Graphs and Functions. Functions

Transcription

1 Graphs and Functions 7A Functions 7-1 The Coordinate Plane 7- Functions 7-3 Graphing Linear Functions 7- Graphing Quadratic Functions 7-5 Cubic Functions 7B Graphs 7-6 Rate of Change and Slope 7-7 Finding Slope of a Line 7-8 Interpreting Graphs 7-9 Direct Variation KEYWORD: MT8CA Ch7 The slope of a line can be used to model the steepness of a ski slope. Heavenl Valle Ski Resort, Lake Tahoe 318 Chapter 7

2 Vocabular Choose the best term from the list to complete each sentence. 1. A(n)? states that two epressions have the same value.. An number that can be written as a fraction is a(n)?. 3. A(n)? serves as a placeholder for a number.. A(n)? can be a whole number or its opposite. algebraic epression equation integer rational number variable Complete these eercises to review skills ou will need for this chapter. Simplif Integer Operations Evaluate Epressions Evaluate each epression for the given value of the variable for for 11. 3( 1) for 1. 3( ) for 1 Solve Multiplication Equations Solve p a s Simplif Ratios Write each ratio in simplest form Solve Two-Step Equations Solve.. 3p 8 5. (a 3) 6. 9 k s Graphs and Functions 319

3 The information below unpacks the standards. The Academic Vocabular is highlighted and defined to help ou understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard AF1.5 Represent quantitative relationships graphicall and interpret the meaning of a specific part of a graph in the situation represented b the graph. (Lesson 7-8) AF3.1 Graph functions of the form n and n 3 and use in solving problems. (Lessons 7-, 7-5) AF3.3 Graph linear functions, noting that the vertical change (change in -value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the slope of a graph. (Lessons 7-3, 7-6, 7-7) AF3. Plot the values of quantities whose ratios are alwas the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. (Lesson 7-7) AF. Solve multi-step problems involving rate, average speed, distance, and time or a direct variation. (Lesson 7-9) Academic Vocabular quantitative can be measured specific eact solving finding the answer to a question vertical straight up or down horizontal parallel to the horizon value(s) amount equals has the same value multi-step more than one part variation difference Chapter Concept You learn how to sketch a graph of a real-world event and describe the situation the graph represents. You graph functions that contain a variable raised to the second or third power. You graph functions that contain a variable whose power is one. You use data to make a graph and learn how to interpret the slope of the line drawn through these points. You plot data points and connect them with a line. Then ou find the slope of the line. You make a graph and compare ratios to determine whether a data set shows direct variation. 30 Chapter 7

4 Writing Strateg: Keep a Math Journal B keeping a math journal, ou can improve our writing and thinking skills. Use our journal to summarize ke ideas and vocabular from each lesson and to analze an questions ou ma have about a concept or our homework. Journal Entr: Read the entr a student made in her journal. Januar 7 I m having trouble with Lesson 6-5. I can find what percent one number is of another number, but I get confused about finding percent increase and decrease. M teacher helped me think it through: Find the percent increase or decrease from 0 to 5. First figure out if it is a percent increase or decrease. It goes from a smaller to a larger number, so it is a percent increase because the number is getting larger, or increasing. Then find the amount of increase, or the difference, between the two numbers Now find what percent the amount of increase, or difference, is of the original number. ount of increase riginal number So it is a 5% increase. am o % Tr This Begin a math journal. Write in it each da this week, using these ideas as starters. Be sure to date and number each page. In this lesson, I alread know... In this lesson, I am unsure about... The skills I need to complete this lesson are... The challenges I encountered were... I handled these challenges b... In this lesson, I enjoed/did not enjo... Reading and Writing Math 31

5 7-1 The Coordinate Plane California Standards Preparation for AF3.3 Graph linear functions, noting that the vertical change (change in -value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the slope of a graph. Wh learn this? You can use a coordinate plane to plot the path of a hurricane. (See Eercise 35.) A coordinate plane is a plane containing a horizontal number line, called the -ais, and a vertical number line, called the -ais. The intersection of these aes is called the origin. The aes divide the coordinate plane into four regions called quadrants. Quadrant II O Origin 3 Quadrant III Quadrant IV 5 6 -ais Quadrant I -ais EXAMPLE 1 Vocabular coordinate plane -ais -ais origin quadrant ordered pair Identifing Quadrants on a Coordinate Plane Identif the quadrant that contains each point. P P lies in Quadrant II. Q Q lies in Quadrant IV. R R lies on the -ais, between Quadrants II and III. P 3 R 1 3 1O Q An ordered pair is a pair of numbers (, ) that can be used to locate a point on a coordinate plane. The two numbers that form the ordered pair are called coordinates. The origin is identified b the ordered pair (0, 0). Ordered pair -coordinate Units right or left from 0 3 Chapter 7 Graphs and Functions (3, ) -coordinate Units up or down from O units up 3 units right

6 EXAMPLE Plotting Points on a Coordinate Plane Plot each point on a coordinate plane. G(, 5) Start at the origin. Move units right and 5 units up. N( 3, ) Start at the origin. Move 3 units left and units down. G(, 5) 3 1 P(0, 0) O 1 3 P(0, 0) Point P is at the origin. EXAMPLE 3 Identifing Points on a Coordinate Plane Give the coordinates of each point. J Start at the origin. Point J is 3 units right and units down. The coordinates of J are (3, ). K Start at the origin. Point K is units left and units up. The coordinates of K are (, ). K 3 L 1 3 1O J 3 L Start at the origin. Point L is 3 units left on the -ais. The coordinates of L are ( 3, 0). Think and Discuss 1. Eplain whether (, 5) and (5, ) name the same point.. Name the -coordinate of a point on the -ais. Name the -coordinate of a point on the -ais. 3. Suppose the equator represents the -ais on a map of Earth and a line called the prime meridian, which passes through England, represents the -ais. Starting at the origin, which of these directions east, west, north, and south are positive? Which are negative? 7-1 The Coordinate Plane 33

7 7-1 Eercises California Standards Practice Preparation for AF3.3; MG3. KEYWORD: MT8CA 7-1 KEYWORD: MT8CA Parent GUIDED PRACTICE See Eample 1 See Eample See Eample 3 Identif the quadrant that contains each point. 1. A. B 3. C. D Plot each point on a coordinate plane. 5. E( 1, ) 6. N(, ) 7. H( 3, ) 8. T(5, 0) Give the coordinates of each point. 9. J 10. P 5 A D 3 P S 1 M 5 3 1O B J C S 1. M See Eample 1 See Eample See Eample 3 INDEPENDENT PRACTICE Identif the quadrant that contains each point. 13. F 1. J 15. K 16. E Plot each point on a coordinate plane. 17. A( 1, 1) 18. M(, ) 19. W( 5, 5) 0. G(0, 3) Give the coordinates of each point. 1. Q. V 3. R E Q O K 3 R P J 5 V L S F. P 5. S 6. L Etra Practice See page EP1. PRACTICE AND PROBLEM SOLVING For Eercises 7 and 8, use graph paper to plot the ordered pairs. Use a different coordinate plane for each eercise. 7. ( 8, 1); (, 3); ( 3, 6) 8. ( 8, ); ( 1, ); ( 1, 3); ( 8, 3) 9. Geometr Connect the points in Eercise 7. Identif the figure and the quadrants in which it is located. 30. Geometr Connect the points in Eercise 8 in the order listed. Identif the figure and the quadrants in which it is located. Identif the quadrant of each point described below. 31. The -coordinate and the -coordinate are both negative. 3. The -coordinate is negative and the -coordinate is positive. 3 Chapter 7 Graphs and Functions

