Name Class Date 1.3 Transformations of Function Graphs Essential Question: What are the was ou can transform the graph of the function f()? Resource Locker Eplore 1 Investigating Translations of Function Graphs You can transform the graph of a function in various was. You can translate the graph horizontall or verticall, ou can stretch or compress the graph horizontall or verticall, and ou can reflect the graph across the -ais or the -ais. How the graph of a given function is transformed is determined b the wa certain numbers, called parameters, are introduced in the function. The graph of ƒ () is shown. Use this graph for the eploration. A First graph g () = ƒ () + k where k is the parameter. Let k = so that g () = ƒ () +. Complete the input-output table and then graph g(). In general, how is the graph of g () = ƒ () + k related to the graph of ƒ () when k is a positive number? - - 0 - f () f () + -1-1 3 - - - 0 5 - Houghton Mifflin Harcourt Publishing Compan B Now tr a negative value of k in g () = ƒ () + k. Let k = -3 so that g () = ƒ () - 3. Complete the inputoutput table and then graph g () on the same grid. In general, how is the graph of g () = ƒ () + k related to the graph of ƒ () when k is a negative number? f() f() - 3-1 - -5 1-1 1-5 Module 1 31 Lesson 3
C Now graph g () = ƒ ( - h) where h is the parameter. Let h = so that g () = ƒ ( - ). Complete the mapping diagram and then graph g (). (To complete the mapping diagram, ou need to find the inputs for g that produce the inputs for ƒ after ou subtract. Work backward from the inputs for ƒ to the missing inputs for g b adding.) In general, how is the graph of g () = ƒ ( - h) related to the graph of ƒ () when h is a positive number? Input for g 1 3 - Input for f -1 1 3 5 Output for f - - Output for g - - - - 0 - D Make a Conjecture How would ou epect the graph of g () = ƒ ( - h) to be related to the graph of ƒ () when h is a negative number? Reflect 1. Suppose a function ƒ () has a domain of 1, and a range of 1,. When the graph of ƒ () is translated verticall k units where k is either positive or negative, how do the domain and range change?. Suppose a function ƒ () has a domain of 1, and a range of 1,. When the graph of ƒ () is translated horizontall h units where h is either positive or negative, how do the domain and range change? 3. You can transform the graph of ƒ () to obtain the graph of g () = ƒ ( - h) + k b combining transformations. Predict what will happen b completing the table. Sign of h Sign of k Transformations of the Graph of f() + + Translate right h units and up k units. + - - + Houghton Mifflin Harcourt Publishing Compan - - Module 1 3 Lesson 3
Eplore Investigating Stretches and Compressions of Function Graphs In this activit, ou will consider what happens when ou multipl b a positive parameter inside or outside a function. Throughout, ou will use the same function ƒ () that ou used in the previous activit. A First graph g () = a ƒ () where a is the parameter. Let a = so that g () = ƒ (). Complete the input-output table and then graph g (). In general, how is the graph of g () = a ƒ () related to the graph of ƒ () when a is greater than 1? f() f() -1 - - 1 3-5 - - 0 - Houghton Mifflin Harcourt Publishing Compan B Now tr a value of a between 0 and 1 in g () = a ƒ (). Let a = 1_ so that g () = 1_ ƒ (). Complete the input-output table and then graph g (). In general, how is the graph of g () = a ƒ () related to the graph of ƒ () when a is a number between 0 and 1? f() 1 f() -1 - -1 1 1 3-5 - - - 0 Module 1 33 Lesson 3
C Now graph g () = ƒ 1 ( b ) where b is the parameter. Let b = so that g () = 1 ƒ ( ). Complete the mapping diagram and then graph g (). (To complete the mapping diagram, ou need to find the inputs for g that produce the inputs for f after ou multipl b 1_. Work backward from the inputs for f to the missing inputs for g b multipling b.) In general, how is the graph of g () = ƒ ( 1 b ) related to the graph of ƒ () when b is a number greater than 1? Input for g - 1 Input for f -1 1 3 Output for f - - Output for g - - - - 0 8 10 5 - D Make a Conjecture How would ou epect the graph of g () = ƒ ( 1 b ) to be related to the graph of ƒ () when b is a number between 0 and 1? Reflect. Suppose a function ƒ () has a domain of 1, and a range of 1,. When the graph of ƒ () is stretched or compressed verticall b a factor of a, how do the domain and range change? 