 3 In this activity, you will use your knowledge and understanding of quadratic functions to do quadratic regression on scatter plots and data sets. In the last activity, you wrote quadratic functions from graphs. In the Linear Functions unit, you drew the best fit line for a scatter plot and determined the equation for the line of best fit. In this activity, you will use the concepts and skills developed in the Linear Functions unit to draw the best fit parabola for given graphs and then determine the equation for the parabola you drew. 1. Which scatter plots below seem to have a quadratic trend? 2. Draw a best fit parabola for the scatter plots you determined had a quadratic trend in Exercise 1. 3. Determine the quadratic functions from the best fit parabolas you drew in Exercise 2.

B

C D

A


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In the Linear Functions unit, you learned to use the graphing calculator to determine the equation for the best fit line from sets of data. We called this linear regression. The graphing calculator can also be used to determine the equation for the best fit parabola from sets of data. We call this quadratic regression. Follow your teacher's instructions on how to use the graphing calculator to determine the quadratic function from sets of data. 4. The table below shows the U. S. population distributed by age (x) and percentage (y).

Under 5 5 to 17 18 to 44 45 to 64 65 and over x 1 2 3 4 5 y 7.4% 18.2% 43.2% 18.6% 12.6%

a. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three

decimal places.

b. Use your graphing calculator to create a scatter plot and graph of the data and sketch the scatter plot and graph on the set of axes.

c. How well does the graph of the best fit parabola fit the data?

5. The students of Mr. G's class were told to record the number of hours spent studying for their mathematics test. For each student, Mr. G wrote an ordered pair (x, y). The x-value represented the number of hours the student spent studying and the y-value represented the student’s test score.

(0.5, 40), (9.3, 75), (8.4, 80), (0.5, 56), (1.0, 60), (8.2, 83), (7.6, 87), (1.0, 47), (1.4, 48), (7.0, 91), (6.5, 94), (1.5,

63), (2.0, 73), (6.2 98), (5.5, 100), (2.3, 78),(2.4, 83), (5.4, 97), (5.4, 98), (2.5, 77), (2.6, 83), (5.2, 95), (5.1, 85), (3.0, 88), (3.0, 86), (4.9, 94), (4.2, 93), (3.5, 91), (3.5, 90), (3.7,89).

a. Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic

equation? Explain.

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b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three

decimal places.

c. Use your graphing calculator to create a scatter plot and graph of the data on the same set of axes.

d. How well does the graph of the best fit parabola fit the data?

6. The table below is the U. S. Census (in millions of people) for the years 1810 through 2000. The x-values represent the year the Census was taken and the y-values represent the population in millions of people. Note: x = 0 for the year 1810, x = 10 for the year 1820, etc.

1810 1820 1830 1840 1850 1860 1870 1880 1890 1990

x 0 10 20 30 40 50 60 70 80 90 y 7.24 9.64 12.87 17.07 23.19 31.44 39.82 50.16 62.95 75.99 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

x 100 110 120 130 140 150 160 170 180 190 y 91.97 105.71 122.78 131.67 151.33 179.32 203.21 226.5 248.71 281.42

a. Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic

equation? Explain. b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three

decimal places. c. Using your equation of best fit, predict the population for the Census in 2010 and 2020.


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Practice Exercises Graph Exercises 1 and 3. Make sure to include the following: a. Identify the values of a, b, and c. b. Vertex coordinates. c. All intercepts. d. Line of symmetry. e. Several points on either side of the vertex. NOTE: Round answers to nearest tenth. 1. = − −25 10 1y x x 2. 23 5 12y x x= − + +

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3. A ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per

second. After 3 seconds it will reach a maximum height of 344 feet. The standard form of a quadratic equation for a projectile is given by = − + +2

0 0( ) 16s t t v t s , where s(t) is the projectiles height at time t, v0 is the initial velocity, and s0 is the initial height. What is the equation of the quadratic function for this problem? What does the y–intercept represent? Graph the quadratic function. Round answers to nearest tenth if necessary.

4. Suppose that in a monopoly market (a market with a downward sloping curve) the total cost per week

of producing a particular product is given by the cost function = + +2( ) 2 100 3600C x x x . The weekly demand for the product is such that the revenue function is = − +2( ) 2 500R x x x . Graph both functions on the same set of axes and shade the region that represents the area in which the company is making a profit. Find the points of intersection for the cost and revenue functions. What do the points of intersection represent?


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5. Determine the quadratic function from the graph at

the right. 6. A ball was dropped from a height of approximately 5 feet and a motion detector was used to measure

the time and height of the ball, relative to the ground, as it was falling. The table below is the height, h, of the ball off the ground in feet after t seconds.

Time t 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 Height

h 4.95 4.86 4.73 4.56 4.34 4.08 3.78 3.43 3.04 2.61 2.13

a. Determine the equation for the parabola of best fit.

b. How long does it take for the ball to hit the ground? Round your answer to the nearest hundredth of a second. HINT: Use the quadratic formula.

7. You run a bicycle rental business for

tourists during the summer in your town. You charge $10 per bike and average 20 rentals a day. An industry journal says that, for every 50–cent increase in rental price, the average business can expect to lose two rentals a day. The graph to the right represents the quadratic equation used to determine how many, if any, 50–cent increases are needed to maximize revenue. Let x represent the number of increases to the current charge rate. Negative values for x represent 50–cent decreases. Use this information and the graph to find the quadratic equation to maximize revenue. What should you charge per bike rental? What is your maximum profit?

