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- 1. Transformations of Graphs http://www.lahc.edu/math/precalculus/math_260a.html
- 2. Transformations of Graphs Function Calisthenics (Origin Unknown)
- 3. eyes, lips, and faces
- 4. Objective: * Vertical stretches and compressions of graphs * Vertical and horizontal translations of graphs Transformations of Graphs
- 5. Using image manipulation software, we can drag and drop or stretch images. Transformations of Graphs
- 6. Using image manipulation software, we can drag and drop or stretch images. For example, from the original image below, Transformations of Graphs y = f(x)
- 7. Using image manipulation software, we can drag and drop or stretch images. For example, from the original image below, we can elongate it, stretch Transformations of Graphs y = f(x)
- 8. Using image manipulation software, we can drag and drop or stretch images. For example, from the original image below, we can elongate it, drag and lower it, stretch lower Transformations of Graphs y = f(x)
- 9. Using image manipulation software, we can drag and drop or stretch images. For example, from the original image below, we can elongate it, drag and lower it, stretch lower vertically reflected and reflect it vertically to create another pattern. Transformations of Graphs y = f(x)
- 10. Using image manipulation software, we can drag and drop or stretch images. For example, from the original image below, we can elongate it, drag and lower it, stretch lower vertically reflected and reflect it vertically to create another pattern. If the original image is the graph of the function y = f(x), then these transformations can be tracked easily with the notation of functions. Transformations of Graphs y = f(x)
- 11. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. x P = (x, f(x)) f(x) = ht y= f(x) x Vertical Translations
- 12. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. Hence expressions in terms of f(x) may be translated precisely into the corresponding manipulation of the graph. x P = (x, f(x)) f(x) = ht y= f(x) x Vertical Translations
- 13. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. Hence expressions in terms of f(x) may be translated precisely into the corresponding manipulation of the graph. Vertical Translations x P = (x, f(x)) f(x) = ht y= f(x) x Vertical Translations
- 14. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. x P = (x, f(x)) f(x) = ht y= f(x) Hence expressions in terms of f(x) may be translated precisely into the corresponding manipulation of the graph. Vertical Translations Changing the y–coordinate to f(x) + 3 moves P vertically up 3 units. (x, f(x)+3) x Vertical Translations
- 15. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. x P = (x, f(x)) f(x) = ht y= f(x) Hence expressions in terms of f(x) may be translated precisely into the corresponding manipulation of the graph. Vertical Translations Changing the y–coordinate to f(x) + 3 moves P vertically up 3 units. (x, f(x)+3) y= f(x) + 3 Hence setting y = f(x) + 3 to all the points on the graph means to move the entire graph 3 units up as shown. x Vertical Translations
- 16. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown, the output f(x) = y represents the height of the point. x P = (x, f(x)) f(x) = ht y= f(x) Hence expressions in terms of f(x) may be translated precisely into the corresponding manipulation of the graph. Vertical Translations Changing the y–coordinate to f(x) + 3 moves P vertically up 3 units. Hence setting y = f(x) + 3 to all the points on the graph means to move the entire graph 3 units up as shown. Likewise changing the y–coordinate to f(x) – 3 corresponds to moving y = f(x) down 3 units. (x, f(x)+3) (x, f(x)–3) y= f(x) – 3 y= f(x) + 3 x Vertical Translations
- 17. The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Vertical Translations Vertical Translations
- 18. The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Vertical Translations P = (x, f(x)) y= f(x) (x, f(x)+c) where c > 0 (x, f(x)+c) where c < 0 Vertical Translations
- 19. The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Vertical Translations P = (x, f(x)) y= f(x) (x, f(x)+c) where c > 0 (x, f(x)+c) where c < 0 Here are the graphs of: y = f(x) = x2 vs. y = f(x) + 5 = x2 + 5 y = x2 y = x2 + 5 (0, 0) (0, 5) Vertical Translations
- 20. Vertical Translations P = (x, f(x)) y= f(x) (x, f(x)+c) where c > 0 (x, f(x)+c) where c < 0 Here are the graphs of: y = f(x) = x2 vs. y = f(x) + 5 = x2 + 5 y = x2 y = x2 + 5 (0, 0) (0, 5) x Vertical Translations The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Assuming c > 0, move the graph (x, f(x)) up to obtain the graph (x, f(x) + c). (0, 5)