8 33. What point is located 9 units right and 3 units up from point (3, )? Weather 3. Reasoning After being moved 6 units right and units down, a point is located at (6, 1). What were the original coordinates of the point? 35. Weather The map shows the path of Hurricane Andrew. Estimate to the nearest integer the coordinates of the storm for each of the times below. 3. Andrew becomes a tropical depression. + Hurricane Andrew August When the wind speed of a tropical storm reaches 7 mi/h, it is classified as a hurricane. Source: National Hurricane Center Andrew makes landfall in Florida. 1. Andrew becomes a hurricane a. when Andrew first became a hurricane b. when Andrew made landfall in Florida c. when Andrew weakened to a tropical depression 36. What s the Error? To plot ( 1, 1), a student started at (0, 0) and moved 1 units right and 1 unit down. What did the student do wrong? 37. Write About It Wh is order important when plotting an ordered pair on a coordinate plane? 38. Challenge Armand and Kala started jogging from the same point. Armand jogged miles south and then 6 miles east. Kala jogged west and then miles south. If the were 11 miles apart when the stopped, how far west did Kala jog? NS1.3, Prep for AF3.3, MG Multiple Choice Which of the following points lie within the circle graphed at right? A (, 6) B (, ) C (0, ) D ( 6, 6) 0. Multiple Choice Which point on the -ais is the same distance from the origin as (0, 3)? O 8 8 A (0, 3) B (3, 0) C (3, 3) D ( 3, 3) Estimate each unit rate. (Lesson 5-) 1. $89 for hours. laps in 13 minutes 3. 7 students and 3 teachers Order the numbers from least to greatest. (Lesson 6-1). 1 3, 0.375, 0.3, 3% , 0.1, 1 7, 15% 6. 1%, 1 8, 0.13, The Coordinate Plane 35

9 7- Functions California Standards Preparation for AF3.3 Graph linear functions, noting that the vertical change (change in - value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the slope of a graph. Vocabular function input output vertical line test Who uses this? A business can use functions to determine how man items it needs to sell to break even. (See Eercise 0.) A function is a set of ordered pairs (, ) so that each -value corresponds to eactl one -value. Some functions can be described b a rule written in words, such as double a number and then add nine to the result, or b an equation with two variables. One variable (often ) represents the input, and the other variable (often ) represents the output. When a function can be written as an equation, the input is the value substituted into the function rule. The output is the result of that substitution. Output variable Function Rule Input variable EXAMPLE 1 The input values of a function are also called the domain. The output values of a function are also called the range. Finding Output Values Find the output for each input., Input: 1, 0, 3 Input Rule Output Make a table. 1 ( 1) 6 Substitute 1 for. Then simplif. 0 (0) Substitute 0 for. Then simplif. 3 (3) 10 Substitute 3 for. Then simplif. The output values are 6,, and 10. 6, Input: 5, 0, 5 Input Rule Output 6 5 6( 5) (0) 0 5 6(5) 150 Make a table. Substitute 5 for. Then simplif. Substitute 0 for. Then simplif. Substitute 5 for. Then simplif. The output values are 150 and Chapter 7 Graphs and Functions

10 Because a function has eactl one output for each input, ou can use the vertical line test to determine whether a graph is a function. If no vertical line intersects the graph at more than one point, then the graph is a function. One wa to perform the vertical line test is to pass a vertical line across a graph. EXAMPLE Identifing Functions Determine if each relationship represents a function Each input value has The input value 1 has two onl one output value. output values, 1 and. The relationship is a function. The relationship is not a function. O O Pass a vertical line across the graph. Man vertical lines intersect the graph at two points. The relationship is not a function. Pass a vertical line across the graph. No vertical lines intersect the graph at more than one point. The relationship is a function. Think and Discuss 1. Describe the possible input and output values for.. Describe how to tell if a relationship is a function. 3. Identif the function rule, the inputs, 3 and the outputs. 1 3( 1) 7 0 3(0) 1 3(1) 1 7- Functions 37

11 7- Eercises California Standards Practice Preparation for AF3.3 KEYWORD: MT8CA 7- See Eample 1 GUIDED PRACTICE KEYWORD: MT8CA Parent Find the output for each input:, 0, See Eample Determine if each relationship represents a function See Eample 1 INDEPENDENT PRACTICE Find the output for each input:, 0, ( 1) (1 ) See Eample Determine if each relationship represents a function O 3 7 O Etra Practice See page EP1. PRACTICE AND PROBLEM SOLVING Determine if each relationship represents a function. Eplain Sports A distance runner trains b running 750 meters at a time. Her coach records the distance covered b the runner ever 0 seconds. The results of one run are presented in the table Time (s) Distance (m) a. Does the relationship represent a function? b. What are the inputs of the function? What are the outputs? 38 Chapter 7 Graphs and Functions

12 Home Economics In 1879, Thomas Edison used a carbonized piece of sewing thread to form a light bulb filament that lasted 13.5 hours before burning out. 0. Health You can burn about 3 calories per minute when paddling a canoe. The function 3 gives the total calories burned when ou have paddled for minutes. 1. a. How man calories will ou burn if ou paddle for 3 min., 10 min., and 30 min.? b. How man calories will ou burn if ou paddle for 1 hour? Home Economics The cost of using a 60-watt light bulb is given b the function The cost is in dollars, and represents the number of hours the bulb is lit. a. How much does it cost to use a 60-watt light bulb 8 hours a da for a week? b. Describe the input values of the function. c. If the total cost of using a 60-watt bulb is $1.98, for how man hours can it be used?. What s the Question? The following set of points defines a function: {(3, 6), (, 1), (5, 5), (9, 6), (10, ), (, 10)}. If the answer is 6, 1, 5, 6,, and 10, what is the question? 3. Write About It Can ou tell if a relationship is a function b just looking at the output? Eplain wh or wh not.. Challenge What values of make the ordered pairs (, 0), (, 3), (, 6) a function? Eplain. 5. Challenge What values of make the ordered pairs ( 1, ), (, ), (3, ) a function? Eplain. NS1.3, Prep for AF3.3, AF.0 6. Multiple Choice Which relationship does NOT represent a function? A (0, 8), (3, 8), (1, 6) C B 3 17 D (0, 3), (, 3), (, 0) 7. Multiple Choice Which function matches the function table? A B C D Solve. Check our answer. (Lesson -7) 8. n m 1 9 Estimate. (Lesson 6-) % of % of % of % of $ % of $ % of $ Functions 39

13 7-3 Graphing Linear Functions California Standards AF3.3 Graph linear functions, noting that the vertical change (change in -value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the slope of a graph. Also covered: AF1.1 Vocabular linear equation linear function Wh learn this? You can use the graph of a linear function to show the relationship between the Celsius and Fahrenheit scales. (See Eample.) Recall that the solution of an equation with one variable is the value of the variable that makes the equation true. The solutions of an equation with two variables are the ordered pairs that make the equation true. When these ordered pairs form a line, the equation is called a linear equation. A function described b a linear equation is a linear function. To graph a linear function, plot some solutions of the related linear equation, then draw a line through them. The line represents all of the ordered pair solutions of the equation. For eample, the function that relates distance d, rate r, and time t is described b the linear equation d rt. This graph shows solutions of this equation when r feet per second. Distance (ft) Time (s) EXAMPLE 1 Not all linear equations describe functions. The graphs of some linear equations are vertical lines, which do not pass the vertical line test. Graphing Linear Functions Graph the linear function 1. Input Rule Output Ordered Pair 1 (, ) 1 ( 1) 1 1 ( 1, 1) 0 (0) 1 1 (0, 1) 1 (1) 1 3 (1, 3) Make a table. Substitute positive, negative, and zero values for. 5 3 (1, 3) Plot each ordered pair on the coordinate grid. Then connect the points to form a line. ( 1, 1) 1 O (0, 1) Chapter 7 Graphs and Functions

14 EXAMPLE Math Builders For more on graphing relationships, see the Graph and Equation Builder on page MB. Science Application For ever degree that temperature increases on the Celsius scale, the temperature increases b 1.8 degrees on the Fahrenheit scale. When the temperature is 0 C, it is 3 F. So a temperature in degrees Fahrenheit is 3 degrees more than 1.8 times a temperature in degrees Celsius. a. Write a linear function that describes the relationship between the Celsius and Fahrenheit scales. Let represent the input, which is the temperature in degrees Celsius. Let represent the output, which is the temperature in degrees Fahrenheit. degrees Fahrenheit is 1.8 times degrees Celsius plus 3 degrees The function is b. Make a graph to show the relationship. Make a function table. Include a column for the rule. Input Rule Output (0) (15) (30) 3 86 Graph the ordered pairs (0, 3), (15, 59), and (30, 86) from our table. Connect the points to form a line. Multipl the input b 1.8, and then add 3. Temperature ( F) Temperature ( C) Think and Discuss 1. Describe how a linear equation is related to a linear graph.. Eplain how to use a graph to find the output value of a linear function for a given input value. 7-3 Graphing Linear Functions 331