5. You can transform the graph of ƒ () to obtain the graph of g () = a ƒ (-h) + k b combining transformations. Predict what will happen b completing the table. Value of a a > 1 0 < a < 1 Transformations of the Graph of f() Stretch verticall b a factor of a, and translate h units horizontall and k units verticall.. You can transform the graph of ƒ () to obtain the graph of g () = ƒ ( 1 b ( - h) ) + k b combining transformations. Predict what will happen b completing the table. Value of b Transformations of the Graph of f() Houghton Mifflin Harcourt Publishing Compan b > 1 Stretch horizontall b a factor of b, and translate h units horizontall and k units verticall. 0 < b < 1 Module 1 3 Lesson 3
Eplore 3 Investigating Reflections of Function Graphs When the parameter in a stretch or compression is negative, another transformation called a reflection is introduced. Eamining reflections will also tell ou whether a function is an even ƒunction or an odd ƒunction. An even function is one for which ƒ (-) = ƒ () for all in the domain of the function, while an odd function is one for which ƒ (-) = -ƒ () for all in the domain of the function. A function is not necessaril even or odd; it can be neither. A First graph g () = a ƒ () where a = -1. Complete the inputoutput table and then graph g () = -ƒ (). In general, how is the graph of g () = -ƒ () related to the graph of ƒ ()? f() -f() -1 - - - 0 1-3 - 5 - B Now graph g () = ƒ ( 1 b ) where b = -1. Complete the input-output table and then graph g () = ƒ (-). In general, how is the graph of g () = ƒ (-) related to the graph of ƒ ()? Input for g (-1) 1-1 Input for f -1 1 3 Output for f - - Output for g - - - - 0 5 - Houghton Mifflin Harcourt Publishing Compan Reflect 7. Discussion Suppose a function ƒ () has a domain of 1, and a range of 1,. When the graph of ƒ () is reflected across the -ais, how do the domain and range change? 8. For a function ƒ (), suppose the graph of ƒ (-), the reflection of the graph of ƒ () across the -ais, is identical to the graph of ƒ (). What does this tell ou about ƒ ()? Eplain. 9. Is the function whose graph ou reflected across the aes in Steps A and B an even function, an odd function, or neither? Eplain. Module 1 35 Lesson 3
Eplain 1 Transforming the Graph of the Parent Quadratic Function You can use transformations of the graph of a basic function, called a parent function, to obtain the graph of a related function. To do so, focus on how the transformations affect reference points on the graph of the parent function. For instance, the parent quadratic function is ƒ () =. The graph of this function is a U-shaped curve called a parabola with a turning point, called a verte, at (0, 0). The verte is a useful reference point, as are the points (-1, 1) and (1, 1). - - 0 - - Eample 1 Describe how to transform the graph of f () = to obtain the graph of the related function g (). Then draw the graph of g (). A g () = -3ƒ ( - ) - Parameter and Its Value a = -3 b = 1 h = k = - Effect on the Parent Graph vertical stretch of the graph of ƒ () b a factor of 3 and a reflection across the -ais Since b = 1, there is no horizontal stretch or compression. horizontal translation of the graph of ƒ () to the right units vertical translation of the graph of ƒ () down units Appling these transformations to a point (, ) on the parent graph results in the point ( +, -3 -). The table shows what happens to the three reference points on the graph of ƒ (). Point on the Graph of f() Corresponding Point on g() (-1, 1) (-1 +, -3 (1) - ) = (1, -7) (0, 0) (0 +, -3 (0) - ) = (, -) (1, 1) (1 +, -3 (1) - ) = (3, -7) Use the transformed reference points to graph g(). - 0 - - Houghton Mifflin Harcourt Publishing Compan -8 Module 1 3 Lesson 3
