PA CORE 8 UNIT 3 - FUNCTIONS FLEX Workbook

Transcription

1 PA CORE 8 UNIT - FUNCTIONS FLEX Workbook LESSON 9. INTRODUCTIONS TO FUNCTIONS 0. WORK WITH LINEAR FUNCTIONS. USE FUNCTIONS TO SOLVE PROBLEMS. USE GRAPHS TO DESCRIBE RELATIONSHIPS. COMPARE RELATIONSHIPS REPRESENTED IN DIFFERENT WAYS QUIZ 8. MODELING RELATIONSHIPS 9. COMPARING FUNCTIONS Name: Period

2 Domain Lesson 9 Introduction to Functions Common Core State Standards: 8.F., 8.F. Getting the Idea A relation is a set of ordered pairs. A function is a relation in which each input value, or -value, corresponds to eactl one output value, or -value. A function or other relation can be represented as a set of ordered pairs in a table, as an equation, or b a graph. Eample Which table below represents a function? Table Table 0 0 Strateg Compare the - and -values. Step Compare the - and -values in Table. The -value corresponds to onl one -value,. The -value corresponds to onl one -value,. The -value 0 corresponds to onl one -value,. The -value corresponds to onl one -value,. Since each -value has eactl one -value, Table shows a function. Duplicating an part of this book is prohibited b law. Step Compare the - and -values in Table. The -value corresponds to onl one -value,. The -value corresponds to two -values, and. Since there is an -value that corresponds to more than one -value, this relation is not a function. Solution Table represents a function. Table represents a relation that is not a function.

3 In a function, the set of all the input values, or -values, is called the domain. The set of all the output values, or -values, is called the range. Braces, h j, are often used when listing the domain and range. Eample Identif the domain and range for the function shown below. Sale Prices Regular Price, $ $0 $ $0 $ Sale Price, $ $ $ $ $ Strateg Step Step Identif the domain and range of the function. Identif the domain. List the -values., 0,, 0, Identif the range. List the -values.,,,, Solution The domain of the function is, 0,, 0,. The range is,,,,. Ever function follows a rule that maps each element in its domain to eactl one element in its range. So, another wa to determine if a relation is a function is to draw a mapping diagram. List the domain elements and the range elements in order. Then draw an arrow from each domain value to its range value. A mapping diagram for the function is shown below. Domain Range Notice that the -values of and both map to, but there is still eactl one -value for each -value in the set. So, this relation is a function. 9 Duplicating an part of this book is prohibited b law. Domain : Functions

4 Lesson 9: Introduction to Functions Eample Create a mapping diagram for the relation below. (, ), (, 9), (, ), (, ), (, 0) Is the relation a function? Strateg Step Step Step Create a mapping diagram. List the domain elements and the range elements in order. List the domain values,,,,, in a bo on the left. List the range values,, 9, 0,,, in a bo on the right. Draw an arrow from each domain value to its range value. (, ) is part of the relation. So, draw an arrow from to. (, 9) and (, ) are part of the relation. So, draw arrows from to 9 and to. Represent (, ) and (, 0) with arrows, too. Domain Range 9 0 Is the relation a function? The domain element maps to two different range elements, 9 and. So, the relation is not a function. Duplicating an part of this book is prohibited b law. Solution The mapping diagram in Step shows that one of the domain elements maps to two range elements. So, the relation is not a function.

5 The graph of a function is the set of ordered pairs consisting of input values and their corresponding output values. To determine whether a graph represents a function, ou can use the vertical line test. Imagine drawing vertical lines through the graph. If no vertical line intersects the graph in more than one point, the graph shows a function. For eample, in the left-hand graph below, no vertical dashed line crosses the graph in more than one point, so the graph shows a function. If ou can draw a vertical line that intersects the graph in two or more points, the graph does not show a function. In the right-hand graph below, the vertical dashed line crosses the graph in two points, so the graph does not show a function. 0 0 function not a function Eample Which graph represents a function? 0 Graph 0 Graph Duplicating an part of this book is prohibited b law. Domain : Functions

6 Lesson 9: Introduction to Functions Strateg Use the vertical line test on each graph. Step Use the vertical line test on Graph. 0 No matter where ou draw a vertical line, it onl crosses the graph once. This graph represents a function. Step Use the vertical line test on Graph. 0 Duplicating an part of this book is prohibited b law. Solution This graph includes a vertical segment at. So, there is more than one -value paired with the -value,. This graph does not represent a function. Graph represents a function.