100

300

200

Page 26 AIIF

Outcome Sentences The vertex is determined by The line of symmetry is used for Applications of quadratic equations really help me to understand Quadratic functions and applications of quadratic functions are graphed by The easiest way for me to write a quadratic equation from a graph is by The most difficult part of graphing is _________________________________________________________ because Quadratic modeling with the graphing calculator

Non–Linear Functions Lesson 4: Power Functions Page 27

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Lesson 4: Power Functions Activity 1 In this activity, you will write and solve power type equations and their applications. Let's look at the following application problem: The area of a cube–shaped box is 64 cubic inches. What are the dimensions of the box? What are the steps necessary to setup and solve problems of this type? We need to start by labeling the known and unknown (variable) information. Next, we need to write an equation with a single variable from the given information. Then, we need to solve the equation and answer the original question or questions. For Exercises 1 through 3, write and solve a power type equation. 1. The cube of a number is 125. What is the number? 2. Six is added to a number that was raised to the sixth power. If the sum is 735, what was the number

that was raised to the sixth power? 3. The difference of a number and six, raised to the fourth power, is 256. What are the numbers? 4. The volume of a spherical weather balloon is 523.3 cubic meters. What is the diameter of the weather

balloon? NOTE: The formula for the volume of a sphere is π= 343

V r where r is the radius. Use 3.14 for

the value of π. 5. A couple plans to invest $25,000 into an account that is compounded annually for 25 years. They hope

to have $75,135.86 after the 25 years. What interest rate will guarantee that their investment of $25,000 will grow to $75,135.86 after the 25 years? NOTE: S = P(1 + r)t, where S is the value of the investment, P is the amount invested, r is the interest rate (as a decimal), and t is the number of years invested.

Page 28

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Activity 2 In this activity, you will investigate the graphs of power and power–like functions. In your description include whether the graph is an even or odd function. Part A: How do different powers affect the graph of ny x= ?

Function ny x=

Describe or Draw General Shape

Describe location of maximum or minimum

Describe similarity or difference to the graph of 2y x=

3y x=

4y x=

5y x=

6y x=

7y x=

Write your overall conclusion as to how different powers affect the graph of ny x= .


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Part B: How do different coefficients affect the graph of ny ax= ?

Function ny ax=



Describe similarity or difference to the graphs of 2y x= or 3y x=

2y x= −

24y x=

29y x= −

212

y x−=

3y x= −

35y x= −

35y x=

312

y x=

Write your overall conclusion as to how different coefficients affect the graph of ny ax= .

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Part C: How does adding or subtracting a constant, k, to ny x= affect the graph of the equation?

Function ny x k= ±




2 1y x= +

2 3y x= +

2 2y x= −

2 4y x= −

3 5y x= +

3 7y x= +

3 6y x= −

Write your overall conclusion as to how adding or subtracting different constants affect the graph of ny x= .


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Part D: How does adding or subtracting a constant, h, to the x–value before completing the power, in the equation ny x= , affect the graph?

Function ( )ny x h= ±




2( 1)y x= +

( )23y x= +

( )22y x= −

( )34y x= +

( )36y x= +

( )33y x= −

Write your overall conclusion to how adding or subtracting a constant to the x–value in the equation ny x= affected the graph. Write your overall conclusion as to what affect the values of a, h, k, and n have on the graph of ( )ny a x h k= ± ± .

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For Exercises 1 through 4, determine the following:

a. Determine if the graph represents a power or power–like function or not. b. Determine if the graph has a maximum or minimum value. If it does, state the value of the

maximum or minimum. c. If the function, represented by the graph, is a power function then determine if it is even, odd,

or neither. 1. 2. 3. 4.


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5. Match the equation with its graph. a. y = 2x4 b. y = x3 c. y = –x3

A.

C.

B.

Page 34

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For Exercises 6 through 10, state any vertical or horizontal translation from the first equation to the second. Sketch a rough graph of the equations showing translation (do not worry about scale). 6. y = x3 and y = x3 + 4. 7. y = x4 and y = x4 – 3. 8. y = x5 and y = (x – 6)5. 9. y = x5 and y = (x – 6)5 – 2. 10. y = x6 and y = (x + 1)6 + 5.


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Practice Exercises For the Exercises 1 through 3, write and solve a power–like equation. 1. The fifth power of a number is 243. What is the number? 2. Nine is subtracted from a number that is raised to the seventh power. If the difference is 119, what

was the number that was raised to the seventh power? 3. The sum of a number and three, raised to the third power, is 1331. What is the number? 4. The volume of a cubic box is approximately 1521 cubic inches. What are the lengths of the sides of the

cubic box? Round your answer to the nearest tenth of an inch. 5. Darnell and Shanice plan to invest $50,000.00 into an account that is compounded annually at a rate of

3.5%. Create a table of values that represents what their investment is worth after 4, 8, 12, and 16 years. NOTE: S = P(1 + r)t, where S is the value of the investment, P is the amount invested, r is the interest rate (as a decimal), and t is the number of years invested. Round the value of the investment to the nearest cent.

6. Darnell and Shanice plan to use the total value of the investment in 16 years for a college education for

their only child. Approximately how much will they have available each year, for four years, for their child's education? Round your answer to the nearest thousand dollars.

Years Invested (t) Value of Investment in dollars (S)

Page 36

AIIF

For Exercises 7 through 9, complete the following:

a. State if it is a power or power–like function or not. b. State if it has a maximum or minimum value and state the value of the maximum or minimum. c. State if the function is even, odd, or neither. d. State any vertical or horizontal translation from the origin. e. Sketch a rough graph of the power or power–like function.

7. y = –x4

8. y = (x + 3)3 9. y = (x – 2)5 – 4


AIIF

10. Determine the power–like function from the given graph.

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AIIF

Outcome Sentences A power function is The difference between an even and an odd function is Applications of power functions really help me to understand When graphing power functions Vertical and horizontal translations from the origin are The most difficult part of power functions is

Non–Linear Functions Lesson 5: Inverse Variation Page 39

AIIF

Lesson 5: Inverse Variation Activity 1 In this activity, you will create a bar graph that represents the equation y = 1/x, for x values 1 through 10. 1. Cut out the grid template. Obtain a piece of construction paper from your teacher and cut a strip that is 1

centimeter wide by 180 millimeters long. Notice the 1–unit location on the vertical axis of the grid. The 1–unit value represents the length of one of the strip cut to 180 millimeters.

2. Place the 1–unit strip at unit 1 on the x–axis. Cut the remaining strips so that their lengths represent the

fractions 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 the length of the unit strip cut in Exercise 1. Place the cut strips along the horizontal axis at the values for 2 through 10. Tape down each strip.

3. Write the fraction values above each strip on the grid and

then calculate their decimal values to the nearest hundredth. Record the fraction and decimal values in the table to the right.