B g () = ƒ ( 1_ ( + 5) ) + Parameter and Its Value a = b = Effect on the Parent Graph Since a = 1, there is no vertical stretch, no vertical compression, and no reflection across the -ais. The parent graph is stretched/compressed horizontall b a factor of. There is no reflection across the -ais. h = The parent graph is translated units horizontall/verticall. k = The parent graph is translated units horizontall/verticall. Appling these transformations to a point on the parent graph results in the point ( - 5, + ). The table shows what happens to the three reference points on the graph of ƒ (). Point on the Graph of f() Corresponding Point on the Graph of g() (-1, 1) ( (-1) - 5, 1 + ) = (, ) (0, 0) ( (0) - 5, 0 + ) = (, ) (1, 1) ( (1) - 5, 1 + ) = (, ) Use the transformed reference points to graph g (). Houghton Mifflin Harcourt Publishing Compan Reflect -10-8 - - 0-10. Is the function ƒ () = an even function, an odd function, or neither? Eplain. 11. The graph of the parent quadratic function ƒ () = has the vertical line = 0 as its ais of smmetr. Identif the ais of smmetr for each of the graphs of g() in Parts A and B. Which transformation(s) affect the location of the ais of smmetr? Module 1 37 Lesson 3
Your Turn 1. Describe how to transform the graph of ƒ () = to obtain the graph of the related function g () = ƒ (- ( - 3) ) + 1. Then draw the graph of g (). 3 1 0-1 1 3 5 Eplain Modeling with a Quadratic Function You can model real-world objects that have a parabolic shape using a quadratic function. In order to fit the function s graph to the shape of the object, ou will need to determine the values of the parameters in the function g () = a ƒ ( 1 b ( - h) ) + k where ƒ () =. Note that because ƒ () is simpl a squaring function, it s possible to pull the parameter b outside the function and combine it with the parameter a. Doing so allows ou to model real-objects using g () = a ƒ ( - h) + k, which has onl three parameters. When modeling real-world objects, remember to restrict the domain of g () = a ƒ ( - h) + k to values of that are based on the object s dimensions. Eample An old stone bridge over a river uses a parabolic arch for support. In the illustration shown, the unit of measurement for both aes is feet, and the verte of the arch is point C. Find a quadratic function that models the arch, and state the function s domain. 0 10 0 30 0 50-10 -0 C (7, -5) A (, -0) B (5, -0) Houghton Mifflin Harcourt Publishing Compan Module 1 38 Lesson 3
Analze Information Identif the important information. The shape of the arch is a. The verte of the parabola is. Two other points on the parabola are and. Formulate a Plan You want to find the values of the parameters a, h, and k in g () = a ƒ ( - h) + k where ƒ () =. You can use the coordinates of point to find the values of h and k. Then ou can use the coordinates of one of the other points to find the value of a. Solve The verte of the graph of g () is point C, and the verte of the graph of ƒ () is the origin. Point C is the result of translating the origin 7 units to the right and 5 units down. This means that h = 7 and k = -5. Substituting these values into g () gives g () = a ƒ ( - 7) - 5. Now substitute the coordinates of point B into g () and solve for a. g () = a ƒ ( - 7) - 5 Write the general function. g ( ) = a ƒ (5-7) - 5 Substitute 5 for. -0 = a ƒ (5-7) - 5 Replace g (5) with -0, the -value of B. -0 = a ƒ ( ) - 5 Simplif. -0 = a (5) - 5 Evaluate ƒ (5) for ƒ () =. a = Solve for a. Houghton Mifflin Harcourt Publishing Compan Substitute the value of a into g (). g () = - 3_ ƒ ( - 7) - 5 15 The arch eists onl between points A and B, so the domain of g () is { 5}. Justif and Evaluate To justif the answer, verif that g () = -0. g () = - 3_ ƒ ( - 7) - 5 15 Write the function. 3_ ƒ ( 15-7) - 5 Substitute for. = - 3_ g ( ) = - = - 15 ƒ ( ) - 5 Subtract. 3_ 15 ( ) - 5 Evaluate ƒ (-5). = -0 Simplif. Module 1 39 Lesson 3
Your Turn 13. The netting of an empt hammock hangs between its supports along a curve that can be modeled b a parabola. In the illustration shown, the unit of measurement for both aes is feet, and the verte of the curve is point C. Find a quadratic function that models the hammock s netting, and state the function s domain. A (-, ) B (8, ) C (3, 3) - 0 8 Elaborate 1. What is the general procedure to follow when graphing a function of the form g () = a ƒ ( - h) + k given the graph of ƒ ()? 15. What are the general steps to follow when determining the values of the parameters a, h, and k in ƒ () = a ( - h) + k when modeling a parabolic real-world object? Houghton Mifflin Harcourt Publishing Compan 1. Essential Question Check-In How can the graph of a function ƒ () be transformed? Module 1 0 Lesson 3