7 Sometimes, the graph of a function or other relation is a set of connected points. Other times, the points are not connected. Coached Eample The points shown in the table below represent a relation. Plot the points and determine if the relation is a function. The data in the table correspond to the points (, ), (, ), (, ), (, ), (, ), and (, ). Plot those points on the coordinate grid below. 0 Draw a vertical line through (, ). Does it pass through more than one point on the graph? Draw a vertical line through (, ). Does it pass through more than one point on the graph? When, the relation corresponds to -value(s). So, the relation a function. Duplicating an part of this book is prohibited b law. Domain : Functions 7

8 Lesson 9: Introduction to Functions Lesson Practice Choose the correct answer.. The table shows that the total amount charged, in dollars, b a hot dog vendor is a function of the number of hot dogs purchased. Vendor Charges Number of Hot Dogs, Total Charge, $ $ $ $8 $0 What is the range of the function? A., 0 B.,,,, C.,,, 8, 0 D.,,,,,, 8, 0. Which table does not represent a function? A. B. C. D Which set of ordered pairs represents a function? Duplicating an part of this book is prohibited b law. A. (, ), (0, ), (, ), (, ) B. (, ), (, ), (, ), (, ) C. (, ), (9, 7), (, 0), (9, 8) D. (, 7), (, ), (, 8), (, ) 8 7

9 . Which graph does not represent a function? A. C. 0 0 B. D The table below shows a relation. 0 0 A. Identif the domain and range for the relation above. Then list the domain and range in the boes below and create a mapping diagram for this relation. Domain Range B. Is the relation also a function? Use our mapping diagram to eplain our answer. Duplicating an part of this book is prohibited b law. 8 Domain : Functions 9

10 Domain Lesson 0 Work with Linear Functions Common Core State Standards: 8.F., 8.F. Getting the Idea A linear function is a special kind of function whose graph is a straight line. It can be represented b a linear equation in the form m b. A nonlinear function is an function that is not linear. In a linear equation, no variable is raised to a power greater than. To review linear equations in that form, look back at Lesson. Eample Is the function linear or nonlinear? Strateg Solution Look at the eponent for each variable in the equation. In a linear equation, no variable is raised to a power greater than. In, the variable is raised to the power of. So, the equation is not a linear equation, and the function is nonlinear. The function represented b is a nonlinear function. You can also determine if a function is linear or nonlinear b graphing it. For eample, the graph of the function from Eample,, is shown below. The points on the graph, (, 8), (, ), (0, 0), (, ), and (, 8), do not lie on a straight line, so the function is nonlinear. Duplicating an part of this book is prohibited b law. 0 (, ) (, 8) (0, 0) 8 (, 8) (, ) 9 0

11 A function can describe a situation in which one quantit determines another. In a function, the output value, or -value, depends on the input value, or -value. That is wh the -value is often called the dependent variable, and the -value is often called the independent variable. Eample Given the linear function 7, find the missing output values in the table below. Then identif the independent and dependent variables. Input () Output ( ) 9 0 7?? Strateg Substitute each -value into the equation and solve for. Then identif the independent and dependent variables. Step Find the output when the input is. 7 () 7 7 Step Find the output when the input is. 7 Step () 7 7 Identif the independent and dependent variables. The output, or -value, depends on the input, or -value. So, the dependent variable is, and the independent variable is. Solution When the input is, the output is. When the input is, the output is. The dependent variable is, and the independent variable is. Duplicating an part of this book is prohibited b law. 0 Domain : Functions

12 Lesson 0: Work with Linear Functions A linear function can be represented in man was. If ou are given a table of values or a graph, ou can use it to write an equation for the function. The equation for a function gives the rule that shows how each input value relates to each output value. You can determine the rate of change and initial value for a linear function from a table of values, a graph, or an equation. The initial value is the value of when equals 0. In a graph, the rate of change is the same as the slope. Since the graph of a linear function is a straight line, the rate of change is constant, and ou can determine it from an two pairs of (, ) values for the function. You can find the initial value of a linear function b identifing the -intercept. Eample The input-output table below represents a linear function. Input () Output ( ) 0 8 Write an equation for the function and identif the rate of change and initial value. Strateg Find the rule that relates each -value to its corresponding -value. Then write the equation. Duplicating an part of this book is prohibited b law. Step Step Step Find a rule that relates the first pair of values in the table. In the first column, the -values are increasing b s. In the second column, each -value is more than the previous -value. Look for a rule that involves multipling b. Consider (0, ): 0? 0, not. But, if ou subtract, ou get 0. So, the rule ma be: multipl each -value b and then subtract. See if the rule works for the other pairs of values in the table.?, and (, ) is in the table.?, and (, ) is in the table.? 9 8, and (, 8) is in the table. Use the rule to write an equation. To find each -value, multipl each -value b and then subtract. So,.