4. Use the grid below to create a scatter plot of the x-values

from the table and then draw a smooth curve connecting the points on your scatter plot.

x y = 1/x Decimal Value 1 2 3 4 5 6 7 8 9

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Page 40

AIIF 5. What do you notice about the values of y, which represent the lengths of the strips, as the values of x

increase? 6. As the values of x get larger and larger, what value does y seem to approach? 7. Using your graphing calculator, determine the values of y- for each x–value in the table below. Write

the y–values in the right column of the table. 8. From your results from Exercise 7, as x gets closer to 0 what value does y get closer to? 9. Can we find the value of y for x = 0? Explain. 10. Investigate variations of the inverse function by using your graphing calculator.

a. Graph y = 2/x. Describe the differences between this graph and the graph y = 1/x.

b. Graph y = 3/x. Describe the differences between this graph and the graph y = 1/x.

c. Graph y = –1/x. Describe the differences between this graph and the graph y = 1/x.

d. Graph y = –2/x. Describe the differences between this graph and the graph y = 1/x.

x y = 1/x 1/2 1/5

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AIIF

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AIIF

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AIIF

Activity 2 In this activity, you will solve inverse variation problems and real–world inverse variation problems. The general form for an inverse variation equation is y = k/x, where k is called the constant of proportionality. Another way of writing this equation is xy = k. You will investigate inverse variation and k by using the xy = k equation. 1. Find five different sets of values (ordered pairs) that make xy = 24 true.

a. As the values of x increase what do you notice about the values of y? b. Why would the values of y have to decrease as x increases to keep the equation true? c. If the x–value doubles what happens to the y–value? d. If the x–value triples what happens to the y–value? e. What happens to the relationship between x and y if we change the constant to a different

number such as 36? f. Why do you think equations in the form of xy =k, where k is constant, are called inverse variation

equations?

For Exercises 2 and 3, use the given information to solve for the constant of proportionality k and then for the unknown value of y. 2. If y varies inversely with x and y = 34 when x = 1/68, what is the value of y when x = 2? 3. If y varies inversely with the cube of x and y = 10 when x = 4, what is the value of y when x = 2?

Page 46

AIIF For exercises 4 through 7, use the information given in the problem to find the constant of proportionality k and answer the question. 4. The number of hours, h, it takes to mow a lawn varies inversely with the

number of people mowing the lawn at the same time.

a. If it takes 3 hours for 3 people to mow the lawn, how long will it take 5 people to mow the same lawn?

b. Write an inverse variation equation for the problem. 5. Boyle's law states that in a perfect gas where mass and temperature are kept

constant, the volume, V, of the gas will vary inversely with the pressure, P. A volume of gas, 550 centimeters cubed, is under a pressure of 1.78 atmospheres. a. If the pressure is increased to 2.5 atmospheres, what is the volume of the

gas? b. Write an inverse variation equation for Boyle's law.

6. In hydraulics, the fluid pressure, P in pounds per square inch, is related directly with the force, f in

pounds, and inversely with the area, A in square inches. The formula is =fPA

. Assume the force is

kept constant. a. If the fluid pressure is 5 pounds per square inch when the area is 20 square inches, what is the

fluid pressure when the area is 40 square inches? b. Write an inverse variation equation for the fluid pressure.


AIIF

7. The weight of a body varies inversely as the square of its distance from

the center of the Earth.

a. If the radius of the Earth is 4000 miles, how much would a 200-pound man weigh 1000 miles above the surface of the earth?

b. Write an inverse variation equation for the weight of a body.

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AIIF Practice Exercises For Exercises 1 through 3, use the given information to solve for the constant of proportionality k and then for the unknown value. Write an inverse variation equation for each exercise. 1. If s varies inversely with t and s = 30 when t = 30, what is the value of s when t = 10? 2. If y varies inversely with the fourth power of x and y = 2 when x = 3, what is the value of y when x = 0.1? 3. If j varies inversely with the square of l and j = 16 when l = 4, what is the value of j when l = 8? 4. The current, I in amps, produced by a battery varies inversely to the resistance, R in ohms, of the circuit

to which the battery is connected.

a. If the current is 0.25 amps when the resistance is 10,000 ohms, what will the current be if the resistance is reduced to 2500 ohms?

b. Write an inverse variation equation for the current of the battery.

5. The intensity, I, of light observed from a source of constant luminosity varies inversely as the square of the distance, d, from the object.

a. If the intensity of a light is 0.1499 lumens when the light source is 1.1 meters away, what is the

intensity of the light if the source is 3 meters away? Round all answers to four decimal places. b. Write an inverse variation equation for the intensity of light, I, a distance d from the source.


AIIF

6. Lengths of radio waves vary inversely with radio wave's frequency.

a. Radio station WJHU broadcasts their FM signal with a frequency of 88.1 MHz and has a wavelength of approximately 3.4 meters. Boston's famous WRKO AM radio station broadcasts their signals with a frequency of 0.680 MHz. What is the wavelength of WRKO's broadcasts? NOTE: Round your k value to the nearest whole number and the wavelength to the nearest tenth of a meter.

b. Write an inverse variation equation for the wavelength of radio waves.

Page 50

AIIF Outcome Sentences Inverse variation is I know when a problem is about inverse variation because For inverse variation, y ________________________________________________as x The opposite of inverse variation is I still need help with

Non-Linear Functions Lesson 6: Exponential Functions Page 51

AIIF

Lesson 6: Exponential Functions Activity 1 Bacterial Growth Respiratory System Model Respiratory sicknesses (infections), such as bronchitis and pneumonia, are caused by bacteria. Once bacteria gets in our lungs, they can duplicate (reproduce) at a certain rate. The following experiment will model the amount of bacteria present over time. In this experimental model, we will use small construction paper squares of one color to represent the bacteria. Experiment Step 1: Cut out 64 red construction paper squares. Each square

should be the same size and shape. The best size is 1 inch by 1 inch or 1 centimeter by 1 centimeter. Use a ruler to draw the squares before cutting.

Experiment Step 2: Cut out the lungs template at the end of the activity. Experiment Step 3: Place one red square on the lung template (any where inside the lung area.) This

represents the initial amount of bacteria, a single cell. Note: Bacteria are actually very small in size. A single cell of bacteria is about 1/10,000th of a centimeter.