15 7-3 Eercises AF1.1, California Standards Practice AF3.3 KEYWORD: MT8CA 7-3 See Eample 1 GUIDED PRACTICE Graph each linear function KEYWORD: MT8CA Parent Ordered Input Rule Output Pair 3 (, ) 0 Ordered Input Rule Output Pair (, ) See Eample See Eample 1 See Eample Etra Practice See page EP1. 3. A water tanker is used to fill a communit pool. The tanker pumps 750 gallons of water per hour, so the amount of water in the pool is 750 times the number of hours. a. Write a linear function that describes the amount of water in the pool over time. b. Make a graph to show the amount of water in the pool over the first 6 hours. INDEPENDENT PRACTICE Graph each linear function Science The temperature of a liquid is increasing at the rate of 3 C per hour. When Joe begins measuring the temperature, it is 0 C, so the temperature of the liquid is 0 C more than 3 times the number of hours. a. Write a linear function that describes the temperature of the liquid over time. b. Make a graph to show the temperature over the first 1 hours. PRACTICE AND PROBLEM SOLVING 9. Business A charter bus service charges a $15 transportation fee plus $8.50 per passenger. This is represented b the function c 8.5p 15, where c is the total cost based on p passengers. What is the total cost of transportation for the following numbers of passengers: 50, 100, 150, 00, and 50? 10. Life Science Trannosaurus re was one of the largest meat-eaters that ever lived. B 1 ears of age, the weight of a T. re was increasing b about pounds ever da. Suppose a T. re was 1 ears old and weighed 5110 pounds. Write a linear function that describes the relationship between its current weight w and how much it would weigh d das later. How much would it weigh after das? after 3.5 das? after 5 das? 33 Chapter 7 Graphs and Functions

16 Environment About 15% of the methane gas in the atmosphere comes from farm animals such as cows and sheep. Graph the function d rt for each value of r. 11. r 35 mi/h 1. r ft/min 13. r 10 ft/s 1. Multi-Step Graph the function 1. Use our graph to find the value of if the ordered pair (, 5) lies on the graph of the function. 15. Environment The graph shows the amount of carbon dioide in the atmosphere from 1958 to 199. a. The graph is approimatel linear. About how man parts per million (ppm) were added each -ear period? Concentration (ppm) b. About how man parts per million do ou predict there will be after five more -ear periods, in 01? 16. The water level in a well is 100 m. Water is seeping into the well and raising the water level b 10 cm per ear. Water is also draining out of the well at a rate of m per ear. What will the water level be in 10 ears? 17. What s the Question? Tron used the equation to track his savings after months. If the answer is $50, what is the question? 18. Write About It Eplain how to graph 5. Carbon Dioide in Atmosphere 19. Challenge Certain bacteria divide ever 30 minutes. You can use the function to find the number of bacteria after each half-hour period, where is the number of half-hour periods. Make a table of values for 1,, 3,, and 5. Plot the points, and then connect them. How does the graph differ from those ou have seen so far in this lesson? Year NS1.3, AF3.3, AF. 0. Multiple Choice The graph of which linear function passes through the origin? A B 3 C 1 D 1. Short Response Simon graphed the linear function 3 at right. Eplain his error. Then graph 3 correctl on a coordinate plane.. Zachar is making 0 decorative place cards for a dinner part. It takes him 15 minutes to make the first 3 place cards. If he continues to work at the same rate, how much longer will it take him to finish making the place cards? (Lesson 5-3) ( 1, ) (0, 3) O (1, ) Find each number to the nearest tenth. (Lesson 6-3) 3. What number is 75% of 3?. What number is 1.5% of 70? 5. What number is 5% of 9? 6. What number is 115% of 57? 7-3 Graphing Linear Functions 333

17 7- Graphing Quadratic Functions California Standards AF3.1 Graph functions of the form n and n 3 and use in solving problems. Vocabular quadratic function parabola EXAMPLE 1 Use the value of n to help ou decide if our graph is reasonable. If n is positive, the graph opens upward. If n is negative, the graph opens downward. Wh learn this? You can use a graph of a quadratic function to find the size of a parabolic mirror. (See Eample.) A quadratic function is a function in which the greatest power of the variable is. The most basic quadratic function is n where n 0. The graphs of all quadratic functions have the same basic shape, called a parabola. Graphing Quadratic Functions Create a table for each quadratic function, and use it to graph the function ( 3) 3 1 ( ) ( 1) 3 0 (0) (1) 3 () (3) ( 3) ( 3) ( ) ( ) 0 1 ( 1) ( 1) 0 (0) 0 1 (1) 1 0 () 3 (3) 3 10 The mirror of this telescope is made of liquid mercur that is rotated to form a parabolic shape. O O Plot the points and connect them with a smooth curve. Plot the points and connect them with a smooth curve. 33 Chapter 7 Graphs and Functions 3RD PRINT

18 EXAMPLE Astronom Application In a liquid mirror, a container of liquid mercur is rotated around an ais. Gravit and centrifugal force cause the liquid to form a parabolic shape. The cross section of a liquid mirror that rotates at 10 revolutions per minute is approimated b the graph of 0.07 where and are measured in meters. If the diameter of the mirror is 3 m, about how much higher are the sides than the center? Spinning mercur forms a parabolic surface. First create a table of values. Then graph the cross section ( ) ( 1) (0) (1) () m O 1 The center of the mirror is at 0 m where the height is 0 m. The diameter of the mirror is 3 m, so the edges (highest points) are at 1.5 m and 1.5 m. The height of the mirror at 1.5 is 0.07(1.5) 0.06 m. The sides are about 0.06 m, or 6 cm, higher than the center. Think and Discuss 1. Compare the graphs of and 1.. Describe the shape of a parabola. 7- Graphing Quadratic Functions 335

19 7- Eercises AF3.1 California Standards Practice KEYWORD: MT8CA 7- See Eample 1 See Eample See Eample 1 See Eample Etra Practice See page EP1. GUIDED PRACTICE Create a table for each quadratic function, and use it to graph the function Sports The function 0.15t.t 5.1 gives the height in feet of a baseball seconds after it was thrown. What was the height of the baseball when it was initiall thrown (t 0)? INDEPENDENT PRACTICE Create a table for each quadratic function, and use it to graph the function Manufacturing The function 300 1,50 gives the cost of manufacturing items per da. Which number of items will give the lowest cost per da: 0, 75, or 90? What will the cost be? PRACTICE AND PROBLEM SOLVING Find when 3, 0, and 3. KEYWORD: MT8CA Parent Match each equation with the correct graph Graph A Graph B Graph C O O O 18. Hobbies The height h of a model airplane launched from the top of a ft hill is given b the function h 0.08t.6t, where t is the time in seconds. Find the height of the airplane after, 8, and 16 seconds. Round to the nearest tenth of a foot. What can ou tell about the direction of the airplane? 19. Science The height h of a to rocket launched straight up with an initial velocit of 8 feet per second is given b the function h 8t 16t. The time t is in seconds. a. Graph the function for t 0, 0.5, 1, 1.5,,.5, and 3. b. How man seconds does it take for the rocket to land? 336 Chapter 7 Graphs and Functions

20 0. Describe the difference between a linear function and a quadratic function in terms of their graphs and their equations. 1. Business A store owner can sell 30 digital cameras a week at a price of $150 each. For ever $5 drop in price, she can sell more cameras a week. If is the number of $5 price reductions, the weekl sales function is Predicted Sales (30 )(150 5). Price $150 $15 $10 a. Find for 3,, 5, 6, and 7. Number Sold b. How man $5 price reductions will Weekl Sales $500 $60 $760 result in the highest weekl sales?. Reasoning The height of an object dropped from the top of a 16 ft ladder is given b the function h t 16. Find h when t is. What does this tell ou about t seconds? Does this equation seem more realistic for dropping a rock or a feather? Eplain. 3. Choose a Strateg Suppose the function gives a compan s profit for producing items. Which of the following numbers of items should be produced to maimize profit? A 5 B 30 C 35 D 0. Write About It Which will grow faster as gets larger, or? Check b testing each function for several values of. 5. Challenge Create a table of values for the quadratic function 3( 1), and then graph the function. At what points does the graph intersect the -ais? NS1.3, AF3.1, MG1. 6. Multiple Choice The height of a tennis ball thrown straight up with an initial velocit of 6 meters per second is given b the function h 6t 16t. The time t is in seconds. At what time does the tennis ball land? A 1 s B s C 16 s D 6 s 7. Gridded Response What is the value of when 1 and is? The scale of a drawing is in. 3 ft. Find the actual measurement for each length in the drawing. (Lesson 5-7) 8. 1 in in in in. Find each number to the nearest tenth. (Lesson 6-) 3. 85% of what number is 150? is.5% of what number? 7- Graphing Quadratic Functions 337

