3 Graphing Linear Functions

Transcription

1 Graphing Linear Functions. Functions. Linear Functions. Function Notation. Graphing Linear Equations in Standard Form.5 Graphing Linear Equations in Slope-Intercept Form. Transformations of Graphs of Linear Functions.7 Graphing Absolute Value Functions Submersible (p. ) SEE the Big Idea Basketball (p. ) Speed of Light (p. 5) Coins (p. ) Tai Ride (p. 9)

2 Maintaining Mathematical Proficienc Plotting Points Eample Plot the point A(, ) in a coordinate plane. Describe the location of the point. Start at the origin. Move units left and units up. Then plot the point. The point is in Quadrant II. A(, ) Plot the point in a coordinate plane. Describe the location of the point.. A(, ). B( 5, ). C(, ). D(, ) 5. E(, ). F(, ) Evaluating Epressions Eample Evaluate 5 when =. 5 = () 5 Substitute for. = 5 Multipl. = 7 Subtract. Eample Evaluate + 9 when = = ( 8) + 9 Substitute 8 for. = + 9 Multipl. = 5 Add. Evaluate the epression for the given value of. 7. ; = ; = ; = 5. 9 ; =. 8; = ; =. ABSTRACT REASONING Let a and b be positive real numbers. Describe how to plot (a, b), ( a, b), (a, b), and ( a, b). Dnamic Solutions available at BigIdeasMath.com

3 , Mathematical Practices Mathematicall profi cient students use technological tools to eplore concepts. Using a Graphing Calculator Core Concept Standard and Square Viewing Windows A tpical graphing calculator screen has a height to width ratio of to. This means that when ou use the standard viewing window of to (on each ais), the graph will not be in its true perspective. WINDOW Xmin=- Xma= Xscl= Ymin=- Yma= Yscl= This is the standard viewing window. To see a graph in its true perspective, ou need to use a square viewing window, in which the tick marks on the -ais are spaced the same as the tick marks on the -ais. WINDOW Xmin=-9 Xma=9 Xscl= Ymin=- Yma= Yscl= This is a square viewing window. Using a Graphing Calculator Use a graphing calculator to graph = + 5. SOLUTION Enter the equation = + 5 into our calculator. Then graph the equation. The standard viewing window does not show the graph in its true perspective. Notice that the tick marks on the -ais are closer together than the tick marks on the -ais. To see the graph in its true perspective, use a square viewing window. = + 5 = + 5 This is the graph in the standard viewing window. 7 This is the graph in a square viewing window. 7 5 Monitoring Progress Determine whether the viewing window is square. Eplain.. 8 7, , Use a graphing calculator to graph the equation. Use a square viewing window.. = + 5. =. = 7. = + 8. = 9. = +. How does the appearance of the slope of a line change between a standard viewing window and a square viewing window? Chapter Graphing Linear Functions

4 . Functions Essential Question What is a function? A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation that pairs each input with eactl one output is a function. Describing a Function Work with a partner. Functions can be described in man was. ANALYZING RELATIONSHIPS To be proficient in math, ou need to analze relationships mathematicall to draw conclusions. b an equation b an input-output table using words b a graph as a set of ordered pairs a. Eplain wh the graph shown represents a function. b. Describe the function in two other was. Identifing Functions Work with a partner. Determine whether each relation represents a function. Eplain our reasoning. 8 8 a. Input, Output, b. Input, Output, c. Input, Output, d e. (, 5), (, 8), (, ), (, ), (, 7) f. (, ), (, ), (, ), (, ), (, ), (, ) g. Each radio frequenc in a listening area has eactl one radio station. h. The same television station can be found on more than one channel. i. = j. = + Communicate Your Answer. What is a function? Give eamples of relations, other than those in Eplorations and, that (a) are functions and (b) are not functions. Section. Functions

5 . Lesson Core Vocabular relation, p. function, p. domain, p. range, p. independent variable, p. 7 dependent variable, p. 7 Previous ordered pair mapping diagram What You Will Learn Determine whether relations are functions. Find the domain and range of a function. Identif the independent and dependent variables of functions. Determining Whether Relations Are Functions A relation pairs inputs with outputs. When a relation is given as ordered pairs, the -coordinates are inputs and the -coordinates are outputs. A relation that pairs each input with eactl one output is a function. Determining Whether Relations Are Functions Determine whether each relation is a function. Eplain. a. (, ), (, ), (, ), (, ), (, ) b. (, ), (8, 7), (, ), (, ), (5, ) c. Input, REMEMBER A relation can be represented b a mapping diagram. d. Output, Input, Output, 5 SOLUTION a. Ever input has eactl one output. So, the relation is a function. b. The input has two outputs, and. So, the relation is not a function. c. The input has two outputs, 5 and. So, the relation is not a function. d. Ever input has eactl one output. So, the relation is a function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the relation is a function. Eplain.. ( 5, ), (, ), (5, ), (5, ). (, 8), (, ), (, ), (5, ). Input, Output,. Input, Output, Chapter Graphing Linear Functions

6 Core Concept Vertical Line Test Words A graph represents a function when no vertical line passes through more than one point on the graph. Eamples Function Not a function Using the Vertical Line Test Determine whether each graph represents a function. Eplain. a. b. SOLUTION a. You can draw a vertical line through (, ) and (, 5). So, the graph does not represent a function. b. No vertical line can be drawn through more than one point on the graph. So, the graph represents a function. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Determine whether the graph represents a function. Eplain Section. Functions 5

7 Finding the Domain and Range of a Function Core Concept The Domain and Range of a Function The domain of a function is the set of all possible input values. The range of a function is the set of all possible output values. input output Finding the Domain and Range from a Graph Find the domain and range of the function represented b the graph. a. b. STUDY TIP A relation also has a domain and a range. SOLUTION a. Write the ordered pairs. Identif the inputs and outputs. inputs (, ), (, ), (, ), (, ) outputs The domain is,,, and. The range is,,, and. b. Identif the - and -values represented b the graph. domain range The domain is. The range is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the domain and range of the function represented b the graph. 9.. Chapter Graphing Linear Functions

8 Identifing Independent and Dependent Variables The variable that represents the input values of a function is the independent variable because it can be an value in the domain. The variable that represents the output values of a function is the dependent variable because it depends on the value of the independent variable. When an equation represents a function, the dependent variable is defined in terms of the independent variable. The statement is a function of means that varies depending on the value of. = + dependent variable, independent variable, Identifing Independent and Dependent Variables The function = + represents the amount (in fluid ounces) of juice remaining in a bottle after ou take gulps. a. Identif the independent and dependent variables. b. The domain is,,,, and. What is the range? SOLUTION a. The amount of juice remaining depends on the number of gulps. So, is the dependent variable, and is the independent variable. b. Make an input-output table to find the range. Input, + Output, () + () + 9 () + () + () + The range is, 9,,, and. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. The function a = b + represents the number a of avocados ou have left after making b batches of guacamole. a. Identif the independent and dependent variables. b. The domain is,,, and. What is the range?. The function t = 9m + 5 represents the temperature t (in degrees Fahrenheit) of an oven after preheating for m minutes. a. Identif the independent and dependent variables. b. A recipe calls for an oven temperature of 5 F. Describe the domain and range of the function. Section. Functions 7

9 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING How are independent variables and dependent variables different?. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Find the range of the function represented b the table. Find the inputs of the function represented b the table. 7 5 Find the -values of the function represented b (, 7), (, 5), and (, ). Find the domain of the function represented b (, 7), (, 5), and (, ). Monitoring Progress and Modeling with Mathematics In Eercises 8, determine whether the relation is a function. Eplain. (See Eample.). (, ), (, ), (, ), (, ), (5, )... (7, ), (5, ), (, 8), (, 5), (, ) 5. Input, Output,. Input, Output, In Eercises, find the domain and range of the function represented b the graph. (See Eample.).. 7. Input, Output, 8. Input, 9 Output, In Eercises 9, determine whether the graph represents a function. Eplain. (See Eample.) MODELING WITH MATHEMATICS The function = represents our monthl rent (in dollars) when ou pa das late. (See Eample.) a. Identif the independent and dependent variables. b. The domain is,,,,, and 5. What is the range? 8 Chapter Graphing Linear Functions

10 8. MODELING WITH MATHEMATICS The function = represents the cost (in dollars) of a tai ride of miles.. MULTIPLE REPRESENTATIONS The function = represents the number of hardcover books and softcover books ou can bu at a used book sale. a. Solve the equation for. b. Make an input-output table to find ordered pairs for the function. c. Plot the ordered pairs in a coordinate plane. 5. ATTENDING TO PRECISION The graph represents a function. Find the input value corresponding to an output of. a. Identif the independent and dependent variables. b. You have enough mone to travel at most miles in the tai. Find the domain and range of the function. ERROR ANALYSIS In Eercises 9 and, describe and correct the error in the statement about the relation shown in the table.. OPEN-ENDED Fill in the table so that when t is the 9.. Input, 5 Output, The relation is not a function. One output is paired with two inputs. independent variable, the relation is a function, and when t is the dependent variable, the relation is not a function. t v 7. ANALYZING RELATIONSHIPS You select items in a vending machine b pressing one letter and then one number. The relation is a function. The range is,,,, and 5. ANALYZING RELATIONSHIPS In Eercises and, identif the independent and dependent variables.. The number of quarters ou put into a parking meter affects the amount of time ou have on the meter.. The batter power remaining on our MP plaer is based on the amount of time ou listen to it.. MULTIPLE REPRESENTATIONS The balance (in dollars) of our savings account is a function of the month. Month, Balance (dollars), a. Describe this situation in words. a. Eplain wh the relation that pairs letter-number combinations with food or drink items is a function. b. Write the function as a set of ordered pairs. b. Identif the independent and dependent variables. c. Plot the ordered pairs in a coordinate plane. c. Find the domain and range of the function. Section. hsnb_alg_pe_.indd 9 Functions 9 //5 :8 PM

11 8. HOW DO YOU SEE IT? The graph represents the height h of a projectile after t seconds. Height (feet) Height of a Projectile 5 h t Time (seconds) a. Eplain wh h is a function of t. b. Approimate the height of the projectile after.5 second and after.5 seconds. c. Approimate the domain of the function. d. Is t a function of h? Eplain.. A function pairs each chaperone on a school trip with students. REASONING In Eercises 5 8, tell whether the statement is true or false. If it is false, eplain wh. 5. Ever function is a relation.. Ever relation is a function. 7. When ou switch the inputs and outputs of an function, the resulting relation is a function. 8. When the domain of a function has an infinite number of values, the range alwas has an infinite number of values. 9. MATHEMATICAL CONNECTIONS Consider the triangle shown. 9. MAKING AN ARGUMENT Your friend sas that a line alwas represents a function. Is our friend correct? Eplain. h. THOUGHT PROVOKING Write a function in which the inputs and/or the outputs are not numbers. Identif the independent and dependent variables. Then find the domain and range of the function. ATTENDING TO PRECISION In Eercises, determine whether the statement uses the word function in a wa that is mathematicall correct. Eplain our reasoning.. The selling price of an item is a function of the cost of making the item.. The sales ta on a purchased item in a given state is a function of the selling price.. A function pairs each student in our school with a homeroom teacher. Maintaining Mathematical Proficienc Write the sentence as an inequalit. (Section.) a. Write a function that represents the perimeter of the triangle. b. Identif the independent and dependent variables. c. Describe the domain and range of the function. (Hint: The sum of the lengths of an two sides of a triangle is greater than the length of the remaining side.) REASONING In Eercises, find the domain and range of the function.. =. =. =. =. A number is less than. 5. Three is no less than a number.. Seven is at most the quotient of a number d and The sum of a number w and is more than. Reviewing what ou learned in previous grades and lessons Evaluate the epression. (Skills Review Handbook) ( ) Chapter Graphing Linear Functions