13 Step Step Solution Determine the rate of change for the function. Let (, ) (, ). Let (, ) (, ). rate of change Note: You could also have determined the rate of change b looking at the equation. In, m. So, the slope of the graph and its rate of change must be. Determine the initial value. The initial value is the value of when equals 0. The -value when 0. The equation describes the linear function. Its rate of change is, and the initial value is. Coached Eample The graph represents a linear function. Find the rate of change for the function. Then write an equation for the function. The rate of change for the function is equal to its, m. Choose two points on the graph to find the rate of change. Let (, ) (0, ). Let (, ) (, ). m 0 The equation for the line graphed above shows the equation of the linear function. The -intercept of the graph is (0, ). That is the initial value. So, b. You alread know that m. Substitute those values into the slope-intercept form, m b. The rate of change for the linear function is, and its equation is. 0 Duplicating an part of this book is prohibited b law. Domain : Functions

14 Lesson 0: Work with Linear Functions Lesson Practice Choose the correct answer.. Which graph represents a linear function? A. B. 0 0 Use the table below for questions and. The table represents the function 8. Input () Output () 8 0 8?? 8?. What is the input when the output is? A. B. C. 0 D. Duplicating an part of this book is prohibited b law. C. D Which statement is not true of the function? A. The independent variable is. B. The dependent variable is. C. When the input is, the output is. D. When the input is 8, the output is.. Which equation does not represent a linear function? A. _ B. C. D.

15 . Which equation represents the linear relationship between and shown in the table?. Which equation represents the function graphed below? 0 7 A. B. C. D. 0 A. _ C. B. _ D. 7. The ordered pairs in the table below represent a linear function. Input () Output ( ) ? 0 A. Write an equation to represent the function shown above. Eplain how ou determined our equation. B. Find the input for this function when the output is 0. Use the equation ou wrote in Part A. Show our work. Duplicating an part of this book is prohibited b law. Domain : Functions

16 Domain Lesson Use Functions to Solve Problems Common Core State Standard: 8.F. Getting the Idea Sometimes, linear functions are used to model and solve real-world problems. Eample The relationship between a side length of a square, s, and its perimeter, P, can be modeled b the function P s. Find the missing perimeters in the table below. Then identif the dependent and independent variables. Side Length (s) Perimeter (P) 8? 9.? Strateg Step Substitute each s-value into the equation and solve for P. Then identif the independent and dependent variables. Find the perimeter when the side length is 8 units. P s P (8) P Duplicating an part of this book is prohibited b law. Step Step Solution Find the perimeter when the side length is 9. units. P s P (9.) P 8 Identif the independent and dependent variables. The perimeter, P, depends on the length of a side of the square, s. So, the dependent variable is P, and the independent variable is s. When a square has sides 8 units long, its perimeter is units. When a square has sides 9. units long, its perimeter is 8 units. In this function, the dependent variable is P, and the independent variable is s.

17 A graph ma also be used to represent a real-world situation that is modeled b a linear function. If so, the slope of the graph shows the rate at which the two quantities in the problem are changing. Eample The graph below shows how the cost of buing gas at Steve s Service Station changes, based on the number of gallons that are purchased. Total Cost (in dollars) Find the price per gallon of gas Gasoline Prices (, ) (, ) Number of Gallons Strateg Find and interpret the rate of change. Step Step Solution Find the rate of change. This is the same as the slope of the graph. Let (, ) (, ). Let (, ) (, ). rate of change Interpret the rate of change. Since the -ais shows number of gallons and the -ais shows total cost in dollars, the rate of change is dollars, or $ per gallon. gallon The gas costs $ per gallon. Duplicating an part of this book is prohibited b law. Domain : Functions 7