Experiment Step 4: Every minute, add enough red squares to double the amount you had previously. This represents the bacteria duplicating (reproducing itself) every hour. While you are waiting for each minute to end, count out the necessary squares that you will be adding for the next minute. Also, record the time and amount of bacteria present in the lungs in the table provided below.

Experiment Step 5: Repeat Step 4 until all 64 squares have been placed "in" your lungs. Experiment Step 6: You should realize that your table matches the table from the Setting the Stage

transparency. Table 1: Bacterial Growth Experiment

Hour Bacteria Count 0 1 1 2 3 4 5 6

Page 52

AIIF 1. Create a scatter plot of the hours

compared to the number of bacteria in the lungs. What type of pattern occurred in the scatter plot graph?

2. What is the rate that the bacteria are

growing?

3. Graph a scatter plot of your data on a

graphing calculator. Set the window range to an x–minimum of –2, x–maximum of 7, x–scale of 1, y–minimum of –2, y–maximum of 100, and y–scale of 10. Is the scatter plot linear? If not describe the shape of the graph.

4. How many bacteria do you expect to be in the lungs after a 24 hour period? How might you calculate

this value? 5. Approximately how many hours will it take until there are 1 trillion (1,000,000,000,000 or 1 x 1012)

bacteria in the lungs? NOTE: The graphing calculator may display 1 trillion as 1.0 E12.

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AIIF

Antibiotic Decay in the Blood Stream Experimental Model To help cure illnesses antibiotics and/or medicines taken into the body are circulated throughout the body by the bloodstream. The kidneys take the drugs out of the blood. We saw, from the first part of the activity, how bacteria can duplicate and create enormous amounts of themselves in a relative short period of time. Bacteria left unchecked can cause major health problems. Sometimes the only way to become healthy again is by the use of antibiotics. The following experiment will model the amount of antibiotics left in the bloodstream over time. In this experimental model, we will use small construction paper squares of one color to represent the blood and small construction paper squares of another color to represent the antibiotics. Experiment Step 1: Cut out 40 red construction paper squares and 20 blue construction paper squares.

Each square should be the same size and shape. The best size is 1 inch by 1 inch. Use a ruler to draw the squares before cutting.

Experiment Step 2: Place 20 red squares and 20 blue squares in a container (bag or box). This represents a bloodstream that is half blood and half antibiotics. Although in real life the blood stream would not consist of 50% antibiotics, this will produce a model quickly that represents the way drugs leave the bloodstream.

Experiment Step 3: Shake the container and randomly remove 10 squares. Replace them with 10 red squares. Determine how many blood squares and antibiotic squares are now in the container. Place this information in Table 1 below. This step models the kidneys randomly cleaning one quarter of the blood each hour.

Experiment Step 4: Repeat Step 3 ten times. Place the information for each cleaning cycle in Table 2, Antibiotics Decay Experiment, below.

Table 2: Antibiotics Decay Experiment

Hour Blood Count Antibiotic Count 0 20 20 1 2 3 4 5 6 7 8 9

10

Page 54

AIIF 6. Create a scatter plot of the hours compared

to the number of antibiotics left in the bloodstream. What type of pattern occurred in the scatter plot graph?

7. Create a transparency copy of your graph.

Place all the transparencies from each group on the overhead at one time and line up the axes. What do you notice about the graph?

8. If no new antibiotics are added, what would the graph do if we continued with the experiment?

9. Graph a scatter plot of your data on a graphing calculator. Set the window range to an x–minimum of

–2, a x–maximum of 12, a y–minimum of –2, and a y–maximum of 24. Is the scatter plot linear? If not describe the shape of the graph.

10. Graph y = 20(0.75)x on the same graph as the scatter plot. Describe how the graph of y = 20(0.75)x fits the data from the scatter plot.

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AIIF

Activity 2 In this activity, you will determine the y–intercept, determine the type of graph, and draw a rough sketch of exponential functions. For Exercises 1 through 4:

a. Determine the coordinates of the y–intercept b. Type of graph: growth or decay c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale

appropriately. 1. 12xy =

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AIIF For Exercises 5 through 7, state the y–intercept and the type of graph. 5.

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which collects interest four times a year, when you were born for your college education. The future value of your college education fund can be determined by the function 410000(1.0375) tS = , where t represents the number of years for the investment. How much money will you have available when you start college? Assume you will be 18 years old when you start college. Draw a rough sketch of the investment; set axis scales accordingly.

2. Viruses can produce many more offspring than bacteria per infection. Some viruses produce at an

exponential rate related to the function (100)thv C= , where v represents the number of viruses, C

represents initial population of viruses, t represents amount of time in hours, and h is the number of hours to produce a new generation. How many viruses will be present after 24 hours if there initially were 5 viruses and the viruses produce a new generation every 4 hours?

3. It has been determined that a certain city has been growing exponentially over the last 20 years

according to the function 0(1 )tP P r= + , where P represents the town's population, P0 is the initial population, r is the rate at which the town's population is increasing, and t is the amount of time in years that the town has been increasing. If the town initially had 450 people 20 years ago and they now have 1,443 people, what was the rate of increase in population over the last 20 years? Round your answer to the nearest whole percent.

4. A local retail store has determined that its sales could grow exponentially based on the amount they

spend on advertising each week by the function (1.15)ws C= , where s represents the number of sales per week, C represents their initial sales before advertising began, w represents the number of consecutive weeks they advertised. If the store averaged 125 sales per week before advertising began, how many sales can they expect to have, each week, after advertising for 4 consecutive weeks? Round your answer down to the nearest whole sale.

Page 60

AIIF

5. The radio active decay of a material is given by the function ( )– 0.693 /T0 tA A e= , where A0 is the initial

amount of the material, t is the amount of time in years, and T is the half–life of the radio active material. Plutonium 240 has a half life of 6540 years. If a nuclear power plant started with 100 pounds of Plutonium 240, how much would be left after 20 years? How many ounces of plutonium decayed during the 20 years? Round your answers to the nearest hundredth pound and ounce.


AIIF

Activity 4 In this activity, you will use exponential regression to obtain an exponential function from real–world data. 1. The following data table represents the daily costs of commuting (driving to work) versus the amount

of commuters (people who drive to work) for a large metropolitan area. Cost (in $) 10 15 20 25 30 35 40 45 50 Commuters 225,000 145,000 110,000 68,000 35,000 13,000 8,000 5,600 2,500

a. What type of graph does the data model?

b. What is the exponential regression function? Round values to three decimal places.

c. How many commuters would you expect if they had to pay $75 each day in commuting expenses? Round your answer to the nearest commuter.