12 . Linear Functions Essential Question How can ou determine whether a function is linear or nonlinear? Finding Patterns for Similar Figures Work with a partner. Cop and complete each table for the sequence of similar figures. (In parts (a) and (b), use the rectangle shown.) Graph the data in each table. Decide whether each pattern is linear or nonlinear. Justif our conclusion. a. perimeters of similar rectangles b. areas of similar rectangles 5 P P 5 A A 8 8 c. circumferences of circles of radius r d. areas of circles of radius r USING TOOLS STRATEGICALLY To be proficient in math, ou need to identif relationships using tools, such as tables and graphs. r 5 C C r 5 A A 8 8 r 8 r Communicate Your Answer. How do ou know that the patterns ou found in Eploration represent functions?. How can ou determine whether a function is linear or nonlinear?. Describe two real-life patterns: one that is linear and one that is nonlinear. Use patterns that are different from those described in Eploration. Section. Linear Functions

13 . Lesson Core Vocabular linear equation in two variables, p. linear function, p. nonlinear function, p. solution of a linear equation in two variables, p. discrete domain, p. continuous domain, p. Previous whole number What You Will Learn Identif linear functions using graphs, tables, and equations. Graph linear functions using discrete and continuous data. Write real-life problems to fit data. Identifing Linear Functions A linear equation in two variables, and, is an equation that can be written in the form = m + b, where m and b are constants. The graph of a linear equation is a line. Likewise, a linear function is a function whose graph is a nonvertical line. A linear function has a constant rate of change and can be represented b a linear equation in two variables. A nonlinear function does not have a constant rate of change. So, its graph is not a line. Identifing Linear Functions Using Graphs Does the graph represent a linear or nonlinear function? Eplain. a. b. SOLUTION a. The graph is not a line. b. The graph is a line. So, the function is nonlinear. So, the function is linear. Identifing Linear Functions Using Tables Does the table represent a linear or nonlinear function? Eplain. a. 9 b REMEMBER A constant rate of change describes a quantit that changes b equal amounts over equal intervals. SOLUTION a As increases b, decreases b. The rate of change is constant. So, the function is linear. b As increases b, increases b different amounts. The rate of change is not constant. So, the function is nonlinear. Chapter Graphing Linear Functions

14 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Does the graph or table represent a linear or nonlinear function? Eplain Identifing Linear Functions Using Equations Which of the following equations represent linear functions? Eplain. =.8, =, =, =, = ( ), and = SOLUTION You cannot rewrite the equations =, =, =, and = in the form = m + b. So, these equations cannot represent linear functions. You can rewrite the equation =.8 as = +.8 and the equation = ( ) as =. So, the represent linear functions. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Does the equation represent a linear or nonlinear function? Eplain. 5. = + 9. = 7. = 5 5 Concept Summar Representations of Functions Words An output is more than the input. Equation = + Input-Output Table Mapping Diagram Graph Input, Output, Input, Output, 5 5 Section. Linear Functions

15 Graphing Linear Functions A solution of a linear equation in two variables is an ordered pair (, ) that makes the equation true. The graph of a linear equation in two variables is the set of points (, ) in a coordinate plane that represents all solutions of the equation. Sometimes the points are distinct, and other times the points are connected. Core Concept Discrete and Continuous Domains A discrete domain is a set of input values that consists of onl certain numbers in an interval. Eample: Integers from to 5 5 A continuous domain is a set of input values that consists of all numbers in an interval. Eample: All numbers from to 5 5 Graphing Discrete Data STUDY TIP The domain of a function depends on the real-life contet of the function, not just the equation that represents the function. Cost (dollars) Museum Tickets 7 (,.8) 5 (, 7.85) (,.9) (, 5.95) (, ) 5 Number of tickets The linear function = 5.95 represents the cost (in dollars) of tickets for a museum. Each customer can bu a maimum of four tickets. a. Find the domain of the function. Is the domain discrete or continuous? Eplain. b. Graph the function using its domain. SOLUTION a. You cannot bu part of a ticket, onl a certain number of tickets. Because represents the number of tickets, it must be a whole number. The maimum number of tickets a customer can bu is four. So, the domain is,,,, and, and it is discrete. b. Step Make an input-output table to find the ordered pairs. Input, 5.95 Output, (, ) 5.95() (, ) 5.95() 5.95 (, 5.95) 5.95().9 (,.9) 5.95() 7.85 (, 7.85) 5.95().8 (,.8) Step Plot the ordered pairs. The domain is discrete. So, the graph consists of individual points. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. The linear function m = 5 9d represents the amount m (in dollars) of mone ou have after buing d DVDs. (a) Find the domain of the function. Is the domain discrete or continuous? Eplain. (b) Graph the function using its domain. Chapter Graphing Linear Functions

16 Graphing Continuous Data STUDY TIP When the domain of a linear function is not specified or cannot be obtained from a real-life contet, it is understood to be all real numbers. A cereal bar contains calories. The number c of calories consumed is a function of the number b of bars eaten. a. Does this situation represent a linear function? Eplain. b. Find the domain of the function. Is the domain discrete or continuous? Eplain. c. Graph the function using its domain. SOLUTION a. As b increases b, c increases b. The rate of change is constant. So, this situation represents a linear function. b. You can eat part of a cereal bar. The number b of bars eaten can be an value greater than or equal to. So, the domain is b, and it is continuous. c. Step Make an input-output table to find ordered pairs. Calories consumed Cereal Bar Calories c 7 (, 5) 5 (, 9) (, ) (, ) (, ) 5 b Number of bars eaten Input, b Output, c (b, c) (, ) (, ) (, ) 9 (, 9) 5 (, 5) Step Plot the ordered pairs. Step Draw a line through the points. The line should start at (, ) and continue to the right. Use an arrow to indicate that the line continues without end, as shown. The domain is continuous. So, the graph is a line with a domain of b. Monitoring Progress 9. Is the domain discrete or continuous? Eplain. Help in English and Spanish at BigIdeasMath.com Input Number of stories, Output Height of building (feet),. A -gallon bathtub is draining at a rate of.5 gallons per minute. The number g of gallons remaining is a function of the number m of minutes. a. Does this situation represent a linear function? Eplain. b. Find the domain of the function. Is the domain discrete or continuous? Eplain. c. Graph the function using its domain. Section. Linear Functions 5

17 Writing Real-Life Problems Writing Real-Life Problems Write a real-life problem to fit the data shown in each graph. Is the domain of each function discrete or continuous? Eplain. a. 8 b SOLUTION a. You want to think of a real-life situation in which there are two variables, and. Using the graph, notice that the sum of the variables is alwas, and the value of each variable must be a whole number from to. 5 5 Discrete domain One possibilit is two people bidding against each other on si coins at an auction. Each coin will be purchased b one of the two people. Because it is not possible to purchase part of a coin, the domain is discrete. b. You want to think of a real-life situation in which there are two variables, and. Using the graph, notice that the sum of the variables is alwas, and the value of each variable can be an real number from to. + = or = + Continuous domain One possibilit is two people bidding against each other on ounces of gold dust at an auction. All the dust will be purchased b the two people. Because it is possible to purchase an portion of the dust, the domain is continuous. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Write a real-life problem to fit the data shown in the graph. Is the domain of the function discrete or continuous? Eplain Chapter Graphing Linear Functions

18 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE A linear equation in two variables is an equation that can be written in the form, where m and b are constants.. VOCABULARY Compare linear functions and nonlinear functions.. VOCABULARY Compare discrete domains and continuous domains.. WRITING How can ou tell whether a graph shows a discrete domain or a continuous domain? Monitoring Progress and Modeling with Mathematics In Eercises 5, determine whether the graph represents a linear or nonlinear function. Eplain. (See Eample.) ERROR ANALYSIS In Eercises 5 and, describe and correct the error in determining whether the table or graph represents a linear function As increases b, increases b a constant factor of. So, the function is linear. In Eercises, determine whether the table represents a linear or nonlinear function. Eplain. (See Eample.) The graph is a line. So, the graph represents a linear function. 9 Section. Linear Functions 7

19 In Eercises 7, determine whether the equation represents a linear or nonlinear function. Eplain. (See Eample.) 7. = + 8. = 7. Input Time (hours), Output Distance (miles), = 8. = (8 ). + = +. =. 8 =. + = 9. Input Rela teams, Output Athletes, 8 5. CLASSIFYING FUNCTIONS Which of the following equations do not represent linear functions? Eplain. A = + B + = C = 8 D = 9 E = 5 F = + ERROR ANALYSIS In Eercises and, describe and correct the error in the statement about the domain... USING STRUCTURE Fill in the table so it represents a linear function is in the domain. In Eercises 7 and 8, find the domain of the function represented b the graph. Determine whether the domain is discrete or continuous. Eplain The graph ends at =, so the domain is discrete. 8 8 In Eercises 9, determine whether the domain is discrete or continuous. Eplain. 9. Input Bags, Output Marbles, 5. MODELING WITH MATHEMATICS The linear function m = b represents the amount m (in dollars) of mone that ou have after buing b books. (See Eample.) a. Find the domain of the function. Is the domain discrete or continuous? Eplain. b. Graph the function using its domain.. Input Years, Output Height of tree (feet), 9 8 Chapter Graphing Linear Functions

20 . MODELING WITH MATHEMATICS The number of calories burned after hours of rock climbing is represented b the linear function = 5. a. Find the domain of the function. Is the domain discrete or continuous? Eplain. b. Graph the function using its domain. WRITING In Eercises 9, write a real-life problem to fit the data shown in the graph. Determine whether the domain of the function is discrete or continuous. Eplain. (See Eample.) MODELING WITH MATHEMATICS You are researching the speed of sound waves in dr air at 8 F. The table shows the distances d (in miles) sound waves travel in t seconds. (See Eample 5.) Time (seconds), t Distance (miles), d a. Does this situation represent a linear function? Eplain. b. Find the domain of the function. Is the domain discrete or continuous? Eplain. c. Graph the function using its domain. 8. MODELING WITH MATHEMATICS The function = + 5 represents the cost (in dollars) of having our dog groomed and buing etra services. Pampered Pups Etra Grooming Services Paw Treatment Teeth Brushing Nail Polish Deshedding Ear Treatment a. Does this situation represent a linear function? Eplain. b. Find the domain of the function. Is the domain discrete or continuous? Eplain. c. Graph the function using its domain.. USING STRUCTURE The table shows our earnings (in dollars) for working hours. a. What is the missing -value that makes the table represent a linear function? b. What is our hourl pa rate?. MAKING AN ARGUMENT The linear function d = 5t represents the distance d (in miles) Car A is from a car rental store after t hours. The table shows the distances Car B is from the rental store. Time (hours), t Time (hours), Distance (miles), d 8 5 Earnings (dollars), a. Does the table represent a linear or nonlinear function? Eplain. b. Your friend claims Car B is moving at a faster rate. Is our friend correct? Eplain. Section. Linear Functions 9