18 Lesson : Use Functions to Solve Problems Eample To bowl at Cavanaugh Lanes, it costs $ per game plus a $ shoe rental. The total cost,, in dollars, depends on, the number of games plaed. Write an equation to represent this situation. Then make a table of values to represent the situation. Strateg Step Write an equation for the situation. Then make a table of values. Translate the words into an equation. $ per game plus $ shoe rental total cost Note: This is a linear equation. The problem situation can be modeled b a linear function. Step Step Decide which -values ou should include in the table. In real life, ou cannot bowl fewer than 0 games. Also, if ou go to the trouble of renting bowling shoes, ou will probabl bowl at least game. So, the initial -value should be. The other -values should be whole numbers because ou can onl pa for a whole number of games. In other words, the domain is limited to,,,,. Make a table of values. Games Bowled () Total Cost in Dollars () () () 7 7 () 9 9 Duplicating an part of this book is prohibited b law. Solution () Note: This table could be continued for other whole number -values. However, when we make tables to represent functions, we accept that the usuall onl represent part of the function. The equation and the table of values in Step represent this problem situation. 8 7

19 Eample A marching band needs to raise $,00 in order to attend the regional marching band festival. The band members are selling tickets to a fundraising breakfast. Tickets are $0 for adults and $ for children. The equation 0,00, where represents the number of adult tickets sold and represents the number of children s tickets sold, can be used to model the situation. The graph of this equation is shown below. Children s Tickets Ticket Sales Adult Tickets Eplain the meaning of the -intercept in terms of the number of adult and children s tickets sold. Strateg Step Step Identif and interpret the -intercept. Identif the -intercept. It is located at (0, 00). Interpret the -intercept. Since represents the number of children s tickets sold, the coordinates of the -intercept, (0, 00), represent a situation where no adult tickets are sold (0 tickets) and onl children s tickets are sold (00 tickets). Solution The -intercept represents a situation in which the marching band sells 00 children s tickets and 0 adult tickets to raise $,00. Duplicating an part of this book is prohibited b law. 8 Domain : Functions 9

20 Lesson : Use Functions to Solve Problems Eample To rent a limousine from Delue Limousines, a customer must pa a set fee plus an additional amount per hour, as shown b the graph below. Total Cost (in dollars) Limousine Rental Number of Hours a. Identif and interpret the initial value and the rate of change. b. Determine the cost of renting a limousine for 8 hours. Strateg Use the graph and the slope formula to identif and interpret the -intercept and the rate of change. Duplicating an part of this book is prohibited b law. Step Step Find and interpret the initial value. The initial value is the value of when is equal to 0. It is the -intercept. The -intercept would be at (0, 0), but the dot at that point is open. The open dot means that (0, 0) is not part of the solution. This makes sense because no one would rent a limousine for 0 hours and pa $0. However, the fact that the cost is $0 if the limousine is rented for 0 hours indicates that $0 is the set fee for a limousine rental before an hourl charges are added, or the initial value. Find and interpret the rate of change. Let (, ) (0, 0). Let (, ) (, 0). rate of change Since the -ais shows number of hours and the -ais shows total cost in dollars, the rate of change is $0, or $0 per hour. hour This is the hourl rate charged for a limousine rental. 0 9

21 Step Solution Determine the cost of renting a limousine for 8 hours. The -ais of the graph does not show 8, so write an equation. The cost,, is a $0 fee plus $0 per hour for hours, so: 0 0 Substitute 8 for and find the value of. 0 0(8) The initial value, or set fee, is $0. The rate of change is $0 per hour. Renting a limousine for 8 hours costs $0. Coached Eample Students have dining cards at a boarding school. Each time a student gets a meal at the dining hall, points are deducted from his or her dining card. Teisha s dining card had a value of 0 points at the beginning of the semester. If her card now has 70 points left on it, how man meals has she eaten at the dining hall? Translate the words into an equation. Let represent the number of meals she has eaten. Let represent the total amount left on the card. value of 0 points each meal, points are deducted total amount left on card Substitute 70 for and solve for Subtract from both sides. Divide both sides b. If Teisha has 70 points left on her card, she has eaten meals at the dining hall. Duplicating an part of this book is prohibited b law. 0 Domain : Functions

