February 14, S2.5q Transformations. Vertical Stretching and Shrinking. Examples. Sep 19 3:27 PM. Sep 19 3:27 PM.

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 2: More on Functions 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 2.2 The Algebra of Functions 2.3 The Composition of Functions 2.4 Symmetry 2.5 Transformations 2.6 Variation and Applications Reflection about x axis or y axis. Reflection about the line y = x. Transformation or translation of functions. Get Transformations Summary at summary.pdf Get Transformations xy tables at xy tables.pdf 2.5 Transformations Given the graph of a function, graph its transformation under translations, reflections, stretchings, and shrinkings. This 8 minute YouTube video gives a very good summary of Function Transformations. Horizontal and Vertical Translations (10 minutes) v=cesxljaq6mk&feature=relmfu Vertical Stretches & Compressions (10 minutes) w&feature=relmfu Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link. Vertical Translation Vertical Translation For b > 0, the graph of y = f(x) + b is up b units; the graph of y = f(x) b is down b units. Transformation or translation of functions. Horizontal Translation Horizontal Translation For d > 0, the graph of y = f(x d) is right d units; the graph of y = f(x + d) is left d units. NOTE: Set (x d) = 0 and solve for x = d which indicates movement to the right because d > 0. Set (x + d) = 0 and solve for x = d which indicates movement to the left because d > 0 or d < 0. Reflections The graph of y = f(x) is the reflection of the graph of y = f(x) across the x axis. The graph of y = f( x) is the reflection of the graph of y = f(x) across the y axis. If a point (x, y) is on the graph of y = f(x), then (x, y) is on the graph of y = f(x), and ( x, y) is on the graph of y = f( x). Example Reflection of the graph y = 3x 3 4x 2 across the x axis. Vertical Stretching and Shrinking The graph of y = af (x) can be obtained from the graph of y = f(x) by stretching vertically for a > 1, or shrinking vertically for 0 < a < 1. For a < 0, the graph is also reflected across the x axis. (The y coordinates of the graph of y = af (x) can be obtained by multiplying the y coordinates of y = f(x) by a.) Example Reflection of the graph y = x 3 2x 2 across the y axis. Stretch y = x 3 x vertically. Each y value of y = 2(x 3 x) is two times the corresponding y value for y = x 3 x. 1

2 Shrink y = x 3 x vertically. Each y value of y = (1/10)(x 3 x) is one tenth times the corresponding y value for y = x 3 x. Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for c > 1, or stretching horizontally for 0 < c < 1. For c < 0, the graph is also reflected across the y axis. Stretch and reflect y = x 3 x across the x axis Each y value of y = 2(x 3 x) is two times the corresponding y value for y = x 3 x and reflected about the x axis. (The x coordinates of the graph of y = f(cx) can be obtained by dividing the x coordinates of the graph of y = f(x) by c.) For example: y 1 = f(4x) and y 2 = f(x). The point (12, 5) on y 2 becomes the point (3, 5) on y 1 because 4x = 12 or x = 3. Shrink y = x 3 x horizontally. Each x value of y = (2x) 3 (2x) is 1/2 times the corresponding x value for y = x 3 x. Stretch horizontally and reflect y = x 3 x. Each x value of y = ( 0.5x) 3 ( 0.5x) is 2 times the corresponding x value for y = x 3 x and reflected about the x axis. Stretch y = x 3 x horizontally. Each x value of y = (0.5x) 3 (0.5x) is 2 times the corresponding x value for y = x 3 x. 210/8. Describe how the graph of the function can be obtained from one of the basic graphs on p Then graph the function by hand or with a graphing calculator. g(x) = 1 / (x 2) 210/18. Describe how the graph of the function can be obtained from one of the basic graphs on p Then graph the function by hand or with a graphing calculator. f(x) = ( x) 3 Sep 22 1:38 PM Sep 22 1:38 PM 2

3 210/36. Describe how the graph of the function can be obtained from one of the basic graphs on p f(x) = 3(x + 4) /26. Describe how the graph of the function can be obtained from one of the basic graphs on p f(x) = x 3 4 Sep 22 1:45 PM Sep 22 1:45 PM The point ( 12, 4) is on the graph of y = f(x). Find the corresponding point on the graph of y = g(x). 210/38. g(x) = f(x 2) 210/40. g(x) = f(4x) Given that f(x) = x 2 + 3, match the function with a transformation of from one of A. f(x 2), B. f(x) + 1, C. 2f(x), D. f(3x) 210/46. g(x) = 9x /48. g(x) = 2x /41. g(x) = f(x) 2 210/44. g(x) = f(x) Sep 22 1:53 PM Write an equation for a function that has a graph with the given characteristics. Given that f(x) = x 2 + 3, match the function with a transformation of from one of A. f(x 2), B. f(x) + 1, C. 2f(x), D. f(3x) 210/50. The shape of y = x, but shifted left 6 units and down 5 units. 210/46. g(x) = 9x /48. g(x) = 2x /52. The shape of y = x 3, but reflected across the x-axis and shifted right 5 units. 3

4 Write an equation for a function that has a graph with the given characteristics. 211/60. y = (1/2)f(x) 210/54. The shape of y = x 2, but shifted right 6 units and up 2 units. 210/56. The shape of y = x, but stretched horizontally by a factor of 2 and shifted down 5 units. 211/62. y = f(2x) 211/63. y = (1/2)f(x 1) /64. y = 3f(x + 1) 4 211/72. g(x) = f(x) + 3 4

5 171S2.5q Transformations 211/74. g(x) = f( x) 211/76. g(x) = (1/3)f(x) 3 212/82. A graph of the function is shown below. Exercises show graphs of functions transformed from this one. Find a formula for the function h(x). 211/78. g(x) = f(x + 2) Reference point (0, 0) stays at (0, 0) and all x values in f(x) remain the same in h(x); no horizontal movement. No vertical shift because f(0) = h(0). Y values in h(x) are 1/2 Y values in f(x); vertical shrink (stretch by 1/2). Thus, h(x) = (1/2)f(x) = 0.5(x3 3x2) = 0.5x3 1.5x2. Sep 22 8:50 PM 212/84. A graph of the function f(x) = x3 3x2 is shown below. Exercises show graphs of functions transformed from this one. Find a formula for each function. Reference point (0, 0) moves to (2, 1); 2 units to right and one unit up. From ( 1, 4) to (2, 4) on f(x) is same as from (1, 3) to (4, 3) on t(x); no horizontal stretch/shrink. From ( 1, 4) to (0, 0) on f(x) is same as from (1, 3) to (2, 1) on t(x); no vertical stretch/shrink. Thus, t(x) = (x 2)3 3(x 2) Sep 22 8:50 PM 5