- 21. Vertical Translations P = (x, f(x)) y= f(x) (x, f(x)+c) where c > 0 (x, f(x)+c) where c < 0 Here are the graphs of: y = f(x) = x2 vs. y = f(x) + 5 = x2 + 5 y = f(x) = x2 vs. y = f(x) – 5 = x2 – 5 y = x2 y = x2 + 5 y = x2 – 5 y = x2 (0, 0) (0, 5) (0, 0) (0, –5) x x Vertical Translations The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Assuming c > 0, move the graph (x, f(x)) up to obtain the graph (x, f(x) + c). (0, 5)
- 22. The graph of (x, y = f(x) + c) is the vertical translation of the graph (x, f(x)). Assuming c > 0, move the graph (x, f(x)) up to obtain the graph (x, f(x) + c). Vertical Translations move the graph (x, f(x)) down to obtain the graph (x, f(x) – c). P = (x, f(x)) y= f(x) (x, f(x)+c) where c > 0 (x, f(x)+c) where c < 0 Here are the graphs of: y = f(x) = x2 vs. y = f(x) + 5 = x2 + 5 y = f(x) = x2 vs. y = f(x) – 5 = x2 – 5 y = x2 y = x2 + 5 y = x2 – 5 y = x2 (0, 0) (0, 5) (0, 0) (0, –5) x x Vertical Translations
- 23. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Vertical Stretches and Compressions
- 24. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Changing the y–coordinate to 3f(x) would triple the height of the point P. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Vertical Stretches and Compressions
- 25. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Changing the y–coordinate to 3f(x) would triple the height of the point P. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Hence setting y = 3f(x) to all the points on the graph means to elongate the entire graph 3 times while the x–intercepts (x, 0)’s remain fixed because 3(0) = 0. y= 3f(x) Vertical Stretches and Compressions
- 26. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Changing the y–coordinate to 3f(x) would triple the height of the point P. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Hence setting y = 3f(x) to all the points on the graph means to elongate the entire graph 3 times while the x–intercepts (x, 0)’s remain fixed because 3(0) = 0. y= 3f(x) Vertical Stretches and Compressions
- 27. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Changing the y–coordinate to 3f(x) would triple the height of the point P. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Hence setting y = 3f(x) to all the points on the graph means to elongate the entire graph 3 times while the x–intercepts (x, 0)’s remain fixed because 3(0) = 0. Likewise setting y = (1/3)f(x) would compress the entire graph to a third of it’s original size while the x–intercepts or (x, 0)’s remain fixed. y= 3f(x) Vertical Stretches and Compressions
- 28. Given a function y = f(x) and P = (x, y = f(x)) a generic point on the graph as shown. Changing the y–coordinate to 3f(x) would triple the height of the point P. Vertical Stretches and Compressions P = (x, f(x)) y= f(x) Hence setting y = 3f(x) to all the points on the graph means to elongate the entire graph 3 times while the x–intercepts (x, 0)’s remain fixed because 3(0) = 0. Likewise setting y = (1/3)f(x) would compress the entire graph to a third of it’s original size while the x–intercepts or (x, 0)’s remain fixed. y= 3f(x) y= f(x)/3 Vertical Stretches and Compressions
- 29. Vertical Stretches and Compressions Assuming c > 0, the graph of y = cf(x) is the vertical-stretch or compression of y = f(x). Vertical Stretches and Compressions
- 30. Vertical Stretches and Compressions Assuming c > 0, the graph of y = cf(x) is the vertical-stretch or compression of y = f(x). Here are the graphs of: y = f(x) = 4 – x2 vs. y = 3f(x) = 3(4 – x2) y = 4 – x2 y = 3(4 – x2) (0, 4) (0, 12) (–2, 0) (2, 0) x Vertical Stretches and Compressions
- 31. Vertical Stretches and Compressions Assuming c > 0, the graph of y = cf(x) is the vertical-stretch or compression of y = f(x). If c > 1, it is a vertical stretch by a factor of c. Here are the graphs of: y = f(x) = 4 – x2 vs. y = 3f(x) = 3(4 – x2) y = 4 – x2 y = 3(4 – x2) (0, 4) (0, 12) (–2, 0) (2, 0) c = 3 x Vertical Stretches and Compressions
- 32. Vertical Stretches and Compressions Assuming c > 0, the graph of y = cf(x) is the vertical-stretch or compression of y = f(x). If c > 1, it is a vertical stretch by a factor of c. Here are the graphs of: y = f(x) = 4 – x2 vs. y = 3f(x) = 3(4 – x2) y = f(x) = 4 – x2 vs. y = f(x)/2 = (4 – x2)/2 y = 4 – x2 y = 3(4 – x2) y = 4 – x2 y = (4 – x2)/2 (0, 4) (0, 12) (0, 4) (0, 2) (–2, 0) (2, 0) (–2, 0) (2, 0) c = 3 c = 1/2 x x Vertical Stretches and Compressions
- 33. Vertical Stretches and Compressions Assuming c > 0, the graph of y = cf(x) is the vertical-stretch or compression of y = f(x). If c > 1, it is a vertical stretch by a factor of c. If 0 < c < 1, it is a vertical compression by a factor of c. Here are the graphs of: y = f(x) = 4 – x2 vs. y = 3f(x) = 3(4 – x2) y = f(x) = 4 – x2 vs. y = f(x)/2 = (4 – x2)/2 y = 4 – x2 y = 3(4 – x2) y = 4 – x2 y = (4 – x2)/2 (0, 4) (0, 12) (0, 4) (0, 2) (–2, 0) (2, 0) (–2, 0) (2, 0) c = 3 c = 1/2 x x Vertical Stretches and Compressions