21 MATHEMATICAL CONNECTIONS In Eercises 5 8, tell whether the volume of the solid is a linear or nonlinear function of the missing dimension(s). Eplain. 5. CLASSIFYING A FUNCTION Is the function represented b the ordered pairs linear or nonlinear? Eplain our reasoning. 5.. (, ), (, ), (5, ), (9, 8), (, ) s 7. cm s h 9 m 8. in. b 5 ft in. 9. REASONING A water compan fills two differentsized jugs. The first jug can hold gallons of water. The second jug can hold gallons of water. The compan fills A jugs of the first size and B jugs of the second size. What does each epression represent? Does each epression represent a set of discrete or continuous values? a. + b. A + B c. A d. A + B 5. THOUGHT PROVOKING You go to a farmer s market to bu tomatoes. Graph a function that represents the cost of buing tomatoes. Eplain our reasoning. r 5. HOW DO YOU SEE IT? You and our friend go running. The graph shows the distances ou and our friend run. Distance (miles) 5 Running Distance You Friend 5 Minutes a. Describe our run and our friend s run. Who runs at a constant rate? How do ou know? Wh might a person not run at a constant rate? b. Find the domain of each function. Describe the domains using the contet of the problem. WRITING In Eercises 5 and 5, describe a real-life situation for the constraints. 5. The function has at least one negative number in the domain. The domain is continuous. 5. The function gives at least one negative number as an output. The domain is discrete. Maintaining Mathematical Proficienc Tell whether and show direct variation. Eplain our reasoning Reviewing what ou learned in previous grades and lessons (Skills Review Handbook) 57. Evaluate the epression when =. (Skills Review Handbook) ( + 5) Chapter Graphing Linear Functions

22 . Function Notation Essential Question How can ou use function notation to represent a function? The notation f(), called function notation, is another name for. This notation is read as the value of f at or f of. The parentheses do not impl multiplication. You can use letters other than f to name a function. The letters g, h, j, and k are often used to name functions. Matching Functions with Their Graphs Work with a partner. Match each function with its graph. ATTENDING TO PRECISION To be proficient in math, ou need to use clear definitions and state the meanings of the smbols ou use. a. f () = b. g() = + c. h() = d. j() = A. B. C. D. Evaluating a Function Work with a partner. Consider the function f() = +. Locate the points (, f()) on the graph. Eplain how ou found each point. 5 f() = + a. (, f( )) b. (, f()) c. (, f()) d. (, f()) Communicate Your Answer. How can ou use function notation to represent a function? How are standard notation and function notation similar? How are the different? Standard Notation Function Notation = + 5 f() = + 5 Section. Function Notation

23 . Lesson Core Vocabular function notation, p. Previous linear function quadrant What You Will Learn Use function notation to evaluate and interpret functions. Use function notation to solve and graph functions. Solve real-life problems using function notation. Using Function Notation to Evaluate and Interpret You know that a linear function can be written in the form = m + b. B naming a linear function f, ou can also write the function using function notation. f() = m + b Function notation The notation f() is another name for. If f is a function, and is in its domain, then f() represents the output of f corresponding to the input. You can use letters other than f to name a function, such as g or h. READING The notation f() is read as the value of f at or f of. It does not mean f times. Evaluating a Function Evaluate f() = + 7 when = and =. SOLUTION f() = + 7 Write the function. f() = + 7 f() = () + 7 Substitute for. f( ) = ( ) + 7 = Multipl. = = Add. = 5 When =, f () =, and when =, f () = 5. Interpreting Function Notation Let f(t) be the outside temperature ( F) t hours after a.m. Eplain the meaning of each statement. a. f() = 58 b. f() = n c. f () < f(9) SOLUTION a. The initial value of the function is 58. So, the temperature at a.m. is 58 F. b. The output of f when t = is n. So, the temperature at noon ( hours after a.m.) is n F. c. The output of f when t = is less than the output of f when t = 9. So, the temperature at 9 a.m. ( hours after A.M.) is less than the temperature at p.m. (9 hours after a.m.). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Evaluate the function when =,, and.. f() = 5. g() =. WHAT IF? In Eample, let f(t) be the outside temperature ( F) t hours after 9 a.m. Eplain the meaning of each statement. a. f() = 75 b. f(m) = 7 c. f() = f (9) d. f () > f() Chapter Graphing Linear Functions

24 Using Function Notation to Solve and Graph Solving for the Independent Variable For h() = 5, find the value of for which h() = 7. SOLUTION h() = 5 Write the function. 7 = 5 Substitute 7 for h() Add 5 to each side. = Simplif. ( ) = Multipl each side b. = When =, h() = 7. Simplif. Graphing a Linear Function in Function Notation Graph f () = + 5. SOLUTION Step Make an input-output table to find ordered pairs. f () Step Plot the ordered pairs. Step Draw a line through the points. STUDY TIP The graph of = f() consists of the points (, f()). 8 f() = + 5 Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of so that the function has the given value.. f() = + 9; f() = 5. g() = + ; g() = Graph the linear function.. f() = 7. g() = + 8. h() = Section. Function Notation

25 Solving Real-Life Problems Modeling with Mathematics Distance (miles) First Flight f() Hours The graph shows the number of miles a helicopter is from its destination after hours on its first flight. On its second flight, the helicopter travels 5 miles farther and increases its speed b 5 miles per hour. The function f() = 5 5 represents the second flight, where f() is the number of miles the helicopter is from its destination after hours. Which flight takes less time? Eplain. SOLUTION. Understand the Problem You are given a graph of the first flight and an equation of the second flight. You are asked to compare the flight times to determine which flight takes less time.. Make a Plan Graph the function that represents the second flight. Compare the graph to the graph of the first flight. The -value that corresponds to f() = represents the flight time.. Solve the Problem Graph f() = 5 5. Step Make an input-output table to find the ordered pairs. f () Step Plot the ordered pairs. Step Draw a line through the points. Note that the function onl makes sense when and f() are positive. So, onl draw the line in the first quadrant. f() = 5 5 From the graph of the first flight, ou can see that when f() =, =. From the graph of the second flight, ou can see that when f() =, is slightl less than. So, the second flight takes less time.. Look Back You can check that our answer is correct b finding the value of for which f() =. f() = 5 5 Write the function. = 5 5 Substitute for f(). 5 = 5 Subtract 5 from each side..8 = Divide each side b 5. So, the second flight takes.8 hours, which is less than. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 9. WHAT IF? Let f() = 5 75 represent the second flight, where f() is the number of miles the helicopter is from its destination after hours. Which flight takes less time? Eplain. Chapter Graphing Linear Functions

26 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE When ou write the function = + as f() = +, ou are using.. REASONING Your height can be represented b a function h, where the input is our age. What does h() represent? Monitoring Progress and Modeling with Mathematics In Eercises, evaluate the function when =,, and 5. (See Eample.). f() = +. g() = 5. h() = + 9. r() = 7 7. p() = + 8. b() = v() = 5. n() = +. INTERPRETING FUNCTION NOTATION Let c(t) be the number of customers in a restaurant t hours after 8 A.M. Eplain the meaning of each statement. (See Eample.) a. c() = b. c() = c(8) c. c(n) = 9 d. c() < c(). INTERPRETING FUNCTION NOTATION Let H() be the percent of U.S. households with Internet use ears after 98. Eplain the meaning of each statement. a. H() = 55 b. H() = k c. H(7) d. H(7) + H() H(9) In Eercises 8, find the value of so that the function has the given value. (See Eample.). h() = 7; h() = In Eercises 9 and, find the value of so that f() = f.. MODELING WITH MATHEMATICS The function C() = 7.5 represents the cost (in dollars) of buing tickets to the orchestra with a $ coupon. a. How much does it cost to bu five tickets? b. How man tickets can ou bu with $?. MODELING WITH MATHEMATICS The function d(t) =,t represents the distance (in kilometers) that light travels in t seconds. a. How far does light travel in 5 seconds? b. How long does it take light to travel million kilometers? f. t() = ; t() = 5. m() = + 5; m() = 7. k() = ; k() = 8 7. q() = ; q() = 8. j() = 5 + 7; j() = 5 In Eercises 8, graph the linear function. (See Eample.). p() =. h() = 5 5. d() =. w() = g() = f () = Section. Function Notation 5

27 9. PROBLEM SOLVING The graph shows the percent p (in decimal form) of batter power remaining in a laptop computer after t hours of use. A tablet computer initiall has 75% of its batter power remaining and loses.5% per hour. Which computer s batter will last longer? Eplain. (See Eample 5.) Power remaining (decimal form) Laptop Batter p t Hours. HOW DO YOU SEE IT? The function = A() represents the attendance at a high school weeks after a flu outbreak. The graph of the function is shown. Number of students Attendance A() 5 5 A Week. PROBLEM SOLVING The function C() = represents the labor cost (in dollars) for Certified Hours Cost $ Remodeling to build a deck, where $ is the number of hours of labor. The table shows sample labor costs $9 from its main competitor, Master Remodeling. The deck is estimated to take 8 hours of labor. Which compan would ou hire? Eplain.. MAKING AN ARGUMENT Let P() be the number of people in the U.S. who own a cell phone ears after 99. Your friend sas that P( + ) > P() for an because + is alwas greater than. Is our friend correct? Eplain.. THOUGHT PROVOKING Let B(t) be our bank account balance after t das. Describe a situation in which B() < B() < B().. MATHEMATICAL CONNECTIONS Rewrite each geometr formula using function notation. Evaluate each function when r = 5 feet. Then eplain the meaning of the result. a. Diameter, d = r b. Area, A = πr c. Circumference, C = πr Maintaining Mathematical Proficienc Solve the inequalit. Graph the solution. (Section.5) a < 5 or a > 9. < k + <. d + 7 < 9 or d > < 7. v or v r a. What happens to the school s attendance after the flu outbreak? b. Estimate A() and eplain its meaning. c. Use the graph to estimate the solution(s) of the equation A() =. Eplain the meaning of the solution(s). d. What was the least attendance? When did that occur? e. How man students do ou think are enrolled at this high school? Eplain our reasoning. 5. INTERPRETING FUNCTION NOTATION Let f be a function. Use each statement to find the coordinates of a point on the graph of f. a. f(5) is equal to 9. b. A solution of the equation f(n) = is 5.. REASONING Given a function f, tell whether the statement f(a + b) = f(a) + f(b) is true or false for all inputs a and b. If it is false, eplain wh. Reviewing what ou learned in previous grades and lessons Chapter Graphing Linear Functions