21 7-5 Cubic Functions California Standards AF3.1 Graph functions of the form n and n 3 and use in solving problems. Vocabular cubic function Wh learn this? You can use a cubic function to approimate the growth of a population. (See Eercise 3.) A cubic function is a function in which the greatest power of the variable is 3. The most basic cubic function is n 3 where n 0. The graph of 3 is shown. When is negative, is negative because odd powers of negative numbers are negative. The graphs of all cubic functions have this same basic shape, curving down, then curving up. O EXAMPLE 1 Graphing Cubic Functions Create a table for each cubic function, and use it to graph the function. 3 3 ( ) ( 1) 3 0 (0) (1) 3 () 3 16 Choose both positive and negative values for. Plot the points and connect them with a smooth curve. 3 3 ( ) ( 1) (0) (1) 3 1 () Choose both positive and negative values for. Plot the points and connect them with a smooth curve Chapter 7 Graphs and Functions 3RD PRINT

22 Create a table for each cubic function, and use it to graph the function. 3 3 ( ) ( 1) (0) 3 1 (1) 3 3 () You can identif different tpes of functions based on their graphs. EXAMPLE The graph of a linear function is a straight line, and the graph of a quadratic function is a parabola. Identifing Tpes of Functions Tell whether each function is linear, quadratic, or cubic. O O The graph is a parabola. Quadratic The graph curves down, then up. Cubic O O The graph is a line. Linear The graph is a parabola. Quadratic Think and Discuss 1. Compare the graph of 5 to the graph of Eplain whether 3 is a cubic function. 7-5 Cubic Functions 339

23 7-5 Eercises California Standards Practice AF3.1, AF3., AF3.3 KEYWORD: MT8CA 7-5 See Eample 1 GUIDED PRACTICE KEYWORD: MT8CA Parent Create a table for each cubic function, and use it to graph the function See Eample Tell whether each function is linear, quadratic, or cubic O O O See Eample 1 INDEPENDENT PRACTICE Create a table for each cubic function, and use it to graph the function See Eample Tell whether each function is linear, quadratic, or cubic O O O Etra Practice See page EP1. PRACTICE AND PROBLEM SOLVING Graph each function Geometr The volume V of a cube with edges inches long is given b the function V 3. Make a table of values and use it to graph this function. (Hint: Would it make sense to include negative values of?) 30 Chapter 7 Graphs and Functions

24 Life Science 3. Reasoning How does the sign of the 3 -term affect the graph of a cubic function?. Life Science A biologist is studing a population of zebra mussels in a river. The number of mussels is approimated b the cubic function 0t 3 10, where t is the number of weeks since the start of the stud. a. Graph the function. b. Use our graph to estimate the number of mussels when t.5 weeks. Zebra mussels have invaded man rivers in the central United States. A single zebra mussel produces up to 1,000,000 eggs per ear. 5. Reasoning The -intercept of a function is the -coordinate of the point where the function s graph crosses the -ais (where 0). a. Graph each cubic function in the table. b. Complete the table b finding the -intercept for each function. c. Look for a pattern in our table. Without graphing, what do ou think is the -intercept of 3 15? Function intercept 6. What s the Error? A student graphed the function 3 so that the graph included the points ( 3, 7), (, 8), and ( 1, 1). Eplain the student s error. 7. Write About It Eplain how the graph of 3 compares to the graph of Challenge Graph the cubic function 3 3. NS1.6, AF3.1, AF Multiple Choice Which equation could be shown in the graph at right? A B D Multiple Choice The price of a stock in dollars is given b the cubic function p 5 3, where is the number of ears since 00. What was the price of the stock in 005? C O A $ B $7 C $137 D $3 Find the percent of increase or decrease to the nearest percent. (Lesson 6-5) 31. from $6 to $88 3. from $95 to $ from $17 to $165 Create a table for each quadratic function, and use it to graph the function. (Lesson 7-) Cubic Functions 31

25 Quiz for Lessons 7-1 Through The Coordinate Plane Plot each point on a coordinate plane. 1. W(1, 5). X(5, 3) 3. Y( 1, 5). Z( 8, ) 7- Functions Determine if each relationship represents a function O O 7-3 Graphing Linear Functions Graph each linear function A freight train travels 50 miles per hour. Write a linear function that describes the distance the train travels over time. Then make a graph to show the distance the train travels in 5 hours. 7- Graphing Quadratic Functions Create a table for each quadratic function, and use it to graph the function The function 300 1,50 gives the cost of manufacturing items per da. Which number of items will give the lowest cost per da, 50, 70, or 85? What will the cost be? 7-5 Cubic Functions Tell whether each function is linear, quadratic, or cubic O O O 3 Chapter 7 Graphs and Functions

26 Make a Plan Choose a method of computation California Standards MR1.1 Analze problems b identifing relationships, distinguishing relevant from irrelevant information, identifing missing information, sequencing and prioritizing information, and observing patterns. Also covered: NS1., NS1.3, NS1.7, AF. When solving problems, ou must decide which calculation method is best: paper and pencil, calculator, or mental math. Your decision will be based on man factors, such as what the problem asks, the numbers involved, and our own number sense. Use the following table as a guideline. Paper and Pencil Calculator Mental Math Use when solving Use when Use when multi-step problems working with large performing basic so ou can see how or difficult operations or the steps relate. numbers. generating simple estimates. For each problem, tell whether ou would use a calculator, mental math, or pencil and paper. Justif our choice, and then solve the problem. 1 3 The local high school radio station has 500 CDs. Each week, the music manager gets 5 new CDs. How man CDs will the station have in 8 weeks? There are 360 deer in a forest. The population each ear is 10% more than the previous ear. How man deer will there be after 3 ears? Heidi works 8-hour shifts frosting cakes. She has frosted 1 cakes so far, and she thinks she can frost cakes an hour during the rest of her shift. How man more hours will it take for her to frost a total of 3 cakes? 5 6 A compan s logo is in the shape of a right triangle. On the compan s stationer, the triangle has legs measuring 5.1 cm and 6.9 cm. On a poster, the shorter leg of the similar logo measures 1.79 cm. Estimate the length of the longer leg on the poster. Margo and her friends decided to hike the Wildcat Rock trail. After hiking 1 of the wa, the turned back because it began to rain. How far did the hike in all? Trail Meadowlark Distance (mi) Kai has $170 in a savings account that earns 3% simple interest each ear. How much interest will he have earned in 1 ears? Ke Lake Wildcat Rock Eagle Lookout 8 Focus on Problem Solving 33

27 7-6 Rate of Change and Slope California Standards AF3.3 Graph linear functions, noting that the vertical change (change in -value) per unit of horizontal change (change in -value) is alwas the same and know that the ratio ( rise over run ) is called the slope of a graph. Vocabular rate of change rise run slope Wh learn this? You can use rates of change to find out how the average price of a movie ticket has increased. (See Eercise 19.) The input of a function is called the independent variable. It is often represented b the letter. The output of a function is called the dependent variable. It is often represented b the letter. A rate of change is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. The rates of change for a set of data ma var or the ma be constant. EXAMPLE 1 Identifing Constant and Variable Rates of Change in Data Determine whether the rates of change are constant or variable The rates of change are variable. Find the difference between consecutive data points. Find each ratio of change in to change in The rates of change are constant. Find the difference between consecutive data points. Find each ratio of change in to change in. 3 Chapter 7 Graphs and Functions