2. The following data table represents the population of the United States from the years 1790 through 2000, where year 0 = 1790, 1 = 1820, etc.

Year 0 (1790) 1 (1820) 2 (1850) 3 (1880) 4 (1910) 5 (1940) 6 (1970) 7 (2000)

Population (in

millions)

3.93 9.64 23.19 50.16 91.97 131.67 204.05 281.42


b. What is the exponential regression function? Round values to four decimal places.

c. Using this exponential equation, what might you predict will be the size of the U. S. population in the year 2060? Round your answer to the nearest ten thousandths. Note: Remember our current units for population is in millions.

Page 62

AIIF 3. The following table represents the early production of crude petroleum in the United States.

Year 0 (1859) 10 (1869) 20 (1879) 30 (1889) 40 (1899) Oil

Production (in barrels)

2,000 4,215,000 19,914,146 35,163,513 57,084,428


b. What is the exponential regression function? Round values to three decimal places.

c. The U. S. oil production peaked in 1970. What could you predict was our country's peak output of oil in 1970? Round your answer to the nearest whole barrel.

d. The actual U. S. oil production in 1970 was approximately 3,500,000,000 barrels. What can you

say about your predicted value of production compared to the actual value of production? e. What suggestion would you make on limiting the use of your exponential regression function?


AIIF

Practice Exercises For Exercises 1 and 3:

a. Determine the coordinates of the y–intercept. b. Type of graph: growth or decay. c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale

appropriately. 1. ( )= 5 3xy .

2. ⎛ ⎞= ⎜ ⎟⎝ ⎠

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3. ( )= 23 4 xy .

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AIIF 4. On January 15th, 2009, the world's population was 6.75 billion people. It is predicted that it will take just

44 years for the world's population to double. What is the rate, per year, at which the world population is increasing? Round your answer to the nearest tenth of a percent. Note: Use 0(1 )tP P r= + . Refer back to Activity 3, Exercise 3.

5. A biologist is conducting an experiment testing a new antibiotic on a certain strain of bacteria cells.

According to the biologist's calculation, the cells are dying (decaying) at a rate given by the function −= 0.223tL ae , where L represents the amount of cells left after time t (in minutes) and a represents the initial amount of bacteria cells present before the antibiotic is applied. How many bacteria cells are present ten minutes after the antibiotic was applied if there initially 10 million bacteria cells? Round your answer to the nearest whole cell. Use 2.178 for the value of e.

6. A person invests $15,000 into an interest bearing account. After 10 years the person's investment is now worth approximately $25,966. Determine the annual interest rate if the future value of an investment can be determined with the function S = P(1+ r/12)12t, where S is the value of the investment after t years, P is the amount invested, and r is the annual interest rate. Round your answer to the nearest tenth of a percent.

7. The intensity of earthquakes is measured by using the Richter scale. We can determine how much more

powerful one earthquake is compared to another earthquake, by the ratios of their intensities. The ratios of the intensities of two earthquakes can be determined by the function I = 10d, where I is the ratio of intensities and d is the absolute value of the difference of the intensities of the earthquakes as measured by the Richter scale. It is estimated that the 2004 Indian Ocean earthquake measured 9.2 on the Richter scale. In comparison, the earthquake that caused Mt. St. Helen's volcano to erupt on May 18th 1980, measured 5.1 on the Richter scale.

a. How much more powerful was the 2004 Indian Ocean earthquake compared to the 1980 Mt. St.

Helen's earthquake? Round your answer to the nearest whole number. b. What can you conclude about the difference in the intensities of two earthquakes?


AIIF

8. For the following graph:

a. State the y–intercept. b. State the type of graph.

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AIIF Outcome Sentences Exponential growth is I know an exponential problem when Exponential decay is When graphing exponential functions The part about exponential functions I don't understand is

Non–Linear Functions Lesson 7: Step Functions Page 67

AIIF

Lesson 7: Step Functions Activity 1 In this activity, you will investigate the graphs of step functions using your graphing calculator and the int( ) function which represents the greatest integer function. Mathematically, the greatest integer function is called the floor function. The int(x) is the greatest integer less than or equal to x. If x = 3.14 then, in function notation, f (3.14) = int(3.14) = 3. Likewise, mathematically f (3.14) = floor (3.14) = 3, or y = floor (3.14) = 3. 1. Evaluate the floor function using your graphing calculator. Write the function down as it was entered

in the calculator.

a. y = floor (23.001)

b. y = 3•floor (15.06) c. y = –6•floor (–34.005)

d. y = 5•floor (–5.045) + 3

e. ( )2( 13.45)y floor= −

2. Your younger sister wants to earn some money. She asks you if you have any chores she can do. Write a step (floor) function for each scenario below.

a. You pay your sister $1.00 for each half hour of work. b. You pay your sister $1.00 for each fifteen minutes of work. c. Your sister wants $2.50 for each hour of work. d. Your sister wants $4.00 for each hour of work. e. Your sister wants $2.50 for each half hour of work.

Page 68 AIIF 3. Create a table of values, which include intervals for x and values for y, and write a step function for

each of the graphs below. a.