28 .. What Did You Learn? Core Vocabular relation, p. function, p. domain, p. range, p. independent variable, p. 7 dependent variable, p. 7 linear equation in two variables, p. linear function, p. nonlinear function, p. solution of a linear equation in two variables, p. discrete domain, p. continuous domain, p. function notation, p. Core Concepts Section. Determining Whether Relations Are Functions, p. Vertical Line Test, p. 5 Section. Linear and Nonlinear Functions, p. Representations of Functions, p. The Domain and Range of a Function, p. Independent and Dependent Variables, p. 7 Discrete and Continuous Domains, p. Section. Using Function Notation, p. Mathematical Practices. How can ou use technolog to confirm our answers in Eercises on page?. How can ou use patterns to solve Eercise on page 9?. How can ou make sense of the quantities in the function in Eercise on page 5? Stud Skills Staing Focused during Class As soon as class starts, quickl review our notes from the previous class and start thinking about math. Repeat what ou are writing in our head. When a particular topic is difficult, ask for another eample. 7

29 .. Quiz Determine whether the relation is a function. Eplain. (Section.). Input,. (, ), ( 8, ), (, 5), ( 8, 8), (, ) Output, 8 Find the domain and range of the function represented b the graph. (Section.) Determine whether the graph, table, or equation represents a linear or nonlinear function. Eplain. (Section.) = ( ) 7 5 Determine whether the domain is discrete or continuous. Eplain. (Section.) 9. Depth (feet), 99 Pressure (ATM),. Hats, Cost (dollars), 5 7. For w() = + 7, find the value of for which w() =. (Section.) Graph the linear function. (Section.). g() = +. p() =. m() = 5. The function m = r represents the amount m (in dollars) of mone ou have after renting r video games. (Section. and Section.) a. Identif the independent and dependent variables. b. Find the domain and range of the function. Is the domain discrete or continuous? Eplain. c. Graph the function using its domain.. The function d() = 75 represents the distance (in miles) a high-speed train is from its destination after hours. (Section.) a. How far is the train from its destination after 8 hours? b. How long does the train travel before reaching its destination? 8 Chapter Graphing Linear Functions

30 . Graphing Linear Equations in Standard Form Essential Question How can ou describe the graph of the equation A + B = C? Using a Table to Plot Points Work with a partner. You sold a total of $ worth of tickets to a fundraiser. You lost track of how man of each tpe of ticket ou sold. Adult tickets are $ each. Child tickets are $ each. FINDING AN ENTRY POINT To be proficient in math, ou need to find an entr point into the solution of a problem. Determining what information ou know, and what ou can do with that information, can help ou find an entr point. adult Number of adult tickets + child Number of child tickets a. Let represent the number of adult tickets. Let represent the number of child tickets. Use the verbal model to write an equation that relates and. b. Cop and complete the table to show the different combinations of tickets ou might have sold. c. Plot the points from the table. Describe the pattern formed b the points. d. If ou remember how man adult tickets ou sold, can ou determine how man child tickets ou sold? Eplain our reasoning. = Rewriting and Graphing an Equation Work with a partner. You sold a total of $8 worth of cheese. You forgot how man pounds of each tpe of cheese ou sold. Swiss cheese costs $8 per pound. Cheddar cheese costs $ per pound. pound Pounds of Swiss + pound Pounds of cheddar = a. Let represent the number of pounds of Swiss cheese. Let represent the number of pounds of cheddar cheese. Use the verbal model to write an equation that relates and. b. Solve the equation for. Then use a graphing calculator to graph the equation. Given the real-life contet of the problem, find the domain and range of the function. c. The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. Use the graph to determine the - and -intercepts. d. How could ou use the equation ou found in part (a) to determine the - and -intercepts? Eplain our reasoning. e. Eplain the meaning of the intercepts in the contet of the problem. Communicate Your Answer. How can ou describe the graph of the equation A + B = C?. Write a real-life problem that is similar to those shown in Eplorations and. Section. Graphing Linear Equations in Standard Form 9

31 . Lesson Core Vocabular standard form, p. -intercept, p. -intercept, p. Previous ordered pair quadrant What You Will Learn Graph equations of horizontal and vertical lines. Graph linear equations in standard form using intercepts. Use linear equations in standard form to solve real-life problems. Horizontal and Vertical Lines The standard form of a linear equation is A + B = C, where A, B, and C are real numbers and A and B are not both zero. Consider what happens when A = or when B =. When A =, the equation becomes B = C, or = C B. Because C is a constant, ou can write = b. Similarl, B when B =, the equation becomes A = C, or = C, and ou can write = a. A Core Concept Horizontal and Vertical Lines = b = a (, b) (a, ) The graph of = b is a horizontal line. The line passes through the point (, b). The graph of = a is a vertical line. The line passes through the point (a, ). Graph (a) = and (b) =. Horizontal and Vertical Lines STUDY TIP For ever value of, the ordered pair (, ) is a solution of =. SOLUTION a. For ever value of, the value of is. The graph of the equation = is a horizontal line units above the -ais. (, ) (, ) (, ) b. For ever value of, the value of is. The graph of the equation = is a vertical line units to the left of the -ais. (, ) (, ) 5 (, ) Monitoring Progress Graph the linear equation.. =.5. = 5 Help in English and Spanish at BigIdeasMath.com Chapter Graphing Linear Functions

32 Using Intercepts to Graph Linear Equations You can use the fact that two points determine a line to graph a linear equation. Two convenient points are the points where the graph crosses the aes. Core Concept Using Intercepts to Graph Equations The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. It occurs when =. The -intercept of a graph is the -coordinate of a point where the graph crosses the -ais. It occurs when =. (, b) O -intercept = b -intercept = a (a, ) To graph the linear equation A + B = C, find the intercepts and draw the line that passes through the two intercepts. To find the -intercept, let = and solve for. To find the -intercept, let = and solve for. Using Intercepts to Graph a Linear Equation Use intercepts to graph the equation + =. SOLUTION Step Find the intercepts. STUDY TIP As a check, ou can find a third solution of the equation and verif that the corresponding point is on the graph. To find a third solution, substitute an value for one of the variables and solve for the other variable. To find the -intercept, substitute for and solve for. + = Write the original equation. + () = Substitute for. = Solve for. To find the -intercept, substitute for and solve for. + = Write the original equation. () + = Substitute for. = Solve for. Step Plot the points and draw the line. The -intercept is, so plot the point (, ). The -intercept is, so plot the point (, ). Draw a line through the points. (, ) (, ) Monitoring Progress Help in English and Spanish at BigIdeasMath.com Use intercepts to graph the linear equation. Label the points corresponding to the intercepts.. =. + = 9 Section. Graphing Linear Equations in Standard Form

33 Solving Real-Life Problems Modeling with Mathematics You are planning an awards banquet for our school. You need to rent tables to seat 8 people. Tables come in two sizes. Small tables seat people, and large tables seat people. The equation + = 8 models this situation, where is the number of small tables and is the number of large tables. a. Graph the equation. Interpret the intercepts. b. Find four possible solutions in the contet of the problem. SOLUTION. Understand the Problem You know the equation that models the situation. You are asked to graph the equation, interpret the intercepts, and find four solutions.. Make a Plan Use intercepts to graph the equation. Then use the graph to interpret the intercepts and find other solutions. STUDY TIP Although and represent whole numbers, it is convenient to draw a line segment that includes points whose coordinates are not whole numbers.. Solve the Problem a. Use intercepts to graph the equation. Neither nor can be negative, so onl graph the equation in the first quadrant. 8 (, 8) The -intercept is 8. So, plot (, 8). + = 8 The -intercept is. So, plot (, ). (, ) 8 8 Check + = 8 () + () =? 8 8 = 8 + = 8 () + () =? 8 8 = 8 The -intercept shows that ou can rent small tables when ou do not rent an large tables. The -intercept shows that ou can rent 8 large tables when ou do not rent an small tables. b. Onl whole-number values of and make sense in the contet of the problem. Besides the intercepts, it appears that the line passes through the points (, ) and (, ). To verif that these points are solutions, check them in the equation, as shown. So, four possible combinations of tables that will seat 8 people are small and 8 large, small and large, small and large, and small and large.. Look Back The graph shows that as the number of small tables increases, the number of large tables decreases. This makes sense in the contet of the problem. So, the graph is reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? You decide to rent tables from a different compan. The situation can be modeled b the equation + = 8, where is the number of small tables and is the number of large tables. Graph the equation and interpret the intercepts. Chapter Graphing Linear Functions

34 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING How are -intercepts and -intercepts alike? How are the different?. WHICH ONE DOESN T BELONG? Which point does not belong with the other three? Eplain our reasoning. (, ) (, ) (, ) (, ) Monitoring Progress and Modeling with Mathematics In Eercises, graph the linear equation. (See Eample.). =. = 5. =. = In Eercises 7, find the - and -intercepts of the graph of the linear equation = 8. + = =. + 9 = 8. =. + 8 = In Eercises, use intercepts to graph the linear equation. Label the points corresponding to the intercepts. (See Eample.). 5 + =. + =. MODELING WITH MATHEMATICS You are ordering shirts for the math club at our school. Short-sleeved shirts cost $ each. Long-sleeved shirts cost $ each. You have a budget of $ for the shirts. The equation + = models the total cost, where is the number of short-sleeved shirts and is the number of long-sleeved shirts. a. Graph the equation. Interpret the intercepts. b. Twelve students decide the want short-sleeved shirts. How man long-sleeved shirts can ou order? ERROR ANALYSIS In Eercises 5 and, describe and correct the error in finding the intercepts of the graph of the equation = + = + () = () + = 5. + =. + = = = 9. + = 7. = =. + =. MODELING WITH MATHEMATICS A football team has an awa game, and the bus breaks down. The coaches decide to drive the plaers to the game in cars and vans. Four plaers can ride in each car. Si plaers can ride in each van. There are 8 plaers on the team. The equation + = 8 models this situation, where is the number of cars and is the number of vans. (See Eample.) a. Graph the equation. Interpret the intercepts. b. Find four possible solutions in the contet of the problem.. = = = 8 = The intercept is at (8, ). + = + = + () = () + = = = = 5 = The -intercept is at (, 5), and the -intercept is at (, ). Section. Graphing Linear Equations in Standard Form