22 Lesson : Use Functions to Solve Problems Lesson Practice Choose the correct answer. Use the table for questions and. An equilateral triangle has sides equal in length. The relationship between a side length of an equilateral triangle, s, and its perimeter, P, can be modeled b the function P s. Side Length (s) Perimeter (P) 9 8?? 8. What is the perimeter of an equilateral triangle whose side length is 8 units? A. units B. units C. units D. units. Ashton earns etra mone b doing odd jobs for his neighbors. He charges a flat fee of $ plus $8 per hour for each job. If he earned $7 for a job he did last week, how man hours did he work? A. C. B.. D.. An online store sells T-shirts for $ each. The store charges a $9 shipping and handling fee no matter how man shirts a customer orders. Which equation best represents, the total cost in dollars, of buing T-shirts from this online store? A. 9 B. 9 C. 9 D. 9 Duplicating an part of this book is prohibited b law.. What is the side length of an equilateral triangle whose perimeter is 8 units? A. 0 units B. units C. units D. units. Mrs. Ames uses a QuickPass device to pa bridge tolls when she drives. Each time she crosses a local toll bridge, $ is automaticall deducted from her QuickPass account. At the beginning of the month, she had a balance of $0 on her QuickPass account. If her current balance is $, how man times has she crossed the bridge since the beginning of the month? A. B. 8 C. D. 08

23 Use the graph for questions and 7. An accountant charges a set fee to complete a client s ta return, plus an additional rate for each hour she works on the return, as shown b the graph below. Total Charge (in dollars) Accountant s Charges Number of Hours. What does the slope of the graph represent? A. the accountant s hourl rate, $ per hour B. the accountant s hourl rate, $7 per hour C. the set fee charged to complete a return, $80 D. the set fee to complete an ta return, $00 7. What is the initial value? A. $0 B. $80 C. $0 D. $0 8. For field trips, a museum charges a flat fee plus an additional rate for each student. The museum uses the equation of a linear function to determine, the total cost in dollars, if students attend the field trip. The table below shows a partial representation of this function. Museum Field Trip Costs Number of Students () Total Cost in Dollars () A. What is the rate of change shown b the table and what does it represent in the problem? Show our work and eplain our answer. B. The initial -value in the table is. Eplain wh it is more appropriate for the table to start at instead of 0. Duplicating an part of this book is prohibited b law. Domain : Functions

24 Domain Lesson Use Graphs to Describe Relationships Common Core State Standard: 8.F. Getting the Idea To represent some real-world situations, ou ma need to break a graph into pieces to show a sequence of events. It ma not be as simple as drawing or interpreting a straight line. Eample On Thursda, Maksim went for a long nature walk, stopping for lunch at one point. The graph below represents his walk. Maksim s Walk Total Distance (in miles) Number of Hours Describe what Maksim did during each interval shown. Strateg Look at the graph piece b piece. Duplicating an part of this book is prohibited b law. Step Look at the first piece of the graph. The first piece is a line segment slanting up from (0, 0) to (, ). Since the -ais shows time, in hours, and the -ais shows total distance, in miles, this segment shows that Maksim walked miles during the first hours. Find and interpret the rate of change for this segment. rate of change miles 0 hours 0 So, Maksim walked at a speed of miles per hour for the first hours.

25 Step Step Solution Look at the second piece of the graph. The second piece is a horizontal line segment from (, ) to (, ). The rate of change for a horizontal segment is 0. So, Maksim walked no additional distance during that hour. That is probabl when he stopped for lunch. Look at the third piece of the graph. The third piece is a line segment slanting up from (, ) to (, )., so Maksim walked more miles during the last hours of his walk. rate of change miles hours. So, Maksim walked at a speed of. miles per hour for the last hours. The graph shows that Maksim walked at a speed of miles per hour during the first hours, stopped for lunch between hours and, and walked at a slightl slower speed of. miles per hour between hours and. Be careful when ou interpret the meaning of the term constant. If a graph is increasing at a constant rate, it is represented b a line segment that slants up. If a graph is decreasing at a constant rate, it is represented b a line segment that slants down. If a piece of a graph is constant, it is represented b a horizontal line segment. Here, constant means that it is neither increasing nor decreasing. Duplicating an part of this book is prohibited b law. Domain : Functions

26 Lesson : Use Graphs to Describe Relationships Eample A function decreases at a constant rate from (, ) to (, ). It then increases at a constant rate from (, ) to (, ). Finall, it is constant from to. Sketch the graph. Strateg Step Step Step Graph the function piece b piece. Graph the first piece. Plot points at (, ) and (, ) on a coordinate grid. Since the graph decreases at a constant rate, draw a line segment to connect the points. Graph the second piece. Plot a point at (, ). Since the graph increases at a constant rate, draw a line segment to connect (, ) to (, ). Graph the final piece. Since the graph is constant from to, it will neither increase nor decrease during that interval. (, ) is on the graph, so plot a point with an -value of and the same -coordinate as (, ). That point is (, ). Connect the two points with a horizontal line segment. Duplicating an part of this book is prohibited b law. Solution 0 The graph in Step fits the verbal description of the function.

