- Transformations of Absolute Value Functions TEKS FOCUS VOCABULARY Compression A compression is a TEKS (6)(C) Analze the effect on the graphs of f() = when f() is replaced b af(), f(b), f( - c), and f() + d for specific positive and negative real values of a, b, c, and d. Transformation A transformation of transformation that decreases the distance between corresponding points of a graph and a line. General form of the absolute value Translation A translation is a function a function of the form f() = a - h + k TEKS ()(E) Create and use representations to organize, record, and communicate mathematical ideas. transformation that shifts a graph verticall, horizontall, or both without changing its shape or orientation. Reflection A reflection is a transformation that flips a graph across a line, such as the - or -ais. Stretch A stretch is a transformation Additional TEKS ()(C), ()(A) a function is a simple change to the equation of the function that results in a change in the graph of the function such as a translation or reflection. that increases the distance between corresponding points of a graph and a line. Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING You can quickl graph absolute value functions b transforming the graph of =. Concept Summar Absolute Value Function Famil Parent Function f() = Translation = + d d 7 shifts up d units d 6 shifts down d units Stretch, Compression, and Reflection = a a 7 vertical stretch 6 a 6 vertical compression (shrink) a 6 reflection across -ais = - c c 7 c 6 shifts to the right c units shifts to the left c units = b b 7 horizontal compression (shrink) 6 b 6 horizontal stretch b 6 reflection across -ais 5
Ke Concept General Form of the Absolute Value Function When a function has a verte, the letters h and k are used to represent the coordinates of the verte. Because an absolute value function has a verte, the general form is = a - h + k. The vertical stretch or compression factor is a, the verte is located at (h, k), and the ais of smmetr is the line = h. Problem TEKS Process Standard ()(E) Analzing the Graph of f () + d When f () = What are the graphs of the absolute value functions = and = +? How are these graphs different from the parent function f () =? Make a table of values that ou can use to compare the -values of each transformed function to the -values of the parent function. + How are ou changing the -coordinates to get the new graphs? For = -, ou are subtracting from each -coordinate of the graph of =, so the graph will move downward. For = +, ou are adding, so the graph will move up. To draw the graph of = -, ou can move the entire graph of = down units without changing its shape. This is a shift or translation down units. The verte is now at (, -) instead of (, ). For the graph of = +, ou can translate the graph of = up unit. The verte of this graph is at (, ). The ais of smmetr is still the same in both graphs. O O 5 Lesson - Transformations of Absolute Value Functions
Problem Analzing the Graph of f ( c) When f () = In Problem ou saw that the graph of = + d translated the graph of = up or down, depending on whether d was positive or negative. Make a table to see what adding or subtracting a number from inside the absolute value does to the graph of the function. What are the graphs of the absolute value functions = and = +? How are these graphs different from the parent function f () =? From the information in the table, ou can see that the verte of = - is at the point (, ). This means ou can draw the graph of = - b translating the graph of = right units. Similarl, if ou translate the graph of = left unit, ou produce the graph of = +. O 6 O 5 + 5 Problem Analzing the Graph of af () When f () = How can ou describe the effect on the graph of f () =? Use the equation to help ou. The -coordinate will be of what it was before. A What is the graph of =? The graph is a vertical compression of the graph of f () = b the factor. Graph the right branch and use smmetr to graph the left branch. Use 6 smmetr to O graph this branch. 6 Starting at (, ), graph. B What is the graph of =? Because the value of a is negative, the graph is reflected across the -ais. Then the graph is a reflection of the graph of f () = followed b a vertical stretch b the factor. O 6 5