35 7. MAKING AN ARGUMENT You overhear our friend eplaining how to find intercepts to a classmate. Your friend sas, When ou want to find the -intercept, just substitute for and continue to solve the equation. Is our friend s eplanation correct? Eplain. 8. ANALYZING RELATIONSHIPS You lose track of how man -point baskets and -point baskets a team makes in a basketball game. The team misses all the -point baskets and still scores 5 points. The equation + = 5 models the total points scored, where is the number of -point baskets made and is the number of -point baskets made. a. Find and interpret the intercepts. b. Can the number of -point baskets made be odd? Eplain our reasoning. c. Graph the equation. Find two more possible solutions in the contet of the problem. MULTIPLE REPRESENTATIONS In Eercises 9, match the equation with its graph =. 5 + =. 5 =. 5 = A. 8 B.. HOW DO YOU SEE IT? You are organizing a class trip to an amusement park. The cost to enter the park is $. The cost to enter with a meal plan is $5. You have a budget of $7 for the trip. The equation + 5 = 7 models the total cost for the class to go on the trip, where is the number of students who do not choose the meal plan and is the number of students who do choose the meal plan. Number of students who do choose the meal plan 8 (, ) Class Trip (9, ) 8 Number of students who do not choose the meal plan a. Interpret the intercepts of the graph. b. Describe the domain and range in the contet of the problem. 5. REASONING Use the values to fill in the equation + = so that the -intercept of the graph is and the -intercept of the graph is 5. 5 C. Maintaining Mathematical Proficienc Simplif the epression. (Skills Review Handbook) 9. 8 ( ) ( ) 8 D MATHEMATICAL CONNECTIONS Graph the equations = 5, =, =, and =. What enclosed shape do the lines form? Eplain our reasoning... THOUGHT PROVOKING Write an equation in standard form of a line whose intercepts are integers. Eplain how ou know the intercepts are integers. 7. WRITING Are the equations of horizontal and vertical lines written in standard form? Eplain our reasoning. 8. ABSTRACT REASONING The - and -intercepts of the graph of the equation + 5 = k are integers. Describe the values of k. Eplain our reasoning. Reviewing what ou learned in previous grades and lessons 9 8 ( 7). 7 5 ( ) Chapter Graphing Linear Functions

36 .5 Graphing Linear Equations in Slope-Intercept Form Essential Question How can ou describe the graph of the equation = m + b? Slope is the rate of change between an two points on a line. It is the measure of the steepness of the line. To find the slope of a line, find the ratio of the change in (vertical change) to the change in (horizontal change). slope = change in change in slope = Finding Slopes and -Intercepts Work with a partner. Find the slope and -intercept of each line. a. b. MAKING CONJECTURES To be proficient in math, ou first need to collect and organize data. Then make conjectures about the patterns ou observe in the data. = + Writing a Conjecture = Work with a partner. Graph each equation. Then cop and complete the table. Use the completed table to write a conjecture about the relationship between the graph of = m + b and the values of m and b. Equation Description of graph Slope of graph -Intercept a. = + Line b. = c. = + d. = Communicate Your Answer. How can ou describe the graph of the equation = m + b? a. How does the value of m affect the graph of the equation? b. How does the value of b affect the graph of the equation? c. Check our answers to parts (a) and (b) b choosing one equation from Eploration and () varing onl m and () varing onl b. Section.5 Graphing Linear Equations in Slope-Intercept Form 5

37 .5 Lesson What You Will Learn Core Vocabular slope, p. rise, p. run, p. slope-intercept form, p. 8 constant function, p. 8 Previous dependent variable independent variable Find the slope of a line. Use the slope-intercept form of a linear equation. Use slopes and -intercepts to solve real-life problems. The Slope of a Line Core Concept Slope The slope m of a nonvertical line passing through two points (, ) and (, ) is the ratio of ( the rise (change in ) to the run (change in )., ) slope = m = rise change in = run change in = (, ) run = O rise = When the line rises from left to right, the slope is positive. When the line falls from left to right, the slope is negative. Finding the Slope of a Line Describe the slope of each line. Then find the slope. a. (, ) b. (, ) STUDY TIP When finding slope, ou can label either point as (, ) and the other point as (, ). The result is the same. (, ) SOLUTION a. The line rises from left to right. So, the slope is positive. Let (, ) = (, ) and (, ) = (, ). (, ) b. The line falls from left to right. So, the slope is negative. Let (, ) = (, ) and (, ) = (, ). READING In the slope formula, is read as sub one and is read as sub two. The numbers and in and are called subscripts. m = = ( ) ( ) = = Monitoring Progress Describe the slope of the line. Then find the slope... (, ) (, ) m = = = = Help in English and Spanish at BigIdeasMath.com (, ) (, ). (5, ) 8 (, ) Chapter Graphing Linear Functions

38 Finding Slope from a Table STUDY TIP As a check, ou can plot the points represented b the table to verif that the line through them has a slope of. The points represented b each table lie on a line. How can ou find the slope of each line from the table? What is the slope of each line? a. b. c SOLUTION a. Choose an two points from the table and use the slope formula. Use the points (, ) = (, ) and (, ) = (7, ). m = = 7 =, or The slope is. b. Note that there is no change in. Choose an two points from the table and use the slope formula. Use the points (, ) = (, ) and (, ) = (5, ). m = = 5 ( ) =, or The change in is. The slope is. c. Note that there is no change in. Choose an two points from the table and use the slope formula. Use the points (, ) = (, ) and (, ) = (, ). m = = ( ) = The change in is. Because division b zero is undefined, the slope of the line is undefined. Monitoring Progress Help in English and Spanish at BigIdeasMath.com The points represented b the table lie on a line. How can ou find the slope of the line from the table? What is the slope of the line? Concept Summar Slope Positive slope Negative slope Slope of Undefined slope O O O O The line rises from left to right. The line falls from left to right. The line is horizontal. The line is vertical. Section.5 Graphing Linear Equations in Slope-Intercept Form 7

39 Using the Slope-Intercept Form of a Linear Equation Core Concept Slope-Intercept Form Words A linear equation written in the form = m + b is in slope-intercept form. The slope of the line is m, and the -intercept of the line is b. Algebra = m + b (, b) = m + b slope -intercept A linear equation written in the form = + b, or = b, is a constant function. The graph of a constant function is a horizontal line. Identifing Slopes and -Intercepts Find the slope and the -intercept of the graph of each linear equation. a. = b. =.5 c. 5 = STUDY TIP For a constant function, ever input has the same output. For instance, in Eample b, ever input has an output of.5. STUDY TIP When ou rewrite a linear equation in slope-intercept form, ou are epressing as a function of. SOLUTION a. = m + b Write the slope-intercept form. slope -intercept = + ( ) Rewrite the original equation in slope-intercept form. The slope is, and the -intercept is. b. The equation represents a constant function. The equation can also be written as = +.5. The slope is, and the -intercept is.5. c. Rewrite the equation in slope-intercept form b solving for. 5 = Write the original equation Add 5 to each side. = 5 = 5 = 5 + Simplif. Divide each side b. Simplif. The slope is 5, and the -intercept is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the slope and the -intercept of the graph of the linear equation.. = + 7. = = 8 Chapter Graphing Linear Functions

40 STUDY TIP You can use the slope to find points on a line in either direction. In Eample, note that the slope can be written as. So, ou could move unit left and units up from (, ) to find the point (, ). REMEMBER You can also find the -intercept b substituting for in the equation + = and solving for. Graph + =. Identif the -intercept. SOLUTION Using Slope-Intercept Form to Graph Step Rewrite the equation in slope-intercept form. = + Step Find the slope and the -intercept. m = and b = Step The -intercept is. So, plot (, ). Step Use the slope to find another point on the line. slope = rise run = Plot the point that is unit right and units down from (, ). Draw a line through the two points. The line crosses the -ais at (, ). So, the -intercept is. (, ) Graphing from a Verbal Description A linear function g models a relationship in which the dependent variable increases units for ever unit the independent variable increases. Graph g when g() =. Identif the slope, -intercept, and -intercept of the graph. SOLUTION Because the function g is linear, it has a constant rate of change. Let represent the independent variable and represent the dependent variable. 5 (, ) (, ) Step Find the slope. When the dependent variable increases b, the change in is +. When the independent variable increases b, the change in is +. So, the slope is, or. Step Find the -intercept. The statement g() = indicates that when =, =. So, the -intercept is. Plot (, ). Step Use the slope to find another point on the line. A slope of can be written as. Plot the point that is unit left and units down from (, ). Draw a line through the two points. The line crosses the -ais at (, ). So, the -intercept is. The slope is, the -intercept is, and the -intercept is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the linear equation. Identif the -intercept. 9. =. + =. + =. A linear function h models a relationship in which the dependent variable decreases units for ever 5 units the independent variable increases. Graph h when h() =. Identif the slope, -intercept, and -intercept of the graph. Section.5 Graphing Linear Equations in Slope-Intercept Form 9

41 Solving Real-Life Problems In most real-life problems, slope is interpreted as a rate, such as miles per hour, dollars per hour, or people per ear. Modeling with Mathematics A submersible that is eploring the ocean floor begins to ascend to the surface. The elevation h (in feet) of the submersible is modeled b the function h(t) = 5t,, where t is the time (in minutes) since the submersible began to ascend. a. Graph the function and identif its domain and range. b. Interpret the slope and the intercepts of the graph. STUDY TIP Because t is the independent variable, the horizontal ais is the t-ais and the graph will have a t-intercept. Similarl, the vertical ais is the h-ais and the graph will have an h-intercept. SOLUTION. Understand the Problem You know the function that models the elevation. You are asked to graph the function and identif its domain and range. Then ou are asked to interpret the slope and intercepts of the graph.. Make a Plan Use the slope-intercept form of a linear equation to graph the function. Onl graph values that make sense in the contet of the problem. Eamine the graph to interpret the slope and the intercepts.. Solve the Problem a. The time t must be greater than or equal to. The elevation h is below sea level and must be less than or equal to. Use the slope of 5 and the h-intercept of, to graph the function in Quadrant IV. The domain is t, and the range is, h. b. The slope is 5. So, the submersible ascends at a rate of 5 feet per minute. The h-intercept is,. So, the elevation of the submersible after minutes, or when the ascent begins, is, feet. The t-intercept is. So, the submersible takes minutes to reach an elevation of feet, or sea level.. Look Back You can check that our graph is correct b substituting the t-intercept for t in the function. If h = when t =, the graph is correct. h = 5(), Substitute for t in the original equation. h = Simplif. Elevation (feet) Elevation of a Submersible Time (minutes) 8 t (, ), 8,, h (,,) Monitoring Progress Help in English and Spanish at BigIdeasMath.com. WHAT IF? The elevation of the submersible is modeled b h(t) = 5t,. (a) Graph the function and identif its domain and range. (b) Interpret the slope and the intercepts of the graph. Chapter Graphing Linear Functions

42 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The of a nonvertical line passing through two points is the ratio of the rise to the run.. VOCABULARY What is a constant function? What is the slope of a constant function?. WRITING What is the slope-intercept form of a linear equation? Eplain wh this form is called the slope-intercept form.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = 5 = 8 = + = + Monitoring Progress and Modeling with Mathematics In Eercises 5 8, describe the slope of the line. Then find the slope. (See Eample.) (, ) (, ) 5 (, ) (, ). 8. (, ) (, ) (, ) (5, ) In Eercises 9, the points represented b the table lie on a line. Find the slope of the line. (See Eample.) ANALYZING A GRAPH The graph shows the distance (in miles) that a bus travels in hours. Find and interpret the slope of the line. Distance (miles) 5 5 Bus Trip (, ) (, ) Time (hours). ANALYZING A TABLE The table shows the amount (in hours) of time ou spend at a theme park and the admission fee (in dollars) to the park. The points represented b the table lie on a line. Find and interpret the slope of the line Time (hours), Admission (dollars), Section.5 Graphing Linear Equations in Slope-Intercept Form