28 Data in a graph is alwas read from left to right. To show rates of change on a graph, plot the data points and connect them with line segments. The graphs and rates of change of the data sets in Eamples 1A and 1B are shown below O 6 8 O 6 8 Steeper segments have rates of change with greater absolute values. A horizontal segment has a rate of change of 0. A segment that slants downward from left to right has a negative rate of change. If all the segments have the same rate of change, then the form a straight line. This is the case when the data set has a constant rate of change. The constant rate of change of a line is called the slope of the line. SLOPE OF A LINE The rise is the difference in the -values of two points on a line. The run is the difference in the -values of two points on a line. The slope of a line is the ratio of rise to run for an two points on the line. slope r ise change in run c hange in (Remember that is the dependent variable and is the independent variable.) 6 Rise 3 Rise 3 Run Rise 3 Run O Run Slope 3_ 6 EXAMPLE Notice that it does not matter which point ou start with. The slope is the same. Finding the Slope of a Line Find the slope of the line. 8 6 Rise (3,3) O Run (5,7) Run Rise 6 8 Begin at one point and count verticall to find the rise. Then count horizontall to the second point to find the run. slope slope The slope of the line is. 7-6 Rate of Change and Slope 35

29 EXAMPLE 3 Finding a Rise or a Run Find the value of a. O slope 3 8 a slope 5 3 O a 0 slope r ise slope r ise run run 3 a a 3 8 a 5a 0 3 a Multipl. 5a 60 Multipl. 6 a Divide both sides b. a 1 Divide both sides b 5. Think and Discuss 1. Eplain how to use a graph to determine whether a data set has a constant or variable rate of change. 7-6 Eercises California Standards Practice AF3.3 KEYWORD: MT8CA 7-6 See Eample 1 GUIDED PRACTICE KEYWORD: MT8CA Parent Determine whether the rates of change are constant or variable See Eample Find the slope of each line (6, 5) (0, 3) O 6 8 (0, ) O (, ) 36 Chapter 7 Graphs and Functions

30 See Eample 3 Find the value of a slope 3 a a 5 O 9 O slope 1 See Eample 1 INDEPENDENT PRACTICE Determine whether the rates of change are constant or variable See Eample Find the slope of each line (, 3) (, 3) O ( 3, 3) O (3, 0) See Eample 3 Find the value of a a O 1 slope 3 slope 3 O 16 a Etra Practice See page EP15. PRACTICE AND PROBLEM SOLVING Graph each set of data. Label the rate of change for each segment. Then tell whether the data set has a constant or variable rate of change Reasoning The data in the table have a constant rate of change. Find the missing value in the table Rate of Change and Slope 37

31 18. Reasoning In a table of data, ever -value is 3 times the corresponding -value. A student makes a conjecture that the data set must have a constant rate of change. Do ou agree or disagree with the student s conjecture? Wh? 19. Entertainment The table shows the average price of a Year ear for 1993 to 1998, for 1998 to 001, and for 001 to 005. movie ticket in different ears. Average a. Find the rate of change per Ticket Price ($) b. During which period did the price increase at the greatest rate? 0. What s the Error? A student was asked to find the slope of the line at right. The student s work is shown below. Eplain the error. slope r is ru e n Write About It Eplain how ou can tell whether the slope of a line is positive, negative, or zero just b looking at the line.. Challenge Draw a line with a slope of that passes through the point (, 1). ( 1, 3) O (, 3) 3. Multiple Choice The slope of the line shown in the graph is. 3 What is the value of k? A 16 C 9 B 9 D 16. Multiple Choice On a winter da, the temperature changed at a constant rate of 5 per hour. Which could be the graph of this situation? O k NS1.3, AF3.3, MG3.3 1 A Temperature Time B Temperature Time C Temperature Time D Temperature Time Solve for the unknown side length to the nearest tenth. (Lesson -9) 5. c a b c 10 3 Estimate. (Lesson 6-) 9. 3% of % of % of % of Chapter 7 Graphs and Functions

32 7-7 Finding Slope of a Line California Standards AF3. Plot the values of quantities whose ratios are alwas the same (e.g., cost to the number of an item, feet to inches, circumference to diameter of a circle). Fit a line to the plot and understand that the slope of the line equals the quantities. Also covered: AF1.5, AF3.3 The small number in 1 is called a subscript. Read 1 as sub one and as sub. Wh learn this? You can use slope to make sure a wheelchair ramp is not too steep. (See Eercise 18.) Recall that lines have constant slope. For a line on the coordinate plane, slope is the following ratios: If ou know an two points on a line, ou can find the slope of the line without graphing. The slope of a line through the points ( 1, 1 ) and (, ) is as follows: When finding slope using the ratio above, it does not matter which point ou choose for ( 1, 1 ) and which point ou choose for (, ). EXAMPLE 1 Finding Slope, Given Two Points Find the slope of the line that passes through each pair of points. (1, 7) and (9, 3) Let ( 1, 1 ) be (1, 7) and (, ) be (9, 3) Substitute 3 for 1 9 1, 7 for 1, 9 for, and 1 for Simplif. The slope of the line that passes through (1, 7) and (9, 3) is 1. (, 5) and (3, 5) Let ( 1, 1 ) be (, 5) and (, ) be (3, 5) Substitute 5 for 1 3 ( ), 5 for 1, 3 for, and for Rewrite subtraction as addition of the opposite Simplif. 0 The slope of the line that passes through (, 5) and (3, 5) is Finding Slope of a Line 39

33 Find the slope of the line that passes through each pair of points. (1, 0) and ( 3, ) Let ( 1, 1 ) be (1, 0) and (, ) be ( 3, ) Substitute for, 0 for, 3 for, and 1 for Simplif. The slope of the line that passes through (1, 0) and ( 3, ) is 1. EXAMPLE You can use an two points to find the slope of the line. Science Application The table shows the volume of water released b Hoover Dam over a certain period of time. Use the data to make a graph. Find the slope of the line and eplain what it shows. Graph the data. Water Released from Hoover Dam Time (s) Volume of Water (m 3 ) 5 75, , , ,000 Volume of Water (m 3 ) 350, ,000 50,000 00, , ,000 50,000 Water Released from Hoover Dam Find the slope of the line , ,000 Substitute , Time (s) 15,000 Simplif. The slope of the line is 15,000. This means that for ever second that passed, 15,000 m 3 of water was released from Hoover Dam. The graph shows that the total amount of water released increased as time passed. The slope of a line ma be positive, negative, zero, or undefined. You can tell which of these is the case b looking at the graph of a line ou do not need to calculate the slope. POSITIVE NEGATIVE ZERO UNDEFINED SLOPE SLOPE SLOPE SLOPE 350 Chapter 7 Graphs and Functions

34 Think and Discuss 1. Eplain wh it does not matter which point ou choose as ( 1, 1 ) and which point ou choose as (, ) when finding slope.. Give an eample of two points on each of the following: a line with zero slope and a line with an undefined slope. 3. Eplain whether ou think it would be more difficult to run up a hill with a slope of 1 3 or a hill with a slope of Eercises California Standards Practice AF1.5, AF3.3, AF3. KEYWORD: MT8CA 7-7 See Eample 1 GUIDED PRACTICE KEYWORD: MT8CA Parent Find the slope of the line that passes through each pair of points. 1. (, 5) and (3, 6). (, 6) and (0, ) 3. (, ) and (6, 6). (, ) and ( 1, ) 5. ( 3, ) and (, ) 6. ( 6, 0) and (, ) See Eample 7. The table shows how much mone Marvin earned while helping his mother with ard work one weekend. Use the data to make a graph. Find the slope of the line and eplain what it shows. Time (hr) Mone Earned $15 $5 $35 $5 See Eample 1 INDEPENDENT PRACTICE Find the slope of the line that passes through each pair of points. 8. (, ) and (, 1) 9. (0, 0) and (, ) 10. (3, 6) and (, 1) 11. (, ) and (0, 5) 1. (, 3) and (, ) 13. (0, ) and ( 7, ) 1. ( 1, 7) and (3, 7) 15. (0, 1) and (5, 0) 16. ( 3, 6) and ( 6, 9) See Eample 17. The table shows how much water was in a swimming pool as it was being filled. Use the data to make a graph. Find the slope of the line and eplain what it shows. Time (min) Amount of Water (gal) Finding Slope of a Line 351