27 Coached Eample The graph below shows Mr. Kowalski s commute home. He used a combination of taking the bus and walking to get home. Number of Miles Mr. Kowalski s Commute Number of Minutes Use that information to describe each part of the graph. Decide which line segment is steeper. The line segment from (0, ) to (0, ) is than the line segment from (0, ) to (0, 0). The steeper line segment shows Mr. Kowalski moving toward home at a faster rate. Since a person travels faster on a bus than on foot, the line segment from (0, ) to (0, ) shows that Mr. Kowalski , so that segment represents minute(s) of his commute., so it represents a distance of mile(s) traveled. Look at the line segment from (0, ) to (0, 0). Does this line segment show Mr. Kowalski taking the bus or walking? 0 0, so that segment represents minute(s) of his commute. 0, so it represents a distance of mile(s) traveled. The graph shows that during his commute, Mr. Kowalski for the first minute(s) and traveled a distance of mile(s). He then for the net minute(s) and traveled a distance of mile(s). Duplicating an part of this book is prohibited b law. Domain : Functions 7

28 Lesson : Use Graphs to Describe Relationships Lesson Practice Choose the correct answer.. For which -values is the interval increasing? 0 A. from to 0 B. from 0 to C. from to D. The graph does not show an increasing interval.. The graph shows the distance that Po-Ting drove during a road trip. At one point during the trip, Po-Ting took a break at a rest stop. During which time period did that occur? Total Distance (in miles) Po-Ting s Road Trip Time (in hours) A. during the first hours B. between hours and C. between hours and D. between hours and Duplicating an part of this book is prohibited b law.. Cara went roller skating. The graph shows the distance she traveled to and from her home. Which best describes what is shown b the graph? A. Cara skated up a hill for 0 minutes, then along the flat top of the hill for 0 minutes, and then down the hill for 0 minutes. B. Cara skated down a hill for 0 minutes, then along the flat top of the hill for 0 minutes, and then up the hill for 0 minutes. C. Cara skated toward her home for 0 minutes, then staed at her home for 0 minutes, and then skated awa from her home for 0 minutes. Distance from House (in miles) D. Cara skated awa from her home for 0 minutes, then took a break for 0 minutes, and then skated toward her home for 0 minutes. 7 0 Cara s Skating Time (in minutes) 7 8

29 . Which statement is true of the interval from to? 0 A. That piece of the graph is nonlinear. C. That piece of the graph is decreasing. B. That piece of the graph is linear. D. That piece of the graph is constant.. Aaron traveled to his grandfather s house. He traveled at a constant rate for the first 0 minutes, for a distance of mile. Then he stopped for 0 minutes to have a snack. Finall, he traveled at a constant rate over the net 0 minutes for a distance of 8 miles. A. Create a graph on the grid below to show Aaron s trip to his grandfather s house. B. For one part of his trip (either before or after his snack), Aaron rode his biccle. For another part, he walked while pushing his biccle. During which part did he ride his biccle and during which part did he walk? Eplain how ou determined our answer. Duplicating an part of this book is prohibited b law. 8 Domain : Functions 9

30 Domain Lesson Compare Relationships Represented in Different Was Common Core State Standards: 8.EE., 8.F. Getting the Idea You alread know that functions can be represented in different was as a set of ordered pairs, in a table, with a verbal or algebraic rule, as an equation, or b a graph. Sometimes, ou ma need to compare two different functions represented in different was. Eample A brother and sister are racing 0 meters to a tree. Since Justin is ounger, his sister Cami lets him have a -meter head start. The graph below shows the distance that Justin runs during the race. Duplicating an part of this book is prohibited b law. Distance (in meters) Justin s Race 8 0 Number of Seconds The equation can be used to represent, the total distance in meters that Cami has run after seconds have passed. Who is running at a faster speed? How much faster? Strateg Step Compare the rate of change shown in the graph to the rate of change shown b the equation. Find Justin s speed. The -ais shows time, in seconds, and the -ais shows distance, in meters. So, the rate of change for the graph compares seconds to meters. Use the points (0, ) and (, 0) to find the rate of change. rate of change seconds 0 meters 0 So, Justin s speed is meters per second. 0 9