43 In Eercises 5, find the slope and the -intercept of the graph of the linear equation. (See Eample.) 5. = +. = 7 7. = 8. = 9. + =. + =. 5 = 8. = + ERROR ANALYSIS In Eercises and, describe and correct the error in finding the slope and the -intercept of the graph of the equation.. = The slope is, and the -intercept is. 5. GRAPHING FROM A VERBAL DESCRIPTION A linear function r models the growth of our right inde fingernail. The length of the fingernail increases.7 millimeter ever week. Graph r when r () =. Identif the slope and interpret the -intercept of the graph.. GRAPHING FROM A VERBAL DESCRIPTION A linear function m models the amount of milk sold b a farm per month. The amount decreases 5 gallons for ever $ increase in price. Graph m when m() =. Identif the slope and interpret the - and -intercepts of the graph. 7. MODELING WITH MATHEMATICS The function shown models the depth d (in inches) of snow on the ground during the first 9 hours of a snowstorm, where t is the time (in hours) after the snowstorm begins. (See Eample.). = The slope is, and the -intercept is. d(t) = t + In Eercises 5, graph the linear equation. Identif the -intercept. (See Eample.) 5. = + 7. = + 7. = 8. = 9. + =. + = =. + = In Eercises and, graph the function with the given description. Identif the slope, -intercept, and -intercept of the graph. (See Eample 5.). A linear function f models a relationship in which the dependent variable decreases units for ever units the independent variable increases. The value of the function at is.. A linear function h models a relationship in which the dependent variable increases unit for ever 5 units the independent variable decreases. The value of the function at is. a. Graph the function and identif its domain and range. b. Interpret the slope and the d-intercept of the graph. 8. MODELING WITH MATHEMATICS The function c() = represents the cost c (in dollars) of renting a truck from a moving compan, where is the number of miles ou drive the truck. a. Graph the function and identif its domain and range. b. Interpret the slope and the c-intercept of the graph. 9. COMPARING FUNCTIONS A linear function models the cost of renting a truck from a moving compan. The table shows the cost (in dollars) when ou drive the truck miles. Graph the function and compare the slope and the -intercept of the graph with the slope and the c-intercept of the graph in Eercise 8. Miles, Cost (dollars), 5 8 Chapter Graphing Linear Functions

44 ERROR ANALYSIS In Eercises and, describe and correct the error in graphing the function... MATHEMATICAL CONNECTIONS The graph shows the relationship between the base length and the side length (of the two equal sides) of an isosceles triangle in meters. The perimeter of a second isosceles triangle is 8 meters more than the perimeter of the first triangle. + = (, ) 8 =. + = (, ) 8 a. Graph the relationship between the base length and the side length of the second triangle. b. How does the graph in part (a) compare to the graph shown? 5. ANALYZING EQUATIONS Determine which of the equations could be represented b each graph.. MATHEMATICAL CONNECTIONS Graph the four equations in the same coordinate plane. = = + 8 = = 7 = = = = = 7 = 9 = = + 5 a. What enclosed shape do ou think the lines form? Eplain. a. b. b. Write a conjecture about the equations of parallel lines.. MATHEMATICAL CONNECTIONS The graph shows the relationship between the width and the length of a rectangle in inches. The perimeter of a second rectangle is inches less than the perimeter of the first rectangle. a. Graph the relationship between the width and length of the second rectangle. b. How does the graph in part (a) compare to the the graph shown? 8 = 8 c. d.. MAKING AN ARGUMENT Your friend sas that ou can write the equation of an line in slope-intercept form. Is our friend correct? Eplain our reasoning. Section.5 Graphing Linear Equations in Slope-Intercept Form

45 7. WRITING Write the definition of the slope of a line in two different was. 8. THOUGHT PROVOKING Your famil goes on vacation to a beach miles from our house. You reach our destination hours after departing. Draw a graph that describes our trip. Eplain what each part of our graph represents. 9. ANALYZING A GRAPH The graphs of the functions g() = + a and h() = + b, where a and b are constants, are shown. The intersect at the point (p, q). (p, q) 5. HOW DO YOU SEE IT? You commute to school b walking and b riding a bus. The graph represents our commute. Distance (miles) d Commute to School 8 t Time (minutes) a. Describe our commute in words. b. Calculate and interpret the slopes of the different parts of the graph. PROBLEM SOLVING In Eercises 5 and 5, find the value of k so that the graph of the equation has the given slope or -intercept. a. Label the graphs of g and h. b. What do a and b represent? c. Starting at the point (p, q), trace the graph of g until ou get to the point with the -coordinate p +. Mark this point C. Do the same with the graph of h. Mark this point D. How much greater is the -coordinate of point C than the -coordinate of point D? Maintaining Mathematical Proficienc Find the coordinates of the figure after the transformation. (Skills Review Handbook) 5. Translate the rectangle units left. A D B C 55. Dilate the triangle with respect to the origin using a scale factor of. X Y Z 5. = k 5; m = 5. = + 5 k; b = 5. ABSTRACT REASONING To show that the slope of a line is constant, let (, ) and (, ) be an two points on the line = m + b. Use the equation of the line to epress in terms of and in terms of. Then use the slope formula to show that the slope between the points is m. Reviewing what ou learned in previous grades and lessons 5. Reflect the trapezoid in the -ais. Q T R S Determine whether the equation represents a linear or nonlinear function. Eplain. (Section.) = 58. = =. = Chapter Graphing Linear Functions

46 . Transformations of Graphs of Linear Functions Essential Question How does the graph of the linear function f() = compare to the graphs of g() = f() + c and h() = f(c)? USING TOOLS STRATEGICALLY To be proficient in math, ou need to use the appropriate tools, including graphs, tables, and technolog, to check our results. Comparing Graphs of Functions Work with a partner. The graph of f() = is shown. Sketch the graph of each function, along with f, on the same set of coordinate aes. Use a graphing calculator to check our results. What can ou conclude? a. g() = + b. g() = + c. g() = d. g() = Comparing Graphs of Functions Work with a partner. Sketch the graph of each function, along with f() =, on the same set of coordinate aes. Use a graphing calculator to check our results. What can ou conclude? a. h() = b. h() = c. h() = d. h() = Matching Functions with Their Graphs Work with a partner. Match each function with its graph. Use a graphing calculator to check our results. Then use the results of Eplorations and to compare the graph of k to the graph of f() =. a. k() = b. k() = + c. k() = + d. k() = A. B. C. D. 8 8 Communicate Your Answer. How does the graph of the linear function f() = compare to the graphs of g() = f() + c and h() = f(c)? Section. Transformations of Graphs of Linear Functions 5

47 . Lesson What You Will Learn Core Vocabular famil of functions, p. parent function, p. transformation, p. translation, p. reflection, p. 7 horizontal shrink, p. 8 horizontal stretch, p. 8 vertical stretch, p. 8 vertical shrink, p. 8 Previous linear function Translate and reflect graphs of linear functions. Stretch and shrink graphs of linear functions. Combine transformations of graphs of linear functions. Translations and Reflections A famil of functions is a group of functions with similar characteristics. The most basic function in a famil of functions is the parent function. For nonconstant linear functions, the parent function is f() =. The graphs of all other nonconstant linear functions are transformations of the graph of the parent function. A transformation changes the size, shape, position, or orientation of a graph. Core Concept A translation is a transformation that shifts a graph horizontall or verticall but does not change the size, shape, or orientation of the graph. Horizontal Translations The graph of = f( h) is a horizontal translation of the graph of = f(), where h. Vertical Translations The graph of = f() + k is a vertical translation of the graph of = f (), where k. = f( h), h < = f() = f( h), h > = f() + k, k > = f() + k, k < = f() Subtracting h from the inputs before evaluating the function shifts the graph left when h < and right when h >. Adding k to the outputs shifts the graph down when k < and up when k >. Horizontal and Vertical Translations Let f() =. Graph (a) g() = f() + and (b) t() = f( + ). Describe the transformations from the graph of f to the graphs of g and t. LOOKING FOR A PATTERN In part (a), the output of g is equal to the output of f plus. In part (b), the output of t is equal to the output of f when the input of f is more than the input of t. SOLUTION a. The function g is of the form = f() + k, where k =. So, the graph of g is a vertical translation units up of the graph of f. g() = f() + f() = b. The function t is of the form = f( h), where h =. So, the graph of t is a horizontal translation units left of the graph of f. 5 t() = f( + ) f() = Chapter Graphing Linear Functions

48 STUDY TIP A reflected point is the same distance from the line of reflection as the original point but on the opposite side of the line. Core Concept A reflection is a transformation that flips a graph over a line called the line of refl ection. Reflections in the -ais The graph of = f() is a reflection in the -ais of the graph of = f(). = f() = f() Reflections in the -ais The graph of = f( ) is a reflection in the -ais of the graph of = f (). = f( ) = f() Multipling the outputs b changes their signs. Multipling the inputs b changes their signs. Reflections in the -ais and the -ais Let f() = +. Graph (a) g() = f() and (b) t() = f( ). Describe the transformations from the graph of f to the graphs of g and t. SOLUTION a. To find the outputs of g, multipl the outputs of f b. The graph of g consists of the points (, f()). f() f() g() = f() b. To find the outputs of t, multipl the inputs b and then evaluate f. The graph of t consists of the points (, f( )). f( ) t() = f( ) f() = + The graph of g is a reflection in the -ais of the graph of f. Monitoring Progress f() = + The graph of t is a reflection in the -ais of the graph of f. Help in English and Spanish at BigIdeasMath.com Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h.. f() = + ; g() = f() ; h() = f( ). f() = ; g() = f(); h() = f( ) Section. Transformations of Graphs of Linear Functions 7

49 STUDY TIP The graphs of = f( a) and = a f() represent a stretch or shrink and a reflection in the - or -ais of the graph of = f(). Stretches and Shrinks You can transform a function b multipling all the -coordinates (inputs) b the same factor a. When a >, the transformation is a horizontal shrink because the graph shrinks toward the -ais. When < a <, the transformation is a horizontal stretch because the graph stretches awa from the -ais. In each case, the -intercept stas the same. You can also transform a function b multipling all the -coordinates (outputs) b the same factor a. When a >, the transformation is a vertical stretch because the graph stretches awa from the -ais. When < a <, the transformation is a vertical shrink because the graph shrinks toward the -ais. In each case, the -intercept stas the same. Core Concept Horizontal Stretches and Shrinks The graph of = f(a) is a horizontal stretch or shrink b a factor of a of the graph of = f(), where a > and a. = f(a), a > = f() = f(a), < a < The -intercept stas the same. Vertical Stretches and Shrinks The graph of = a f() is a vertical stretch or shrink b a factor of a of the graph of = f(), where a > and a. = a f(), a > = f() = a f(), < a < The -intercept stas the same. Horizontal and Vertical Stretches Let f() =. Graph (a) g() = f ( ) and (b) h() = f(). Describe the transformations from the graph of f to the graphs of g and h. SOLUTION f() = g() = f ( ) a. To find the outputs of g, multipl the inputs b. Then evaluate f. The graph of g consists of the points (, f ( ) ). The graph of g is a horizontal stretch of the graph of f b a factor of =. () f ( ) b. To find the outputs of h, multipl the outputs of f b. The graph of h consists of the points (, f()). f() f() = The graph of h is a vertical stretch of the graph of f b a factor of. f() h() = f() 8 Chapter Graphing Linear Functions