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AIIF

Activity 2 In this activity, you will investigate evaluating another form of step function, the ceiling function, using your graphing calculator and the int( ) function. Mathematically, the ceiling of x is called the least integer function. It represents the smallest integer not less than x. Remember from Activity 1, the int(x) function, mathematically the floor function, is the greatest integer less than or equal to x. If x = 3.14 then the int(3.14) = 3. However, the ceiling of 3.14, or ⎡ ⎤⎢ ⎥3.14 , equals 4. 1. Evaluate the ceiling function using your graphing calculator. Write the function down as it was entered

in the calculator.

a. ⎡ ⎤⎢ ⎥35.001

b. –4• −⎡ ⎤⎢ ⎥6.43 c. 8• ⎡ ⎤⎢ ⎥0.045

d. 3.5• −⎡ ⎤⎢ ⎥7.89 – 2 e. ⎡ ⎤⎢ ⎥

34.28 2. Check the results from the ceiling functions below. If any of the results are incorrect, give the correct

result and state what may have caused the incorrect results.

a. =⎡ ⎤⎢ ⎥23.15 23

b. − = −⎡ ⎤⎢ ⎥7.45 8 c. − =⎡ ⎤⎢ ⎥4 34.678 140 d. + =⎡ ⎤⎢ ⎥5.65 2 4

Page 70 AIIF In this part of the activity, you will investigate real–world examples of step functions. In Activity 1, if the “run” was expanded by dividing x by a certain amount inside the step function and contracted (shortened) by multiplying x by a certain amount inside the step function. Also, the “run” was increased by multiplying by a certain amount outside the step function and decreased by dividing by a certain amount outside the step function. For example, Exercise 2 in Activity 1, if your sister was paid every half hour, you had to multiply x inside the floor function by 2 to decrease the run, y = floor (2x). Also, when your sister wanted $2.50 for each hour of work, you had to multiply the floor function by 2.5, y = 2.5 floor (x). 3. You have recently graduated from college and have taken a job with a company. Your starting salary is

$30,000.00 per year. The company pays its employees once a month.

a. Write a step function equation based on the information in the exercise. b. Create a table of values for an appropriate x interval.

x (months) y value ($)

c. Graph your step function equation.

Note: To expand a “run” by a certain amount, we divide x by that amount inside our step function symbol. To

increase the “rise” by a certain amount, we multiply by that amount outside of

the step function symbol.

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

5,000

10,000

15,000

20,000

25,000

Months

x

y


AIIF

4. From Exercise 1, the company has decided to pay its employees weekly. Note: Graph a portion of your

function.

a. Write a step function equation based on the information in the exercise. Note: There are 52 weeks in a year.

b. Create a table of values for an appropriate x interval.

c. Graph your step function equation.

d. Do you think there are any weeks where the pay could be different? Explain you answer.

5. After graduating from college with a degree in meteorology, you have taken up a position with NOAA, the National Oceanic and Atmospheric Administration. Your first assignment is to introduce a new tornado scale to replace the current Fujita Scale table shown below. Due to temperature changes over the past several decades, NOAA has decided to make a more consistent range of wind values for tornados. A gale force tornado will now start at 50 miles per hour (mph) and the new scale will have increments of 50 mph. The scale will still go from F0 through F6. Note: This problem deals with a hypothetical situation.

The Fujita Scale

Wind Speed (MPH)

F–Scale Number Tornado Classification

40–72 F0 Gale tornado 73–112 F1 Moderate tornado

113–157 F2 Significant tornado 158–206 F3 Severe tornado 207–260 F4 Devastating tornado 261–318 F5 Incredible tornado 319–379 F6 Inconceivable tornado

x (weeks) y value ($)

10 20

5,000

10,000

15,000

x

y

Weeks

Page 72 AIIF

1

Voltsx

y

9 18

Bina

ry V

alue

a. Write a step function equation based on the information in the exercise. b. Create a table of values for an appropriate x interval.

c. Graph your step function equation. d. What is the name you have given to your new tornado scale? Write the name of the new scale in

the table above.

6. Computers store data using the binary number system, which has only two values, 0 and 1.

Computers use voltages to record data as a 0 or a 1. Low voltages represent a 0 and high voltages represent a 1. Some computers use a RISC (Reduced Instruction Set Computer) microcontroller which operates in the range of voltages 0 to 18 volts. Assume that half the voltages represent a 0 (low voltages) and the other half of the voltages (high voltages) represent a 1.

a. What is the interval of voltage values that would

represent a binary value of 0 for a RISC based computer?

b. What is the interval of voltage values that would represent a binary value of 1 for a RISC based computer?

c. Write and graph a step function representing the two binary values of 0 and 1 for the range of voltages.

x value (wind speed in mph)

y (scale)

Wind Speed in MPH

4

x64 128 192

y

256 320 384

Scal

e 1

2

5

6

3


AIIF

Practice Exercises For Exercises 1 and 4, evaluate the step function and state the type of step function, floor or ceiling. 1. y = ⎢ ⎥⎣ ⎦5.8 /2 2. y = −⎡ ⎤⎢ ⎥

32.567 3. y = +⎢ ⎥⎣ ⎦5 14.689 4 4. y = ⎡ ⎤

⎢ ⎥37.89

For Exercises 5 through 7, create a table of appropriate x- and y–values based on the “run” and “rise” for the given step functions and graph the step function. Pick appropriate scales for your axes. 5. = ⎢ ⎥⎣ ⎦/5y x

x y

6. = ⎢ ⎥⎣ ⎦5y x

x y

Page 74 AIIF 7. = ⎢ ⎥⎣ ⎦6 /3y x

x y

For Exercises 8 and 9, create a table of appropriate x- and y-values for the given graph, write a step function, and state the type of step function, ceiling or floor. 8.

x y


AIIF

# 9.

For Exercises 10 and 11:

a. Write a step function equation based on the information in the exercise. b. Create a table of values for an appropriate x interval. c. Graph your step function equation.

10. You have a part–time job after school making $12 per hour. Your boss gives you partial pay for every 6 minutes that you work.

x y

x y ($)

Page 76 AIIF 11. A company pays its employees a salary based on the number of years of employment with the company.

New employees start with a salary of $25,000 a year. The company increases the employee’s salary by $4,000 for each completed year of employment.

x (Years) y (Salary in dollars)


AIIF

Outcome Sentences A step function is A floor step function is A ceiling step function is To increase the run of a step function To increase the rise of a step function I still need help with

Page 78

AIIF

Lesson 8: Miscellaneous Non-Linear Functions Activity 1 In the modeled exercises, you attempted to determine how to translate, vertically and horizontally, the absolute function y = |x|. In these exercises, you will continue your investigation of translations of y = |x| using the graphing calculator. For Exercises 1 through 8, you will investigate the transformations of the graphs of the absolute value function from the origin by adding, subtracting, multiplying, and dividing inside and outside the absolute value brackets. Using your graphing calculator, write an absolute value function for each absolute value situation, create a table of ordered pairs, and draw the graph on the provided grid. State the type and value of the transformation on the graph; vertical translation or horizontal translation compared to the graph of y = |x|. Also state if the graph has been dilated (contracted or expanded) compared to y = |x|. The first exercise has been completed for you. 1. Add 5 inside the absolute value brackets. The function is y = |x + 5|; the graph has been

horizontally translated 5 units to the left. Grid scale is one horizontal and one vertical unit.