31 Step Step Solution Find Cami s speed. The equation is in the form m. Since m, the rate of change is meters per second. Find the person running at a faster speed. How much faster? meters per second. meters per second, and. So, Cami is running meter per second faster than Justin. Cami s speed is meter per second faster than Justin s speed. Eample Compare the rates of change for the two linear functions described below. Function : An -value can be found using the epression. Function : 0.. Which function has a greater rate of change, or are the the same? Strateg Identif the rate of change for each function. Then compare them. Step Find the rate of change for Function. An -value can be found using the epression, so find two ordered pairs for the function. If 0, then the -value is (0), or. If, then the -value is (), or. Use the points (0, ) and (, ) to find the rate of change. rate of change 0 Note: You might notice that this function could also be represented as. In that case, ou can see that the rate of change is because m. Step Find the rate of change for Function. Step Use the points (0, ) and (, ) to find the rate of change. rate of change 0 Compare the rates of change.., so Function has the greater rate of change. Solution The rate of change for Function is greater than for Function. Duplicating an part of this book is prohibited b law. 70 Domain : Functions

32 Lesson : Compare Relationships Represented in Different Was Eample The price of buing baseball caps at Sport s is shown b the graph below. The advertisement shows the cost of buing the same baseball caps at Hats R Us. Brand B Caps at Sport s Cost (in dollars) Hats R Us Brand B Caps $0 for the first cap! $ for each cap after that! 0 Number of Caps When does it make sense to bu caps at Sport s? at Hats R Us? Strateg Step List ordered pairs for each store and compare them. List ordered pairs for each store. It ma help to list the pairs side b side in a table. The ordered pairs for Sport s can be found b looking at the graph. To find the ordered pairs for Hats R Us, use the words in the advertisement. The cost is $0 for the first cap and $ for each cap after that. Duplicating an part of this book is prohibited b law. Number of Caps () Price at Sport s () Price at Hats R Us () $7.0 $0.00 $.00 $0 $ $.00 $.0 $0 $() $.00 $0.00 $0 $() $8.00 7

33 Step Solution Compare the -values, or prices, in the table. $7.0, $0.00, so buing cap is cheaper at Sport s. $.00, $.00, so buing caps is also cheaper at Sport s. $.0. $.00, so buing caps is cheaper at Hats R Us. The price of buing caps continues to be cheaper at Hats R Us for an number of caps over. If a customer wants to bu or caps, it is cheaper to go to Sport s. If a customer wants or more caps, Hats R Us is the better choice. Coached Eample Compare the rates of change for the two linear functions represented below. Function Function 0 Which function has a greater rate of change, or are the the same? Find the rate of change for Function. Use the points (0, ) and (, 0). m 0 0 Find the rate of change for Function. Use the points (, ) and (, ). m Which function has a greater rate of change, or are the the same? Function has a rate of change of and Function has a rate of change of, so. Duplicating an part of this book is prohibited b law. 7 Domain : Functions

34 Lesson : Compare Relationships Represented in Different Was Lesson Practice Choose the correct answer.. Compare the rates of change for the two functions represented below. For Function, an -value can be found using the rule: multipl b and then add. The graph below represents Function. Function. Compare the rates of change for the two linear functions represented below. Function Function 0 0 Duplicating an part of this book is prohibited b law. Which statement about the rates of change for the two functions is true? A. Function has a greater rate of change than Function. B. Function and Function have the same rate of change. C. Function has a negative rate of change, and Function has a positive rate of change. D. Function has a positive rate of change, and Function has a negative rate of change. Which statement about the rates of change for the two functions is true? A. Function has a greater rate of change than Function. B. Function has a greater rate of change than Function. C. Function and Function have the same rate of change. D. Function does not have a constant rate of change, but Function does. 7