50 Horizontal and Vertical Shrinks Let f() = +. Graph (a) g() = f() and (b) h() = f(). Describe the transformations from the graph of f to the graphs of g and h. SOLUTION a. To find the outputs of g, multipl the inputs b. Then evaluate f. The graph of g consists of the points (, f () ). g() = f() 5 f() = + f() The graph of g is a horizontal shrink of the graph of f b a factor of. b. To find the outputs of h, multipl the outputs of f b. The graph of h consists of the points (, f() ). f() f() f() = + h() = f() The graph of h is a vertical shrink of the graph of f b a factor of. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Using f, graph (a) g and (b) h. Describe the transformations from the graph of f to the graphs of g and h.. f() = ; g() = f ( ) ; h() = f(). f() = + ; g() = f(); h() = f() STUDY TIP You can perform transformations on the graph of an function f using these steps. Combining Transformations Core Concept Transformations of Graphs The graph of = a f( h) + k or the graph of = f(a h) + k can be obtained from the graph of = f() b performing these steps. Step Translate the graph of = f() horizontall h units. Step Use a to stretch or shrink the resulting graph from Step. Step Reflect the resulting graph from Step when a <. Step Translate the resulting graph from Step verticall k units. Section. Transformations of Graphs of Linear Functions 9

51 Combining Transformations Graph f() = and g() = +. Describe the transformations from the graph of f to the graph of g. SOLUTION ANOTHER WAY You could also rewrite g as g() = f( ) +. In this case, the transformations from the graph of f to the graph of g will be different from those in Eample 5. Note that ou can rewrite g as g() = f() +. Step There is no horizontal translation from the graph of f to the graph of g. Step Stretch the graph of f verticall b a factor of to get the graph of h() =. Step Reflect the graph of h in the -ais to get the graph of r() =. Step Translate the graph of r verticall units up to get the graph of g() = +. g() = + 5 f() = Solving a Real-Life Problem A cable compan charges customers $ per month for its service, with no installation fee. The cost to a customer is represented b c(m) = m, where m is the number of months of service. To attract new customers, the cable compan reduces the monthl fee to $ but adds an installation fee of $5. The cost to a new customer is represented b r(m) = m + 5, where m is the number of months of service. Describe the transformations from the graph of c to the graph of r. SOLUTION Note that ou can rewrite r as r(m) = c(m) + 5. In this form, ou can use the order of operations to get the outputs of r from the outputs of c. First, multipl the outputs of c b to get h(m) = m. Then add 5 to the outputs of h to get r(m) = m + 5. c(m) = m r(m) = m m The transformations are a vertical shrink b a factor of and then a vertical translation 5 units up. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Graph f() = and h() =. Describe the transformations from the graph of f to the graph of h. 5 Chapter Graphing Linear Functions

52 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. WRITING Describe the relationship between f() = and all other nonconstant linear functions.. VOCABULARY Name four tpes of transformations. Give an eample of each and describe how it affects the graph of a function.. WRITING How does the value of a in the equation = f(a) affect the graph of = f()? How does the value of a in the equation = af() = f() affect the graph of = f()? = g(). REASONING The functions f and g are linear functions. The graph of g is a vertical shrink of the graph of f. What can ou sa about the -intercepts of the graphs of f and g? Is this alwas true? Eplain. Monitoring Progress and Modeling with Mathematics In Eercises 5, use the graphs of f and g to describe the transformation from the graph of f to the graph of g. (See Eample.) 5. g() = f() +. f() = 7. f() = + ; g() = f() 8. f() = + ; g() = f() + 9. f() = ; g() = f( + 5). f() = 5; g() = f( ) g() = f( + ) f() =. MODELING WITH MATHEMATICS The total cost C (in dollars) to cater an event with p people is given b the function C(p) = 8p + 5. The set-up fee increases b $5. The new total cost T is given b the function T(p) = C(p) + 5. Describe the transformation from the graph of C to the graph of T. CC Pricing $5 set-up fee + $8 per person In Eercises, use the graphs of f and h to describe the transformation from the graph of f to the graph of h. (See Eample.). MODELING WITH MATHEMATICS You and a friend start biking from the same location. Your distance d (in miles) after t minutes is given b the function d(t) = t. Your friend starts biking 5 5 minutes after ou. Your friend s distance f is given b the function f(t) = d(t 5). Describe the transformation from the graph of d to the graph of f.. f() = + h() = f(). 5. f() = 5 ; h() = f( ). f() = ; h() = f() h() = f( ) f() = + Section. Transformations of Graphs of Linear Functions 5

53 In Eercises 7, use the graphs of f and r to describe the transformation from the graph of f to the graph of r. (See Eample.) 7. f() = r() = f() f() = ; r() = f ( ). f() = + 5; r() = f ( ). f() = + ; r() = f(). f() = ; r() = f() f() = r() = f ( ) In Eercises 8, use the graphs of f and h to describe the transformation from the graph of f to the graph of h. (See Eample.). 5 h() = f(). f() = + 5. f() = ; h() = f(). f() = + ; h() = f() 7. f() = ; h() = f(5) 8. f() = + 8; h() = f() h() = f() 5 f() = In Eercises 9, use the graphs of f and g to describe the transformation from the graph of f to the graph of g. 9. f() = ; g() = f(). f() = + 8; g() = f(). f() = 7; g() = f( ). f() = + 8; g() = f ( ). f() = ; g() = f(). f() = ; g() = f() In Eercises 5 8, write a function g in terms of f so that the statement is true. 5. The graph of g is a horizontal translation units right of the graph of f.. The graph of g is a reflection in the -ais of the graph of f. 7. The graph of g is a vertical stretch b a factor of of the graph of f. 8. The graph of g is a horizontal shrink b a factor of 5 of the graph of f. ERROR ANALYSIS In Eercises 9 and, describe and correct the error in graphing g. 9.. g() = f( ) 5 f() = f() = + g() = f( ) In Eercises, graph f and h. Describe the transformations from the graph of f to the graph of h. (See Eample 5.). f() = ; h() = +. f() = ; h() =. f() = ; h() =. f() = ; h() = + 5. f() = ; h() = 5. f() = ; h() = 7 5 Chapter Graphing Linear Functions

54 7. MODELING WITH MATHEMATICS The function t() = + 7 represents the temperature from 5 P.M. to P.M., where is the number of hours after 5 P.M. The function d() = + 7 represents the temperature from A.M. to P.M., where is the number of hours after A.M. Describe the transformation from the graph of t to the graph of d. 5. USING STRUCTURE The graph of h() = a f(b c) + d is a transformation of the graph of the linear function f. Select the word or value that makes each statement true. vertical horizontal stretch shrink 5 -ais -ais 5 a. The graph of h is a shrink of the graph of f when a =, b =, c =, and d =. b. The graph of h is a reflection in the of the graph of f when a =, b =, c =, and d =. 8. MODELING WITH MATHEMATICS A school sells T-shirts to promote school spirit. The school s profit is given b the function P() = 8 5, where is the number of T-shirts sold. During the pla-offs, the school increases the price of the T-shirts. The school s profit during the pla-offs is given b the function Q() =, where is the number of T-shirts sold. Describe the transformations from the graph of P to the graph of Q. (See Eample.) c. The graph of h is a horizontal stretch of the graph of f b a factor of 5 when a =, b =, c =, and d =. 5. ANALYZING GRAPHS Which of the graphs are related b onl a translation? Eplain. A B C D $8 E 9. USING STRUCTURE The graph of g() = a f( b) + c is a transformation of the graph of the linear function f. Select the word or value that makes each statement true. reflection stretch left -ais translation shrink right -ais F 5. ANALYZING RELATIONSHIPS A swimming pool is a. The graph of g is a vertical of the graph of f when a =, b =, and c =. filled with water b a hose at a rate of gallons per hour. The amount v (in gallons) of water in the pool after t hours is given b the function v(t) = t. How does the graph of v change in each situation? b. The graph of g is a horizontal translation of the graph of f when a =, b =, and c =. a. A larger hose is found. Then the pool is filled at a rate of gallons per hour. c. The graph of g is a vertical translation unit up of the graph of f when a =, b =, and c =. b. Before filling up the pool with a hose, a water truck adds gallons of water to the pool. Section. hsnb_alg_pe_.indd 5 $ Transformations of Graphs of Linear Functions 5 //5 : PM

55 5. ANALYZING RELATIONSHIPS You have $5 to spend on fabric for a blanket. The amount m (in dollars) of mone ou have after buing ards of fabric is given b the function m() = How does the graph of m change in each situation? Fabric: $9.98/ard a. You receive an additional $ to spend on the fabric. b. The fabric goes on sale, and each ard now costs $ THOUGHT PROVOKING Write a function g whose graph passes through the point (, ) and is a transformation of the graph of f() =. In Eercises 55, graph f and g. Write g in terms of f. Describe the transformation from the graph of f to the graph of g. 55. f() = 5; g() = 8 5. f() = + ; g() =. HOW DO YOU SEE IT? Match each function with its graph. Eplain our reasoning. f 5 a. a() = f( ) b. g() = f() c. h() = f() + d. k() = f() REASONING In Eercises, find the value of r.. f() = + 5 g() = f( r). D A B C g() = f(r) f() = f() = + 9; g() = f() = ; g() = 59. f() = + ; g() = +. f() = ; g() =. REASONING The graph of f() = + 5 is a vertical translation 5 units up of the graph of f() =. How can ou obtain the graph of f() = + 5 from the graph of f() = using a horizontal translation? Maintaining Mathematical Proficienc Solve the formula for the indicated variable. (Section.5) 8. Solve for h. 9. Solve for w. r 5. f() =. 5 g() = rf() f() = + 8 Reviewing what ou learned in previous grades and lessons g() = f() + r 7. CRITICAL THINKING When is the graph of = f() + w the same as the graph of = f( + w) for linear functions? Eplain our reasoning. h w V = πr h P = + w Solve the inequalit. Graph the solution, if possible. (Section.) > < Chapter Graphing Linear Functions

56 .7 Graphing Absolute Value Functions Essential Question How do the values of a, h, and k affect the graph of the absolute value function g() = a h + k? The parent absolute value function is f() =. The graph of f is V-shaped. Parent absolute value function Identifing Graphs of Absolute Value Functions Work with a partner. Match each absolute value function with its graph. Then use a graphing calculator to verif our answers. a. g() = b. g() = + c. g() = + d. g() = e. g() = f. g() = + + A. B. C. D. E. F. LOOKING FOR STRUCTURE To be proficient in math, ou need to look closel to discern a pattern or structure. Communicate Your Answer. How do the values of a, h, and k affect the graph of the absolute value function g() = a h + k?. Write the equation of the absolute value function whose graph is shown. Use a graphing calculator to verif our equation. Section.7 Graphing Absolute Value Functions 55