2. Add 5 outside the absolute value brackets.

x y –9 4 –7 2 –5 0 –3 2 –1 4 1 6 3 8

x y –6 –4 –2 0 2 4 6

y

x

y

x

Non–Linear Functions Lesson 8: Miscellaneous Non-Linear Functions Page 79

AIIF

3. Subtract 3 inside the absolute value brackets.

4. Subtract 3 outside the absolute value brackets.

5. Multiply by 2 inside the absolute value brackets.

x y

x y

x y

y

x

y

x

y

x

Page 80

AIIF 6. Multiply by 2 outside the absolute value brackets.

7. Multiply by –2 inside the absolute value brackets.

8. Multiply by –2 outside the absolute value brackets.

x y

x y

x Y

y

xy

x

y

x


AIIF

9. Was there a difference between Exercises 5 and 6? Explain your answer. 10. Was there a difference between Exercises 7 and 8? Explain your answer. 11. What would you expect the results to be if we divided inside the absolute value brackets and outside

the absolute value brackets by a positive constant? Explain your answer. 12. What would you expect the results to be if we divided inside the absolute value brackets and outside

the absolute value brackets by a negative constant? Explain your answer. 13. When dividing an absolute value function, inside or outside of the absolute value brackets, by a

positive constant, what type of transformation would you expect on the graph: horizontal translation, vertical translation, dilation (contraction or expansion)? Explain your answer.

For Exercises 14 and 15, write a function for the given situation and draw its graph on the grid provided. 14. The function y = |x| has been horizontally translated left by 2 units and vertically translated up by 2

units from the origin.

Page 82

AIIF 15. The function y = |x| has been expanded by a factor of 2 and vertically translated down by 4 units from

the origin. For Exercises 16 and 17, write an absolute value function from the given graph 16. 17.


AIIF

Activity 2 In this activity, you will investigate graphing the equations of a circle. For Exercises 1 through 6 (draw two circles per grid):

a. State the radius and center of the circle. b. Draw a rough sketch of the circle equation. Use only

four points to draw the rough sketch of the circle. State the coordinates of the four points.

1. 2 2 81x y+ =

2. 2 2 121x y+ =

3. ( )2 26 36x y− + =

4. ( )22 5 25x y+ + =

General Equation of a Circle with Center at (0, 0) and Radius r 2 2 2x y r+ = General Equation of a Circle with Center at (h, k) and Radius r ( ) ( )22 2x h y k r− + − =

Page 84

AIIF

5. ( ) ( )224 3 49x y+ + − =

6. ( ) ( )226 3 64x y− + + =

7. Explain the technique you used to find the coordinates of the four points for circles that had a center at

any location other than (0, 0). For Exercise 8, pick any three equations from Exercises 1 through 6 and solve them for y. 8. 9. Using the general equation of a circle, ( ) ( )22 2x h y k r− + − = , with center at (h, k) and radius r, solve the

equation for y to obtain a function equation for a circle.


AIIF

For exercises 10 through 13, write the equation, in general form, for the graphed circle. State the center of the circle and the four points used to define its graph. 10. 11. 12. 13.

Page 86

AIIF Activity 3 In this activity, you will investigate piece–wise functions and their graphs. You will also use the graphing calculator to graph piece–wise equations. Follow your teacher's directions for graphing piece–wise equations on the classroom graphing calculator. For exercises 1 through 4, graph the piece–wise equation on the grid provided. Also, graph the piece–wise equation on your graphing calculator and write the format of the equation as it was entered into the graphing calculator.

1. 5 3 for 0

2 7 for 0x x

yx x+ <⎧ ⎫

= ⎨ ⎬− + ≥⎩ ⎭

2. 22 for 0

3 6 for 0x xyx x

⎧ ⎫<⎪ ⎪= ⎨ ⎬− ≥⎪ ⎪⎩ ⎭

3. 2 for 28 for 2

x xy

xx

⎧ ⎫<⎪ ⎪= ⎨ ⎬≥⎪ ⎪⎩ ⎭


AIIF

4. 34 for 2

for 2 24 for 2

x

y x xx

< −⎧ ⎫⎪ ⎪

= − ≤ <⎨ ⎬⎪ ⎪− ≥⎩ ⎭

For Exercises 5 and 6, write a piece–wise equation for the given graph. 5. 6.

Page 88

AIIF For Exercises 7 through 9, write a piece–wise equation from the given information and then graph your piece–wise equation. Set axes scales accordingly. 7. The a local electric company charges $0.0968

per kilowatt hour (KWH) for the first 200 KWH used and then $0.0762 per kilowatt hour used beyond the initial 200 KWH. What does the value of y represent?

8. A cell phone company charges a $39.99

monthly fee that includes 500 anytime cell minutes. If you use more than 500 cell minutes, the cell phone company charges $0.40 for each additional minute. What does the value of y represent?

9. The Reel Time movie theater charges $4.50 for

children younger than 12 and for adults 65 and older. Everybody else must pay the full price of $10.00.

200

20

200

30

12


AIIF

Activity 4: Courting the Graphing Calculator In this activity, you will use your knowledge and understanding of equations for horizontal lines, half circles, and piece–wise functions to draw half a basketball court on your graphing calculator. Follow the instructions given by your teacher to draw vertical lines as needed on your calculator screen. The information and diagram below show requirements of a basketball court. The dimensions of an NBA basketball court are:

• Length of court: 94 feet • Width 50 feet • Diameter of rim: 18 inches • Distance from backboard to free throw line: 19 feet • Distance from backboard (boundary line) to rim: 6 inches • Width of the key: 12 feet • Three-point line/arc: From the center of the rim (basketball hoop) to the three-point line is 22.5 feet.