35 . Two trucks are driving on the same highwa at the same time. Truck is 0 miles from Smithtown. Its distance from Smithtown, in miles, after hours can be found using the epression 0. The distance of Truck from Smithtown is represented b the graph below. Number of Miles Distance of Truck From Smithtown Number of Hours Eactl how man hours will pass before the two trucks are each the same distance from Smithtown? A. C. B. D.. Michelle planted two plants. After each plant had grown a little, she began using them for a science eperiment. The table below shows the growth of Plant over several das. Number of Das () Height in Centimeters () The equation. represents, the height in centimeters, of Plant over das. Which statement accuratel compares the growth of the plants? A. Plant is growing at a faster rate than Plant. B. Plant is growing at a faster rate than Plant. C. Plant and Plant are both growing at the same rate. D. Plant was growing at a faster rate than Plant at first, but then Plant began to grow at a faster rate.. Two brothers, Leo and William, are racing their bikes around a 9.-kilometer loop. Since Leo is ounger, he is given a.-kilometer head start. The graph below shows Leo s progress. The epression 9 can be used to determine, the distance in kilometers that William has traveled after hours have passed. Which statement about this situation is not true? A. William is traveling at a faster speed than Leo. B. The difference between their speeds is eactl. kilometers per hour. C. Leo traveled onl 8 kilometers during the race. D. Both brothers will finish the race in _ hour. Distance (in miles) Leo s Biking 0. Number of Hours Duplicating an part of this book is prohibited b law. 7 Domain : Functions

36 Lesson : Compare Relationships Represented in Different Was. The price of buing pairs of Brand B socks at Shoewa is shown b the graph below. The advertisement shows the cost of buing several pairs of the same brand of socks at Foot World. Shoewa Brand B Socks Cost (in dollars) Number of Pairs Foot World Bu pair of Brand B socks for $.00. Pa $.0 for each pair after that. A. Determine the costs of buing,,, and pairs of socks at each store. Show our work. Duplicating an part of this book is prohibited b law. B. Eplain when a customer should bu socks at Shoewa and when a customer should bu socks at Foot World. 7

37 LESSON 8 Quiz Write not a function, linear function, or nonlinear function to describe each relationship Choose the best answer.. Which of the following is not true of? A. It is a function. B. It has a slope of. C. It has a -intercept of. D. Its graph is a line. Graph the line A plumber charges $ plus $0 per hour for a repair job. Which function gives c, the cost of a job lasting h hours? A. c 0h B. c h 0 C. c h 0 D. c 0h 8. 0 Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC 8 7

38 Lesson 8 Quiz Each situation can be modeled b a linear function. Find the rate of change and initial value. 9. A scuba diver was 0 meters below the lake s surface. She then descended toward the bottom of the lake at a rate of meters per second. Rate of change 0. An empt rain gauge fills with water at a rate of 00 millimeters per hour. Rate of change Initial value Lesson Quizzes Initial value Write the equation for the linear function from the indicated problem.. A room with 0 people emptied at a rate of 7 people per minute.. An accountant charges $7 plus $0 per hour to prepare a ta return.. A group of friends started a chess club at school. After months, there were members. After 7 months, there were members.. A local hot dog-eating champion ate hot dogs per minute after warming up with hot dogs. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Solve.. Song Hee walked to the librar from her house. After walking minutes, she was,70 feet from the librar. After walking a total of minutes, she was,700 feet from the librar. How man feet per minute does Song Hee walk? How far does she live from the librar? How far will she be from the librar after walking 0 minutes? 8 9

39 LESSON 9 Quiz Use the graph to describe the functional relationship. Include what the - and -coordinates and intercepts represent. Determine if the graph is linear or nonlinear, and what the slope of each linear segment represents.. Celia made a graph to show the total number of miles she ran while practicing for weeks with the track team. Miles Track Team Practice 0 7 Time (in weeks). Mrs. Jackson bought a refrigerator b making a down pament, and each month she paid a set amount to the store. She made a graph to show her debt for the refrigerator over time. Amount Owed (in dollars) Refrigerator Debt,00,00, Time (in months) Sketch a graph that represents the functional relationship. Give a title and label the aes.. When Minkoung took a loaf of bread out of the oven, it was 0 F. It then cooled at a rate of per minute for minutes until it reached room temperature, 7 F. Once it reached room temperature, it remained at that temperature Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC 0 9

40 Lesson 9 Quiz Use the four functions below for questions and. Choose the correct answer. Function : The output of a function is equal to less than the input multiplied b. Function : 0 Function : Function : Lesson Quizzes. Which function has the greatest rate of change? A. Function B. Function C. Function D. Function. Which function has the greatest -intercept? A. Function B. Function C. Function D. Function Solve. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC. Ali and Michael opened bank accounts on the same da. The amount of mone in Ali s account, in dollars, is described b the equation 0 0, where is the number of months since he opened the account. Michael made the table below showing his monthl balance. Month 0 Balance $7 $00 $ $0 $7 Who made a larger initial deposit? Who contributed more mone each month? 0