Evaluate: Homework and Practice Write g () in terms of ƒ () after performing the given transformation of the graph of ƒ (). Online Homework Hints and Help Etra Practice 1. Translate the graph of ƒ () to the left 3 units.. Translate the graph of ƒ () up units. - - 0 - - - - 0-3. Translate the graph of ƒ () to the right units.. Translate the graph of ƒ () down 3 units. - - 0 - - - - 0 - - Houghton Mifflin Harcourt Publishing Compan 5. Stretch the graph of ƒ () horizontall b a factor of 3. - - 0 -. Stretch the graph of ƒ () verticall b a factor of. - - 0 - - - Module 1 1 Lesson 3
7. Compress the graph of ƒ () horizontall b a factor of 3. 8. Compress the graph of ƒ () verticall b a factor of. - - 0 - - - 0-9. Reflect the graph of ƒ () across the -ais. 10. Reflect the graph of ƒ () across the -ais. - - 0 - - - - 0 - - 11. Reflect the graph of ƒ () across the -ais. 1. Reflect the graph of ƒ () across the -ais. - - 0 - - - - 0 - - Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 3
13. Determine if each function is an even function, an odd function, or neither. a. b. c. - - 0 - - - 0 - - - 0-1. Determine whether each quadratic function is an even function. Answer es or no. a. ƒ () = 5 b. ƒ () = ( - ) c. ƒ () = ( _ 3) d. ƒ () = + Describe how to transform the graph of f () = to obtain the graph of the related function g (). Then draw the graph of g (). ƒ ( + ) 15. g () = - _ 3 1. g () = ƒ () + Houghton Mifflin Harcourt Publishing Compan - - 0 - - - - 0 - Module 1 3 Lesson 3
17. Architecture Fling buttresses were used in the construction of cathedrals and other large stone buildings before the advent of more modern construction materials to prevent the walls of large, high-ceilinged rooms from collapsing. The design of a fling buttress includes an arch. In the illustration shown, the unit of measurement for both aes is feet, and the verte of the arch is point C. Find a quadratic function that models the arch, and state the function s domain. 1 10 8 C(, 1) B(8, ) 0 8 10 18. A red velvet rope hangs between two stanchions and forms a curve that can be modeled b a parabola. In the illustration shown, the unit of measurement for both aes is feet, and the verte of the curve is point C. Find a quadratic function that models the rope, and state the function s domain. A (1, ) B (7, ) C (, 3.5) 0 8 Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 3
H.O.T. Focus on Higher Order Thinking 19. Multiple Representations The graph of the function g () = ( 1 + ) is shown. Use the graph to identif the transformations of the graph of ƒ () = needed to produce the graph of g (). (If a stretch or compression is involved, give it in terms of a horizontal stretch or compression rather than a vertical one.) Use our list of transformations to write g () in the form g () = ƒ ( 1_ b ( - h) ) + k. Then show wh the new form of g () is algebraicall equivalent to the given form. -8 - - 0 - - Houghton Mifflin Harcourt Publishing Compan Image Credits: Michael Dwer/Alam 0. Represent Real-World Situations The graph of the ceiling function, ƒ() =, is shown. This function accepts an real number as input and delivers the least integer greater than or equal to as output. For instance, ƒ1.3 = because is the least integer greater than or equal to 1.3. The ceiling function is a tpe of step function, so named because its graph looks like a set of steps. Write a function g() whose graph is a transformation of the graph of ƒ() based on this situation: A parking garage charges $ for the first hour or less and $ for ever additional hour or fraction of an hour. Then graph g(). - 0 - - Parking fee (dollars) 8 0 f 1 3 Time (h) t Module 1 5 Lesson 3
Lesson Performance Task You are designing two versions of a chair, one without armrests and one with armrests. The diagrams show side views of the chair. Rather than use traditional straight legs for our chair, ou decide to use parabolic legs. Given the function ƒ () =, write two functions, g () and h (), whose graphs represent the legs of the two chairs and involve transformations of the graph of ƒ (). For the chair without armrests, the graph of g () must touch the bottom of the chair s seat. For the chair with armrests, the graph of h () must touch the bottom of the armrest. After writing each function, graph it. Vertical dimensions (in.) 3 30 8 0 18 1 1 1 10 8 0 8 1 1 0 Vertical dimensions (in.) 3 30 8 0 18 1 1 1 10 8 0 8 1 1 0 Horizontal dimensions (in.) Horizontal dimensions (in.) Houghton Mifflin Harcourt Publishing Compan Module 1 Lesson 3