35 Etra Practice See page EP15. PRACTICE AND PROBLEM SOLVING 18. Safet For safet reasons, the slope of a wheelchair ramp should never eceed the ratio of 1 to 1. A wheelchair ramp rises 1.5 feet for ever 0 feet of horizontal distance it covers. Is this wheelchair ramp considered to be safe? Eplain our answer. 19. Architecture The Luor Hotel in Las Vegas, Nevada, has a 350-foot-tall glass pramid. The elevator of the pramid moves at an incline such that its rate of change is feet in the vertical direction for ever 5 feet in the horizontal direction. Graph the line that describes the path it travels. (Hint: The point (0, 350) is the top of the pramid.) 0. Manufacturing A factor produces widgets at a constant rate. After 3 hours, 50 widgets have been produced. After 8 hours, 670 widgets have been produced. At what rate are the widgets being produced? How long will it take to produce 10,080 widgets? 1. Construction The angle, or pitch, of a roof is the number of inches it rises verticall for ever 1 inches it etends horizontall. Morgan s roof has a pitch of 0. What does this mean?. A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left. At what rate is the water leaking? After how man minutes will the container be empt? 3. What s the Error? The slope of the line through the points (, 5) and (, 5) is ( ) 5 ( 5). 5 What is the error in this statement?. Write About It A vertical line passes through the points (3, ) and (3, 7). Graph the line. Eplain wh the slope of the line is undefined. 5. Challenge Find the slope of the line containing the points (w, z) and (w, 3z). AF.1, AF3.3, MG Multiple Choice Which best describes the slope of the line that passes through the points (, ) and (9, )? A positive B negative C zero D undefined 7. Gridded Response What is the slope of the line that passes through the points ( 5, ) and ( 7, )? Use conversion factors to find each of the following. (Lesson 5-) 8. The number of movie tickets sold in three hours at a rate of 1 ticket per minute 9. The number of miles walked in 1 hour at an average rate of 35 feet per minute Simplif. (Lesson -1) 30. (3 ) (5 ) 3. 7 ( 3) 35 Chapter 7 Graphs and Functions

36 7-8 Interpreting Graphs California Standards AF1.5 Represent quantitative relationships graphicall and interpret the meaning of a specific part of a graph in the situation represented b the graph. Wh learn this? You can use graphs to analze the speed of a horse over time. You can use graphs to show real-world situations visuall. For eample, the graph below shows the varing speeds at which Emma eercises her horse. The horse walks at a constant speed for the first 10 minutes. Its speed increases over the net 7 minutes. It gallops at a constant rate for 0 minutes. Then it slows down over the net 3 minutes. It walks at a constant pace for 10 minutes Time (min) Read the graph from left to right. A horizontal line represents no change, a line that slants upward represents an increase, and a line that slants downward represents a decrease. Recall that a steeper line represents a rate of change with a greater absolute value. Speed (mi/h) EXAMPLE 1 Relating Graphs to Situations Jenn leaves home and drives to the beach. She stas at the beach all da before driving back home. Which graph best shows the situation? Graph A Graph B Graph C Distance from home Time Distance from home Time Distance from home Time As Jenn drives to the beach, her distance from home increases, so the segment slants upward. While she is at the beach, her distance from home does not change, so the segment is horizontal. As she drives home, her distance from home decreases, so the segment slants downward. The answer is Graph B. 7-8 Interpreting Graphs 353

37 EXAMPLE Sketching Graphs for Situations Maili and Katrina traveled 10 miles from Maili s house to the movie theater. The watched a movie, and then the traveled 5 miles farther to a restaurant to eat lunch. After eating the returned to Maili s house. Sketch a graph to show the distance that the two friends are from Maili s house compared to time. Step 1 List the different actions ou need to show on the graph. Then describe the segment for each action. Went to the movies Watched the movie Went to the restaurant Ate lunch Went to Maili s house Distance from home increased: segment slants upward. Distance does not change: horizontal segment. Distance from home increased: segment slants upward. Distance does not change: horizontal segment. Distance from home decreased: segment slants downward. Step Draw the graph. Distance from Maili s house (mi) Watched the movie Ate lunch Went to the restaurant Went to the movies Went to Maili s Time Think and Discuss 1. Eplain the meaning of a horizontal segment on a graph that compares distance to time.. Describe a real-world situation that could be represented b a graph that has connected lines or curves. 35 Chapter 7 Graphs and Functions

38 7-8 Eercises AF1.5 California Standards Practice KEYWORD: MT8CA 7-8 See Eample 1 GUIDED PRACTICE KEYWORD: MT8CA Parent 1. The temperature of an ice cube increases until it starts to melt. While it melts, its temperature stas constant. Which graph best shows the situation? Temperature Graph A Time Temperature Graph B Time Temperature Graph C Time See Eample See Eample 1 See Eample. Mike and Claudia rode a bus 15 miles to a wildlife park. The waited in line to ride a train, which took them on a 3-mile ride around the park. After the train ride, the ate lunch, and then the rode the bus home. Sketch a graph to show the distance Mike and Claudia are from home compared to time. INDEPENDENT PRACTICE 3. The ink in a printer is used until the ink cartridge is empt. The cartridge is refilled, and the ink is used up again. Which graph best shows the situation? Amount of Ink Graph A Time Amount of Ink Graph B Time. On her wa from home to the grocer store, a 6-mile trip, Veronica stopped at a gas station to bu gas. After filling her tank, she continued to the grocer store. She then returned home after shopping. Sketch a graph to show the distance Veronica traveled compared to time. Amount of Ink Graph C Time Etra Practice See page EP15. PRACTICE AND PROBLEM SOLVING 5. Describe a situation that fits the graph at right. 6. Lnn jogged for.5 miles. Then she walked a little while before stopping to stretch. Sketch a graph to show Lnn s speed compared to time. Elevation Time 7. On his wa to the librar, Jeff runs two blocks and then walks three more blocks. Sketch a graph to show the distance Jeff travels compared to time. 7-8 Interpreting Graphs 355

39 The graphs below show the speeds of three dogs at given times during an obstacle course race. Tell which graph corresponds to each situation in Eercises Graph A Graph B Graph C Speed (mi/h) Time Speed (mi/h) Time Speed (mi/h) Time 8. Brand increases her speed throughout the race. 9. Bruno decreases his speed earl in the race to run around cones on the course. After this, he steadil increases his speed. 10. Ma gets off to a fast start and picks up speed for several seconds. He slows down to run through a tunnel but then increases his speed right afterward. 11. Write About It A driver sets his car s cruise control to 55 mi/h. Describe a graph that shows the car s speed compared to time. Then describe a second graph that shows the distance traveled compared to time. 1. Challenge The graph at right shows the temperature of an oven after the oven is turned on. Eplain what the graph shows. Temperature NS., Time NS.5, AF Multiple Choice How does speed compare to time in the graph at right? A It increases. It stas the same. B It decreases. D It fluctuates. Time 1. Short Response Keisha takes a big drink from a bottle of water. She sets the bottle down to tie her shoe and then picks up the bottle to take a small sip of water. Sketch a graph to show the amount of water in the bottle over time. Find each absolute value. (Lesson 1-3) C Add or subtract. (Lesson -6) A tabletop has a thickness of 7 8 of an inch. A glass cover for the table is 3 of an inch 16 thick. Find the total thickness of the tabletop with the glass cover. Speed 356 Chapter 7 Graphs and Functions

40 7-9 Direct Variation California Standards AF. Solve multistep problems involving rate, average speed, distance, and time or a direct variation. Also covered: AF3.3, AF3. Vocabular direct variation constant of variation Wh learn this? You can use a direct variation to determine reaction distance. (See Eample 3.) A direct variation is a linear function that can be written as k, where k is a nonzero constant called the constant of variation. Solve k for k. k k Divide both sides b. k The value of k is the ratio of to. This ratio is the same for all ordered pairs that are solutions of a direct variation. Since the rate of change k is constant for an direct variation, the graph of a direct variation is alwas linear. The graph of an direct variation alwas contains the point (0, 0) because for an value of k, 0 k0. EXAMPLE 1 Determining Whether a Data Set Varies Directl Determine whether the data set shows direct variation. Method 1 Make a graph. Shoe Sizes U.S. Size European Size European Size U.S. Size The graph is not linear. Method Compare ratios The ratios are not equivalent. Both methods show the relationship is not a direct variation. 7-9 Direct Variation 357