57 .7 Lesson What You Will Learn Core Vocabular absolute value function, p. 5 verte, p. 5 verte form, p. 58 Previous domain range Translate graphs of absolute value functions. Stretch, shrink, and reflect graphs of absolute value functions. Combine transformations of graphs of absolute value functions. Translating Graphs of Absolute Value Functions Core Concept Absolute Value Function An absolute value function is a function that contains an absolute value epression. The parent absolute value function is f() =. The graph of f() = is V-shaped and smmetric about the -ais. The verte is the point where the graph changes direction. The verte of the graph of f() = is (, ). The domain of f() = is all real numbers. The range is. The graphs of all other absolute value functions are transformations of the graph of the parent function f() =. The transformations presented in Section. also appl to absolute value functions. Graphing g() = + k and g() = h Graph each function. Compare each graph to the graph of f() =. Describe the domain and range. a. g() = + b. m() = verte f() = g() = + SOLUTION a. Step Make a table of values. b. Step Make a table of values. g() 5 5 m() Step Plot the ordered pairs. Step Draw the V-shaped graph. Step Plot the ordered pairs. Step Draw the V-shaped graph. m() = 5 The function g is of the form = f() + k, where k =. So, the graph of g is a vertical translation units up of the graph of f. The domain is all real numbers. The range is. The function m is of the form = f( h), where h =. So, the graph of m is a horizontal translation units right of the graph of f. The domain is all real numbers. The range is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of f() =. Describe the domain and range.. h() =. n() = + 5 Chapter Graphing Linear Functions

58 Stretching, Shrinking, and Reflecting Graphing g() = a Graph each function. Compare each graph to the graph of f() =. Describe the domain and range. a. q() = b. p() = SOLUTION a. Step Make a table of values. Step Plot the ordered pairs. Step Draw the V-shaped graph. q() q() = STUDY TIP A vertical stretch of the graph of f() = is narrower than the graph of f() =. The function q is of the form = a f(), where a =. So, the graph of q is a vertical stretch of the graph of f b a factor of. The domain is all real numbers. The range is. b. Step Make a table of values. Step Plot the ordered pairs. Step Draw the V-shaped graph. p() STUDY TIP A vertical shrink of the graph of f() = is wider than the graph of f() =. p() = The function p is of the form = a f(), where a =. So, the graph of p is a vertical shrink of the graph of f b a factor of and a reflection in the -ais. The domain is all real numbers. The range is. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Compare the graph to the graph of f() =. Describe the domain and range.. t() =. v() = Section.7 Graphing Absolute Value Functions 57

59 STUDY TIP The function g is not in verte form because the variable does not have a coefficient of. Core Concept Verte Form of an Absolute Value Function An absolute value function written in the form g() = a h + k, where a, is in verte form. The verte of the graph of g is (h, k). An absolute value function can be written in verte form, and its graph is smmetric about the line = h. Graphing f() = h + k and g() = f(a) Graph f() = + and g() = +. Compare the graph of g to the graph of f. SOLUTION Step Make a table of values for each function. f() g() Step Plot the ordered pairs. Step Draw the V-shaped graph of each function. Notice that the verte of the graph of f is (, ) and the graph is smmetric about =. g() = + 5 f() = + Note that ou can rewrite g as g() = f(), which is of the form = f(a), where a =. So, the graph of g is a horizontal shrink of the graph of f b a factor of. The -intercept is the same for both graphs. The points on the graph of f move halfwa closer to the -ais, resulting in the graph of g. When the input values of f are times the input values of g, the output values of f and g are the same. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Graph f() = and g() =. Compare the graph of g to the graph of f.. Graph f() = + + and g() = + +. Compare the graph of g to the graph of f. 58 Chapter Graphing Linear Functions

60 Combining Transformations Graphing g() = a h + k REMEMBER You can obtain the graph of = a f( h) + k from the graph of = f() using the steps ou learned in Section.. Let g() = +. (a) Describe the transformations from the graph of f() = to the graph of g. (b) Graph g. SOLUTION a. Step Translate the graph of f horizontall unit right to get the graph of t() =. Step Stretch the graph of t verticall b a factor of to get the graph of h() =. Step Reflect the graph of h in the -ais to get the graph of r() =. Step Translate the graph of r verticall units up to get the graph of g() = +. b. Method Step Make a table of values. Step Plot the ordered pairs. g() Step Draw the V-shaped graph. g() = + Method Step Identif and plot the verte. (h, k) = (, ) Step Plot another point on the graph, such as (, ). Because the graph is smmetric about the line =, ou can use smmetr to plot a third point, (, ). Step Draw the V-shaped graph. g() = + (, ) (, ) (, ) Monitoring Progress Help in English and Spanish at BigIdeasMath.com 7. Let g() = + +. (a) Describe the transformations from the graph of f() = to the graph of g. (b) Graph g. Section.7 Graphing Absolute Value Functions 59

61 .7 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. COMPLETE THE SENTENCE The point (, ) is the of the graph of f() =.. USING STRUCTURE How do ou know whether the graph of f() = a h + k is a vertical stretch or a vertical shrink of the graph of f() =?. WRITING Describe three different tpes of transformations of the graph of an absolute value function.. REASONING The graph of which function has the same -intercept as the graph of f () = + 5? Eplain. g() = + 5 h() = + 5 Monitoring Progress and Modeling with Mathematics In Eercises 5, graph the function. Compare the graph to the graph of f() =. Describe the domain and range. (See Eamples and.) 5. d() =. r() = m() = + 8. v() = 9. p() =. j() =. f() =. p() = a f() = w() = a. a() = 5. q() = In Eercises, graph the function. Compare the graph to the graph of f() =.. h() = +. n() = 5. k() =. g() = In Eercises 7 and 8, graph the function. Compare the graph to the graph of f() = () = + 8. b() = + + In Eercises 9, compare the graphs. Find the value of h, k, or a. 9. f() =. g() = + k t() = h f() = In Eercises, write an equation that represents the given transformation(s) of the graph of g() =.. vertical translation 7 units down. horizontal translation units left 5. vertical shrink b a factor of. vertical stretch b a factor of and a reflection in the -ais In Eercises 7, graph and compare the two functions. (See Eample.) 7. f() = ; g() = 8. h() = + 5 ; t() = p() = + ; q() = +. w() = + ; () = 5 +. a() = + + ; b() = + +. u() = + ; v() = + Chapter Graphing Linear Functions

62 In Eercises, describe the transformations from the graph of f() = to the graph of the given function. Then graph the given function. (See Eample.). r() = +. c() = d() = + 5. v() = m() = + 8. s() = 9. j() = + 5. n() = + +. MODELING WITH MATHEMATICS The number of pairs of shoes sold s (in thousands) increases and then decreases as described b the function s(t) = t 5 + 5, where t is the time (in weeks).. USING STRUCTURE Eplain how the graph of each function compares to the graph of = for positive and negative values of k, h, and a. a. = + k b. = h c. = a d. = a ERROR ANALYSIS In Eercises 5 and, describe and correct the error in graphing the function. 5. = 5 a. Graph the function. b. What is the greatest number of pairs of shoes sold in week?. MODELING WITH MATHEMATICS On the pool table shown, ou bank the five ball off the side represented b the -ais. The path of the ball is described b the function p() = 5. ( 5, 5) (5, 5). = MATHEMATICAL CONNECTIONS In Eercises 7 and 8, write an absolute value function whose graph forms a square with the given graph. 7. ( 5, ) (, ) 5 (5, ) = a. At what point does the five ball bank off the side? b. Do ou make the shot? Eplain our reasoning. 8. = +. USING TRANSFORMATIONS The points A (, ), B(, ), and C(, ) lie on the graph of the absolute value function f. Find the coordinates of the points corresponding to A, B, and C on the graph of each function. a. g() = f() 5 b. h() = f( ) c. j() = f() d. k() = f() 9. WRITING Compare the graphs of p() = and q() =. Section.7 Graphing Absolute Value Functions

63 5. HOW DO YOU SEE IT? The object of a computer game is to break bricks b deflecting a ball toward them using a paddle. The graph shows the current path of the ball and the location of the last brick. 8 brick paddle 8 BRICK FACTORY a. You can move the paddle up, down, left, and right. At what coordinates should ou place the paddle to break the last brick? Assume the ball deflects at a right angle. b. You move the paddle to the coordinates in part (a), and the ball is deflected. How can ou write an absolute value function that describes the path of the ball? In Eercises 5 5, graph the function. Then rewrite the absolute value function as two linear functions, one that has the domain < and one that has the domain. 5. = 5. = 5. = = Maintaining Mathematical Proficienc In Eercises 55 58, graph and compare the two functions. 55. f() = + ; g() = s() = 5 ; t() = v() = + + ; w() = c() = + ; d() = REASONING Describe the transformations from the graph of g() = + + to the graph of h() =. Eplain our reasoning.. THOUGHT PROVOKING Graph an absolute value function f that represents the route a wide receiver runs in a football game. Let the -ais represent distance (in ards) across the field horizontall. Let the -ais represent distance (in ards) down the field. Be sure to limit the domain so the route is realistic.. SOLVING BY GRAPHING Graph = + and = in the same coordinate plane. Use the graph to solve the equation + =. Check our solutions.. MAKING AN ARGUMENT Let p be a positive constant. Your friend sas that because the graph of = + p is a positive vertical translation of the graph of =, the graph of = + p is a positive horizontal translation of the graph of =. Is our friend correct? Eplain.. ABSTRACT REASONING Write the verte of the absolute value function f() = a h + k in terms of a, h, and k. Reviewing what ou learned in previous grades and lessons Solve the inequalit. (Section.). 8a 7 (a ) 5. (p + ) > p 5. (h +.5) (h ) 7. ( + ) < ( 9) Find the slope of the line. (Section.5) 8. (, ) 9. (5, ) 7. (, ) (, ) (, ) 5 (, ) Chapter Graphing Linear Functions

64 ..7 What Did You Learn? Core Vocabular standard form, p. -intercept, p. -intercept, p. slope, p. rise, p. run, p. slope-intercept form, p. 8 constant function, p. 8 famil of functions, p. parent function, p. transformation, p. translation, p. reflection, p. 7 horizontal shrink, p. 8 horizontal stretch, p. 8 vertical stretch, p. 8 vertical shrink, p. 8 absolute value function, p. 5 verte, p. 5 verte form, p. 58 Core Concepts Section. Horizontal and Vertical Lines, p. Using Intercepts to Graph Equations, p. Section.5 Slope, p. Slope-Intercept Form, p. 8 Section. Horizontal Translations, p. Vertical Translations, p. Reflections in the -ais, p. 7 Reflections in the -ais, p. 7 Horizontal Stretches and Shrinks, p. 8 Vertical Stretches and Shrinks, p. 8 Transformations of Graphs, p. 9 Section.7 Absolute Value Function, p. 5 Verte Form of an Absolute Value Function, p. 58 Mathematical Practices. Eplain how ou determined what units of measure to use for the horizontal and vertical aes in Eercise 7 on page.. Eplain our plan for solving Eercise 8 on page 5. The Cost of a T-Shirt You receive bids for making T-shirts for our class fundraiser from four companies. To present the pricing information, one compan uses a table, one compan uses a written description, one compan uses an equation, and one compan uses a graph. How will ou compare the different representations and make the final choice? To eplore the answers to this question and more, go to BigIdeasMath.com. Performance Task