From the center of the rim to the arc is 23.75 feet.

three point line

Page 90

AIIF Practice Exercises For each exercises 1 and 4, create a table of ordered pairs based on the given absolute value function, graph the absolute value function, and state any horizontal or vertical translation from the origin and whether the graph has been expanded or contracted. Label the axis and units on the graph 1. 6y x= +

x y

2. 4 6y x= − −

x y

3. /3y x=

x y

y

x

y

x

y

x


AIIF

4. 2 3 1y x= + +

x y

For Exercises 5 and 6, create a table of ordered pairs for the given graphs, write an absolute value equation, and state any horizontal or vertical translation from the origin and whether the graph has been expanded or contracted. 5. 6.

x y

x y

y

x

y

x–5

–5

5

5

y

x

5

5 –5

–5

Page 92

AIIF For exercises 7 through 10 (draw two circles per grid):

a. State the radius and center of the circle. b. Draw a rough sketch of the circle represented by the equation. Use only four points to draw the rough

sketch of the circle. State the coordinates of the four points. c. Solve the equation for y.

7. 2 2 4x y+ = . 8. ( )2 24 64x y+ + = . 9. ( ) ( )226 4 25x y− + + = . 10. ( ) ( )221 2 81x y+ + + = .

y

x

y

x


AIIF

For each of the Exercises 11 through 14, graph the given piece–wise function on the provided grid. Also, graph the piece–wise equation on your graphing calculator and write the format of the equation as it was entered into the graphing calculator.

11. 1 for 0

3 1 for 0x

yx x

− <⎧ ⎫= ⎨ ⎬− ≥⎩ ⎭

12. | 4| for 5

/5 for 5x x

yx x

+ < −⎧ ⎫= ⎨ ⎬≥ −⎩ ⎭

13. 2( 2) for 2

4(2 ) for 2xx x

yx

⎧ ⎫− + < −⎪ ⎪= ⎨ ⎬≥ −⎪ ⎪⎩ ⎭

14. 2

3

2

( 2) 2 2 2

2<

x x

y x x

x x

− − >⎧ ⎫⎪ ⎪⎪ ⎪= − − − ≤ ≤⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

Page 94

AIIF 15. A cell phone company offers broadband wireless internet access at a cost of $50 per month for the first

1 GB (giga byte) of usage. After the first 1 GB of usage, the company charges $0.05 per 1MB (mega byte). Write a piece–wise equation representing the total monthly cost for broadband wireless internet. Note: 1 GB = 1,000 MB. Label the independent variable and state what it represents.

50

100

150

200

400 800 1200 1600 x (MB)


AIIF

Outcome Sentences Graphing absolute value functions was similar to When graphing a circle equation on the graphing calculator When solving a circle equation for y Piece–wise graphing was hard to understand because When graphing a piece–wise equation I would like to know more about

Page 96

AIIF References & Resources The authors and contributors of Algebra II Foundations gratefully acknowledges the following resources: Donovan, Suzanne M.; Bransford, John D. How Students Learn Mathematics in the Classroom. Washington, DC: The

National Academies Press. 2005. Driscoll, Mark. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999. Eves, Howard. An Introduction to the History of Mathematics (5th Edition) Philadelphia, PA: Saunders College Publishing,

1983. Harmin, Merrill. Inspiring Active Learning: A Handbook for Teachers. Alexandria, VA: Association for Supervision and

Curriculum Development, 1994. Harshbarger , Ronald J. and Reynolds, James J., Mathematical Applications for the Management, Life, and Social Sciences Eighth

Edition, Houghton Mifflin Boston, MA 2007. Hoffman, Mark S, ed. The World Almanac and Book of Facts 1992. New York, NY: World Almanac. 1992. Kagan, Spencer. Cooperative Learning. San Clemente, CA: Resources for Teachers. 1994. Karush, William. Webster’s New World Dictionary of Mathematics. New York: Simon & Schuster. 1989. McIntosh, Alistair, Barbara Reys, and Robert Reys. Number Sense: Simple Effective Number Sense Experiences. Parsippany,

New Jersey: Dale Seymour Publications. 1997. McTighe, Jay; Wiggins, Grant. Understand by Design. Alexandria, VA: Association for Supervision and Curriculum

Development. 2004. Marzano, Robert J. Building Background Knowledge for Academic Achievement. Alexandria, VA: Association for

Supervision and Curriculum Development. 2004. Marzano, Robert J.; Pickering, Debra J.; Jane E. Pollock. Classroom Instruction that Works. Alexandria, VA: Association

for Supervision and Curriculum Development. 2001. National Research Council. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

2001. Ogle, D.M. (1986, February). “K-W-L: A Teaching Model That Develops Active Reading of Expository Text.” The Reading

Teacher, 39(6), 564–570. The National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: The

National Council of Teachers of Mathematics. 2000. Van de Walle, Jon A. Elementary and Middle School Mathematics: Teaching Developmentally (4th Edition). New York: Addison

Wesley Longman, Inc. 2001. The authors and contributors Algebra II Foundations gratefully acknowledges the following internet resources: http://www.metalprices.com http://nationalzoo.si.edu/Animals/AsianElephants/factasianelephant.cfm http://hypertextbook.com/facts/1998/JuanCancel.shtml http://www.conservationinstitute.org/ocean_change/predation/salmonsharks.htm http://blogs.payscale.com/ask_dr_salary/2007/03/starting_salari.html http://www.mpaa.org/FlmRat_Ratings.asp (December 2008) http://federaljobs.net/fbijobs.htm (December 2008) http://www-pao.ksc.nasa.gov/kscpao/release/2000/103-00.htm (December 2008) www.seaworld.org http://www.dailyherald.com/story/?id=92571 http://www.infoplease.com/ipa/A0004598.html http://www.washingtonpost.com/wp-dyn/content/article/2006/10/05/AR2006100501782.html http://www.dxing.com/frequenc.htm http://www.wjhuradio.com/ http://www.wrko.com/


AIIF

http://www.census.gov http://www.ehs.washington.edu/rso/calculator/chelpdk.shtm http://en.wikipedia.org/wiki/Petroleum http://en.wikipedia.org/wiki/2004_Indian_Ocean_earthquake http://www.fs.fed.us/gpnf/mshnvm/ http://www.popularmechanics.com/home_journal/workshop/4224738.html http://www.fitness.gov/exerciseweight.htm http://www.economagic.com/em-cgi/data.exe/cenc25/c25q07 http://www.ndbc.noaa.gov/hurricanes/1999/floyd.shtml

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