41 Determine whether the data set shows direct variation. Number of Watts of Sound for Watts of Power Input Signal Power (W) Output Sound Intensit m W Method 1 Make a graph. Watts of Sound for Watts of Power Output Intensit ( W ) m The points lie in a straight line. (0, 0) is on the line Input Power (W) Method Compare ratios The ratio is constant. Both methods show the relationship is a direct variation. EXAMPLE Chelsea s income varies directl with the number of hours she babsits. She earns $11 for hours of babsitting. Write a direct variation function for this situation. If Chelsea babsits for 6 hours, how much will she make? Step 1 Write the direct variation function. and 11 Think: Her pa varies directl with time worked. k 11 k Substitute 11 for and for. 5.5 k Solve for k. 5.5 Substitute 5.5 for k in the original equation. Step Find how much Chelsea will make if she babsits for 6 hours. 5.5(6) Substitute 6 for in the direct variation function. 33 Multipl. Chelsea will make $33 if she babsits for 6 hours. 358 Chapter 7 Graphs and Functions

42 EXAMPLE 3 Science Application When a driver applies the brakes, a car s total stopping distance is the sum of the reaction distance and the braking distance. The reaction distance is the distance the car travels between the time the driver decides to appl the brakes and the time the driver actuall presses the brake pedal. The braking distance is the distance the car travels after the brakes have been applied. Determine whether there is a direct variation between each of the following. Distance (ft) Reaction Distance Speed (mi/h) Reaction distance Braking distance 68 reaction distance and speed reactio n d spe istance ed reactio n d spe istance ed The first two pairs of data result in a common ratio. In fact, all of the reaction distance to speed ratios are equivalent to.. reactio n d spe istance ed The variables are related b a constant ratio of. to 1. braking distance and speed braking distance speed 15 braking distance speed If an of the ratios are not equal, then there is no direct variation. It is not necessar to compute additional ratios. Think and Discuss 1. Describe the slope of a direct variation function.. Compare and contrast proportional and non-proportional linear relationships. 7-9 Direct Variation 359

43 7-9 Eercises See Eample 1 GUIDED PRACTICE California Standards Practice AF3.3, AF3., AF. KEYWORD: MT8CA 7-9 KEYWORD: MT8CA Parent 1. The table shows an emploee s pa per number of hours worked. Determine whether the data set shows direct variation. Hours Worked Pa ($) See Eample See Eample 3. Life Science A moose can be a ver fast animal. It can move at speeds up to 80 feet per minute for short periods of time. Write a direct variation function for the distance a moose can travel in minutes. How far could a moose travel in 1 min? 3. The table shows how man hours it takes to travel 600 miles, depending on our speed in miles per hour. Determine whether there is direct variation between the two data sets. Speed (mi/h) Time (h) See Eample 1 INDEPENDENT PRACTICE. The table shows the amount of current flowing through a 1-volt circuit with various resistances. Determine whether the data set shows direct variation. Resistance (ohms) Current (amps) See Eample See Eample 3 5. Science Weight varies directl with gravit. Apollo 11 weighed 5898 kg on Earth but onl 983 kg on the Moon. Apollo 1 weighed 5806 kg on Earth. Write a direct variation function for this situation. About how much did Apollo 1 weigh on the Moon? Round our answer to the nearest kilogram. 6. The table shows how man hours it takes to drive certain distances at a speed of 30 miles per hour. Determine whether there is direct variation between the two data sets. Distance (mi) Time (h) Chapter 7 Graphs and Functions

44 Etra Practice See page EP15. Life Science PRACTICE AND PROBLEM SOLVING Tell whether each equation represents direct variation between and k 10. π 11. Reasoning Is ever linear relationship a direct variation? Is ever direct variation a linear relationship? Eplain. Find each equation of direct variation, given that varies directl with. 1. is 10 when is 13. is 16 when is 1. is 1 when is is 3 when is is 0 when is 17. is 5 when is 0 Although snakes shed their skins all in one piece, most reptiles shed their skins in much smaller pieces. 18. is 8 when is 19. is 5 when is 0. Life Science The weight of a person s skin is related to bod weight b the equation s 1 1 w, 6 where s is skin weight and w is bod weight. a. Does this equation show direct variation between bod weight and skin weight? b. If a person s skin weight is 9 3 lb, what is the person s bod weight? 1. Write a Problem The perimeter P of a square varies directl with the length l of a side. Write a direct variation problem about the perimeter of a square.. Write About It Describe how the constant of variation k affects the appearance of the graph of a direct variation function. 3. Challenge Watermelons are being sold at 79 a pound. What condition would have to eist for the price paid and the number of watermelons sold to represent a direct variation? NS1.7, AF.1, AF.. Multiple Choice Given that varies directl with, what is the equation of direct variation if is 16 when is 0? A B 5 C D Gridded Response If varies directl with, what is the value of when 1 and k 1? 6. The school track team is selling pizzas to make mone for travel. The supplier charges $100 plus $ per pizza. If the team sells the pizzas for $10 each, how man pizzas will the need to sell to make a $150 profit? (Lesson 3-8) Find the simple interest and the total amount to the nearest cent. (Lesson 6-7) 7. $775 at.5% for 10 ears 8. $1595 at 8% for 9 months 7-9 Direct Variation 361

45 Quiz for Lessons 7-6 Through Rate of Change and Slope Determine whether the rates of change are constant or variable Find the slope of each line O O O 7-7 Finding Slope of a Line Find the slope of the line that passes through each pair of points. 5. (6, 3) and (, ) 6. (1, ) and ( 1, 3) 7. (0, 3) and (, 0) 7-8 Interpreting Graphs 8. Raj climbs to the top of a cliff. He descends a little bit to another cliff and then begins to climb again. Which graph best shows the situation? Elevation Graph A Elevation Graph B Elevation Graph C Time Time Time 9. T walks 1 mile to the mall. An hour later, he walks 1 mile farther to a park and eats lunch. Then he walks home. Sketch a graph to show the distance T is from home compared to time. 7-9 Direct Variation 10. The table shows an emploee s pa per number of hours worked. Determine whether the data set shows direct variation. Time Worked (hr) Amount Earned ($) Chapter 7 Graphs and Functions

46 Beset b Beavers Greg and Maria are wildlife biologists. The are studing beaver population trends in a national forest. There are currentl 00 beavers in the forest. The table shows Greg s and Maria s predictions for the beaver population in future ears. 1. Write a function based on Greg s prediction that gives the beaver population in ear. Then use the function to find the population in ear 8.. According to our function, in what ear will the beaver population be 500? Eplain. 3. Write a function based on Maria s prediction that gives the beaver population in ear. Then use the function to find the population in ear 8.. A third biologist, Amir, makes his predictions using the function 5 195, where is the ear number. Use the function to find the beaver population that Amir predicts in ear Which of the three biologists predicts the greatest beaver population in ear 1? What is this population? Beaver Population Predictions Year Greg s Maria s Predictions Predictions Concept Connection 363

47 Squared Awa How man squares can ou find in the figure at right? Did ou find 30 squares? There are four different-sized squares in the figure. Size of Square Number of Squares 3 3 squares squares Total 30 The total number of squares is Draw a 5 5 grid and count the number of squares of each size. Can ou see a pattern? What is the total number of squares on a 6 6 grid? a 7 7 grid? Can ou come up with a general formula for the sum of squares on an n n grid? What s Your Function? One member from the first of two teams draws a function card from the deck, and the other team tries to guess the rule of the function. The guessing team gives a function input, and the card holder must give the corresponding output. Points are awarded based on the tpe of function and number of inputs required. Complete rules and function cards are available online. KEYWORD: MT8CA Games 36 Chapter 7 Graphs and Functions

48 Materials small paper bag scissors tape graph paper stapler PROJECT Graphing Tri-Fold A Use this organizer to hold notes, vocabular, and practice problems related to graphing. Directions 1 Hold the bag flat with the flap facing ou at the bottom. Fold up the flap. Cut off the part of the bag above the flap. Figure A B 3 5 Unfold the bag. Cut down the middle of the top laer of the bag until ou get to the flap. Then cut across the bag just above the flap, again cutting onl the top laer of the bag. Figure B Open the bag. Cut awa the sides at the bottom of the bag. These sections are shaded in the figure. Figure C Unfold the bag. There will be three equal sections at the bottom of the bag. Fold up the bottom section and tape the sides to create a pocket. Figure D Trim several pieces of graph paper to fit in the middle section of the bag. Staple them to the bag to make a booklet. C D Taking Note of the Math Write definitions of vocabular words behind the doors at the top of our organizer. Graph sample linear equations on the graph paper. Use the pocket at the bottom of the organizer to store notes on the chapter. 365

