3.6. Transformations of Graphs of Linear Functions

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1 . 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)? Comparing Graphs of Functions USING TOOLS STRATEGICALLY To be proicient in math, ou need to use the appropriate tools, including graphs, tables, and technolog, to check our results. 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

2 . Lesson What You Will Learn Translate and reflect graphs of linear functions. Stretch and shrink graphs of linear functions. Core Vocabul Vocabular larr Combine transformations of graphs of linear functions. famil of functions, p. parent function, p. transformation, p. translation, p. relection, p. 7 horizontal shrink, p. 8 horizontal stretch, p. 8 vertical stretch, p. 8 vertical shrink, p. 8 Previous linear function 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 Vertical Translations The graph of = f( h) is a horizontal translation of the graph of = f(), where h. The graph of = f () + k is a vertical translation of the graph of = f (), where k. = f() = f() + k, k> = f( h), h< = f() = f() + k, k< = f( h), h> 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 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. 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. Chapter 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. g() = f() + t() = f( + ) f() = f() = Graphing Linear Functions

3 Core Concept A reflection is a transformation that lips a graph over a line called the line of reflection. 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. Reflections in the -ais Reflections in the -ais The graph of = f() is a relection in the -ais of the graph of = f(). The graph of = f ( ) is a relection in the -ais of the graph of = f (). = f() = 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. a. To ind the outputs of g, multipl the outputs of f b. The graph of g consists of the points (, f()). f () f () b. To ind the outputs of t, multipl the inputs b and then evaluate f. The graph of t consists of the points (, f( )). f ( ) g() = f() t() = f( ) f() = + The graph of g is a relection in the -ais of the graph of f. Monitoring Progress f() = + The graph of t is a relection 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

4 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 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(). Horizontal Stretches and Shrinks Vertical Stretches and Shrinks The graph of = f(a) is a horizontal stretch or shrink b a factor of of a the graph of = f(), where a > and a. 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. = f(a), a> = a f(), a> = f() = f() = a f(), <a< = f(a), <a< The -intercept stas the same. 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. f() = ( ( )) points, f. ( ) g() = f f() = 8 The graph of g is a horizontal stretch of the graph of f b a factor of =. b. To ind the outputs of h, multipl the outputs of f b. The graph of h consists of the points (, f()). a. To ind the outputs of g, multipl the inputs b. Then evaluate f. The graph of g consists of the The graph of h is a vertical stretch of the graph of f b a factor of. h() = f() Chapter Graphing Linear Functions () f f () f () ( )

5 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. a. To ind the outputs of g, multipl the inputs b. Then evaluate f. The graph of g consists of the points (, f () ). f () g() = f() f() = + The graph of g is a horizontal shrink of the graph of f b a factor of. b. To ind 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 Relect 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

6 Combining Transformations Graph f () = and g() = +. Describe the transformations from the graph of f to the graph of g. Note that ou can rewrite g as g() = f() +. 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. Step There is no horizontal translation from the graph of f to the graph of g. g() = + Step Stretch the graph of f verticall b a factor of to get the graph of h() =. Step Relect 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() = +. 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 $. The cost to a new customer is represented b r(m) = m +, where m is the number of months of service. Describe the transformations from the graph of c to the graph of r. Note that ou can rewrite r as r(m) = c(m) +. 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 to the outputs of h to get r(m) = m +. c(m) = m r(m) = m m The transformations are a vertical shrink b a factor of and then a vertical translation units up. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. Graph f() = and h() =. Describe the transformations from the graph of f to the graph of h. Chapter Graphing Linear Functions

7 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 = f() graph of = f()? How does the value of a in the equation = af() 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, use the graphs of f and g to describe the transformation from the graph of f to the graph of g. (See Eample.). g() = f() +. C (in dollars) to cater an event with p people is given b the function C(p) = 8p +. The set-up fee increases b $. The new total cost T is given b the function T( p) = C(p) +. Describe the transformation from the graph of C to the graph of T. g() = f( + ) f() =. MODELING WITH MATHEMATICS The total cost CC f() = 7. f() = Pricing $ set-up fee + $8 per person + ; g() = f() 8. f() = + ; g() = f() + 9. f() = ; g() = f( + ). f() = ; g() = f( ) 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 minutes after ou. Your friend s distance f is given b the function f(t) = d(t ). Describe the transformation from the graph of d to the graph of f. f() = +. h() = f() h() = f( ) f() = +. f() = ; h() = f ( ). f() = ; h() = f() Section. Transformations of Graphs of Linear Functions

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

9 7. MODELING WITH MATHEMATICS The function t() = + 7 represents the temperature from P.M. to P.M., where is the number of hours after 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.. 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 -ais -ais 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 relection 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 proit is given b the function P() = 8, where is the number of T-shirts sold. During the pla-offs, the school increases the price of the T-shirts. The school s proit 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 when a =, b =, c =, and d =.. 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. relection stretch left translation shrink right -ais -ais F. ANALYZING RELATIONSHIPS A swimming pool is a. The graph of g is a vertical of the graph of f when a =, b =, and c =. illed 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 illed 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 illing up the pool with a hose, a water truck adds gallons of water to the pool. Section. Transformations of Graphs of Linear Functions

10 . ANALYZING RELATIONSHIPS You have $ to spend. HOW DO YOU SEE IT? Match each function with its 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? graph. Eplain our reasoning. A f B Fabric: $9.98/ard C a. You receive an additional $ to spend on the fabric. b. The fabric goes on sale, and each ard now costs $.99. D a. a() = f( ) b. g() = f() c. h() = f() + d. k() = f(). THOUGHT PROVOKING Write a function g whose graph passes through the point (, ) and is a transformation of the graph of f() =. REASONING In Eercises, ind the value of r.. In Eercises, graph f and g. Write g in terms of f. Describe the transformation from the graph of f to the graph of g. f() = +.. f() = ; g() = 8 g() = f( r). f() = + ; g() = 7. f() = + 9; g() = +.. f() = + 8. f() = ; g() = 7. CRITICAL THINKING When is the graph of = f() + w the same as the graph of = f( + w) for linear functions? Eplain our reasoning. Maintaining Mathematical Proficienc Reviewing what ou learned in previous grades and lessons Solve the formula for the indicated variable. (Section.) 9. Solve for w. r h w V = πr h P = ℓ+ w Solve the inequalit. Graph the solution, if possible. Chapter 7. + > Graphing Linear Functions g() = f() + r translation units up of the graph of f() =. How can ou obtain the graph of f() = + from the graph of f() = using a horizontal translation? g() = rf(). REASONING The graph of f() = + is a vertical 8. Solve for h. 9. f() = + ; g() = + f() = + f() = 8. f() = ; g() = 7. g() = f(r) (Section.) <