Problem TEKS Process Standard ()(C) Analzing the Graph of f (b) When f () Which answer choices will help ou consider graphs? You can use paper and pencil or a graphing calculator to graph functions. Consider how different values of b affect the graph of the function f (b) when f (). Use these values for b:,,, and. A Which tool would ou use to analze how changes in the value of b affect the graph of the function: real objects, manipulatives, paper and pencil, or technolog? Wh? Use technolog. With a graphing calculator ou can graph all of the functions at once to quickl analze how the different values of b affect the graph. B Describe the effect on the graph b changing the value of b. When b 7, the graph of f () = is compressed horizontall to produce the graph of f () = b. When 6 b 6, the graph of f () = is stretched horizontall to produce the graph of f () = b. The sign of b does not affect the graph, since the absolute value of an epression is alwas nonnegative. O O O Problem 5 f() = or f() = - f() = f() = Identifing Transformations To what should ou compare? Compare it to the general form, = a - h + k. Without graphing, what are the verte, ais of smmetr, maimum or minimum, and - and -intercepts of the graph of? How is the parent function transformed? Compare = - + with the general form = a - h + k. a =, h =, and k =. The verte is (, ) and the ais of smmetr is =. Because the value of a is positive, the -coordinate of the verte is the minimum point of the graph. The minimum is. To find the -intercept(s), set =. = - + - = - The equation = - + has no solutions, so there are no -intercepts. continued on net page 5 Lesson - Transformations of Absolute Value Functions
Problem 5 continued To find the -intercept, set =. Problem 6 = - + = The -intercept is (, ) or. The parent function = is translated units to the right, verticall stretched b the factor, and translated units up. Check Check b graphing the equation on a graphing calculator. Writing an Absolute Value Function What is the equation of the absolute value function? What does the graph tell ou about a? The upside-down V suggests that a 6. Step Identif the verte. The verte is at (-, ), so h = - and k =. Step Identif a. The slope of the branch to the right of the verte is -, so a = -. O Step Write the equation. Substitute the values of a, h, and k into the general form = a - h + k. The equation that describes the graph is = - + +. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Make a table of values for each equation. Then graph the equation. Analze the effect on the graph of the parent function f () =. For additional support when completing our homework, go to.. = +. = -. = -. = + 5. = + 6. = -.5 7. = - + 8. = + 6-9. = -.5 +.5 Graph each equation. Then analze the effect on the graph of the parent function f () =.. =. = -. = -. =. = 5. = - Without graphing, identif the verte, ais of smmetr, and transformations from the parent function f ()=. 6. = + - 7. = - 6 8. = + 6 9. = - +. = - - 5. = - - 6 55
Write an absolute value equation for each graph... O O 6 6. Displa Mathematical Ideas ()(G) Graph = - + +. List the - and -intercepts, if an. 5. Graph = - +. List the verte and the - and -intercepts, if an. 6. A classmate sas that the graphs of = - and = - are identical. Graph each function and eplain wh our classmate is not correct. 7. The graphs of the absolute value functions f () and g () are given. a. Describe a series of transformations that ou can use to transform f () into g(). b. Eplain Mathematical Ideas ()(G) If ou change the order of the transformations ou found in part(a), could ou still transform f () into g()? Eplain. 8. Graph each pair of equations on the same coordinate grid. a. = + ; = + b. = 5 - ; = 5 - c. Eplain Mathematical Ideas ()(G) Eplain wh each pair of graphs in parts (a) and (b) are different. Select Tools to Solve Problems ()(C) Use paper and pencil or a graphing calculator to compare the given graph to the parent function f ()=. 5 9. f () = -. f () =. f () =. O f() g() 6. Consider how different values of b affect the graph of the function f (b) when f () =. What aspects of the parent function do not change for an value of b? What aspects change for particular values of b? Graph each absolute value equation. Analze the effect on the graph of the parent function f () =.. = - -. = 5-5. = + 6. = - 6 + 7. = - - 8. = + 6 9. = + -. = 6 -. = 6 - + 56 Lesson - Transformations of Absolute Value Functions
. The graph at the right models the distance between a roadside stand and a car traveling at a constant speed. The -ais represents time and the -ais represents distance. Which equation best represents the relation shown in the graph? A. = 6 C. = + 6 B. = D. = +. a. Graph the equations f () = - - and g () = - ( - ) on the same set of aes. b. Analze Mathematical Relationships ()(F) Describe the similarities and differences in the graphs.. a. Use a graphing calculator. Graph = k and = k for some positive value of k. b. Graph = k and = k for some negative value of k. c. What conclusion can ou make about the graphs of = k and = k? Graph each absolute value equation. 5. = - 6. = Miles From Roadside Stand + - 7. = + - 8 O Hours Before Roadside Stand Hours After Roadside Stand TEXAS Test Practice 8. The graph at the right shows which equation? A. = - + C. = - - B. = - + D. = - - 9. How are the graphs of = and = + related? F. The graph of = + is the graph of = translated down two units. G. The graph of = + is the graph of = translated up two units. H. The graph of = + is the graph of = translated to the left two units. J. The graph of = + is the graph of = translated to the right two units. 5. What is the equation of a line parallel to = that passes through the point (, )? A. = + B. = + C. = - D. = - 5. Is = a function? Eplain. 57