49 Vocabular constant of variation coordinate plane cubic function direct variation function input linear equation linear function ordered pair origin output parabola quadrant quadratic function rate of change vertical line test ais -ais Complete the sentences below with vocabular words from the list above. 1. Two variables related b a constant ratio are in?.. A(n)? gives eactl one output for ever input. 3. A(n)? is a function whose graph is a nonvertical line. 7-1 The Coordinate Plane (pp. 3 35) Prep for AF3.3 EXAMPLE Give the coordinates of each point and tell which quadrant contains it. A( 3, ); II B(, 3); IV C(, 3); III D(3, ); I A 3 D 3 O C 3 B 3 EXERCISES Give the coordinates of each point and tell which quadrant contains it.. J 5. K 6. L 7. M K M O L J 7- Functions (pp ) Prep for AF3.3 EXAMPLE EXERCISES Find the output for each input. Find the output for each input: 1, 0, 1,, 3. Input Rule 3 Output Determine if the relationship represents a function Chapter 7 Graphs and Functions

50 7-3 Graphing Linear Functions (pp ) AF3.3 EXAMPLE Graph the linear function O EXERCISES Graph each linear function Graphing Quadratic Functions (pp ) AF3.1 EXAMPLE EXERCISES Graph the quadratic function 1. Graph each quadratic function O Cubic Functions (pp ) AF3.1 EXAMPLE EXERCISES Graph the cubic function 3 1. Graph each cubic function O Tell whether the function is linear, quadratic, or cubic. Tell whether each function is linear, quadratic, or cubic. The graph curves down, then up. 3.. Cubic Stud Guide: Review 367

51 7-6 Rate of Change and Slope (pp. 3 38) AF3.3 EXAMPLE Determine whether the rates of change are constant or variable The data set has a variable rate of change. EXERCISES Determine whether the rates of change are constant or variable Finding Slope of a Line (pp ) AF3.3, AF3. EXAMPLE Find the slope of the line that passes through ( 1, ) and (1, 3) ( 1) EXERCISES Find the slope of the line that passes through each pair of points. 7. (, ) and (8, 5) 8. (, 3) and (5, 1) 9. ( 3, 3) and (, ) 7-8 Interpreting Graphs (pp ) AF1.5 EXAMPLE Ari drives 5 miles. Then he returns home, stopping for gas along the wa. Sketch a graph to show Ari s distance from home compared to time. EXERCISES 30. Joel rides his bike 1 miles. He then rides an additional 6 miles and returns home. Sketch a graph to show Joel s distance from home compared to time. Distance (mi) Time 7-9 Direct Variation (pp ) AF. EXAMPLE Determine whether the data set shows a direct variation. Time (h) Distance (mi) Yes; the ratio is constant. EXERCISE 31. Ben earns $10 for working hours at his after-school job. The total amount of his pacheck varies directl with the amount of time he works. If he works for.5 hours, how much will he make? 368 Chapter 7 Graphs and Functions 3RD PRINT

52 Plot each point and identif the quadrant in which it lies. 1. L(, 3). M( 5, ) 3. N(7, 1). O( 7, ) Find the output for each input:, 1, 0, 1, Graph each function Tell whether each function is linear, quadratic, or cubic O O O Graph each set of data. Label the rate of change for each segment. Then tell whether the data set has a constant or variable rate of change Find the slope of the line that passes through each pair of points. 19. (0, 8) and ( 1, 10) 0. (0, ) and ( 5, 0) 1. (3, 1) and (0, 3). Ian jogs miles to the lake and then rests for 30 minutes before jogging home. Sketch a graph to show Ian s distance from home compared to time. 3. Make a graph to determine whether the data set shows direct variation. Miles per Gallon in a Hbrid Car Gallons (gal) Miles (mi) The number of gallons of water used b a washing machine varies directl as the number of loads of laundr. A new washer saves 5 gallons of water after 3 loads of laundr. Find the number of gallons of water saved after 7 loads of laundr. Chapter 7 Test 369

53 Etended Response: Write Etended Responses Etended response test items often consist of multi-step problems to evaluate our understanding of a math concept. Etended response questions are scored using a -point scoring rubric. Etended Response Julianna bought a shirt marked down 0%. She had a coupon for an additional 0% off the sale price. Is this the same as getting 0% off the regular price? Eplain our reasoning. -point response: No, the prices are not the same. Suppose the shirt originall cost $0. 0% off a 0% markdown: $0 0% $8; $0 $8 $3; $3 0% $6.0; $3 $6.0 $5.60 0% off: $0 0% $16; $0 $16 $ The student answers the question correctl and shows all work. 3-point response: Yes, it is the same. If the shirt originall cost $5, it would cost $15 after taking 0% off of a 0% discount. A 0% discount off $0 is $15. Shirt original price $5 Shirt at 0% off $0 $5 0% $5; $5 $5 $0 Shirt at 0% off sales price $15 $0 0% $; $0 $ $15 Shirt at 0% off $15 $5 0% $10; $5 $10 $15 The student makes a minor computation error that results in an incorrect answer. -point response: No, it is not the same. A $30 shirt with 0% off and then an additional 0% off is $6. A $30 shirt at 0% off is $1. The student makes major computation errors and does not show all work. 1-point response: It is the same. The student shows no work and has the wrong answer. Scoring Rubric points: The student answers all parts of the question correctl, shows all work, and provides a complete and correct eplanation. 3 points: The student answers all parts of the question, shows all work, and provides a complete eplanation that demonstrates understanding, but the student makes minor errors in computation. points: The student does not answer all parts of the question but shows all work and provides a complete and correct eplanation for the parts answered, or the student correctl answers all parts of the question but does not show all work or does not provide an eplanation. 1 point: The student gives incorrect answers and shows little or no work or eplanation, or the student does not follow directions. 0 points: The student gives no response. 370 Chapter 7 Graphs and Functions

54 To receive full credit, make sure all parts of the problem are answered. Be sure to show all of our work and to write a neat and clear eplanation. Read each test item and answer the questions that follow. Item A Janell has two job offers. Job A pas $500 per week. Job B pas $00 per week plus 15% commission on her sales. She epects to make $7500 in sales per month. Which job pas better? Eplain our reasoning. 1. A student wrote this response: Job A pas better. What score should the student s response receive? Eplain our reasoning.. What additional information, if an, should the student s response include in order to receive full credit? 3. Add to the response so that it receives a score of -points.. How much would Janell have to make in sales per month for job A and job B to pa the same amount? Item C Three houses were originall purchased for $15,000. After each ear, the value of each house either increased or decreased. Which house had the least value after the third ear? What was the value of that house? Eplain our reasoning. Percent Change in Value House Original Year 1 Year Year 3 Cost ($) A 15,000 1% 1% 1% B 15,000 % % 1% C 15,000 3% % % 7. A student wrote this response: House A increased 3% over three ears. House B increased 1% over three ears. House C increased 3% over three ears. So, House B had the least value after the third ear. Its value increased 1% of $15,000, or $150, for a total value of $16,50. What score should the student s response receive? Eplain our reasoning. 8. What additional information, if an, should the student s response include in order to receive full credit? Item B A new MP3 plaer normall costs $ This week, it is on sale for 15% off its regular price. In addition to this, Jasmine receives an emploee discount of 0% off the sale price. Ecluding sales ta, what percent of the original price will Jasmine pa for the MP3 plaer? 5. What information needs to be included in a response to receive full credit? 6. Write a response that would receive full credit. Item D Kara is tring to save $500 to bu a used car. She has $3000 in an account that earns a earl simple interest of 5%. Will she have enough mone in her account after 3 ears to bu a car? If not, how much more mone will she need? Eplain our reasoning. 9. What information needs to be included in a response to receive full credit? 10. Write a response that would receive full credit. Strategies for Success 371