- 34. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) x Vertical Stretches and Compressions
- 35. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) Q = (x, –f(x)) x Vertical Stretches and Compressions
- 36. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) Hence setting y = –f(x) to all the points on the graph means to reflect the entire graph vertically across the x–axis. y= –f(x)Q = (x, –f(x)) x Vertical Stretches and Compressions
- 37. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) Hence setting y = –f(x) to all the points on the graph means to reflect the entire graph vertically across the x–axis. Hence setting y = –cf(x) = c*(–f(x)) to all the points means to reflect the entire graph (x, f(x)) vertically, then stretch the reflection by a factor of c. y= –f(x)Q = (x, –f(x)) x Vertical Stretches and Compressions
- 38. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) Hence setting y = –f(x) to all the points on the graph means to reflect the entire graph vertically across the x–axis. Hence setting y = –cf(x) = c*(–f(x)) to all the points means to reflect the entire graph (x, f(x)) vertically, then stretch the reflection by a factor of c. y= –f(x) y= –2f(x) Q = (x, –f(x)) (x, –2f(x)) x Vertical Stretches and Compressions
- 39. Changing the y-coordinate to –f(x) reflects the point P vertically across the x–axis to Q(x, –f(x)) as shown. P = (x, f(x)) y= f(x) Hence setting y = –f(x) to all the points on the graph means to reflect the entire graph vertically across the x–axis. Hence setting y = –cf(x) = c*(–f(x)) to all the points means to reflect the entire graph (x, f(x)) vertically, then stretch the reflection by a factor of c. The order of applying stretching vs. reflecting does not matter, “reflect then stretch” or “stretch then reflect” yields the same result. This is not the case for “stretch” vs. “vertical shift”. y= –f(x) y= –2f(x) Q = (x, –f(x)) (x, –2f(x)) x Vertical Stretches and Compressions
- 40. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) x Vertical Stretches and Compressions
- 41. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. x Vertical Stretches and Compressions
- 42. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. x Vertical Stretches and Compressions
- 43. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. x Vertical Stretches and Compressions
- 44. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. (–3, 1) x Vertical Stretches and Compressions y = g(x) = –2f(x) + 3y = f(x)
- 45. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. (–3, 1) (–3, –2f(–3) + 3 = 1) x Vertical Stretches and Compressions y = g(x) = –2f(x) + 3y = f(x)
- 46. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. (–3, 1) (–3, –2f(–3) + 3 = 1) (–1, –1) (–1, –2f(–1) + 3 = 5) x Vertical Stretches and Compressions y = g(x) = –2f(x) + 3y = f(x)
- 47. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. (–3, 1) (–3, –2f(–3) + 3 = 1) (–1, –1) (–1, –2f(–1) + 3 = 5) (1, 1) (1, –2f(1) + 3 = 1) (2, 1) (2, –2f(2) + 3 = 1) x Vertical Stretches and Compressions y = g(x) = –2f(x) + 3y = f(x)
- 48. Example A. a. Given the graph of y = f(x), graph y = g(x) = –2f(x) + 3 (–3, 1) (–1, –1) (1, 1) (2, 1) “–2f(x)" corresponds stretching the graph by a factor of 2, then reflecting the entire graph across the x axis. Afterwards move the graph vertically up by 3. To draw the graph, track the “important points” on the graph. Plug in their x–values to find the corresponding y–values, then plot their destinations. (–3, 1) (–3, –2f(–3) + 3 = 1) (–1, –1) (–1, –2f(–1) + 3 = 5) (1, 1) (1, –2f(1) + 3 = 1) (2, 1) (2, –2f(2) + 3 = 1) (–3, 1) (–1, 5) (1, 1) (2, 1) x x Vertical Stretches and Compressions y = g(x) = –2f(x) + 3y = f(x) Graph of y = g(x)
- 49. Horizontal Translations y= f(x) x Let y = f(x) be as shown.
- 50. Horizontal Translations y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x).
- 51. Horizontal Translations x y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1),
- 52. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), x+1
- 53. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). x+1
- 54. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). x+1 (x, f(x +1))
- 55. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). ux+1 (x, f(x +1)) Likewise if the input is u,
- 56. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). u (u+1, f(u+1)) ht = f(u+1) u+1x+1 (x, f(x +1)) Likewise if the input is u, then y = g(u) = f(u + 1)
- 57. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). u (u+1, f(u+1)) ht = f(u+1) u+1x+1 (x, f(x +1)) (u, f(u +1)) Likewise if the input is u, then y = g(u) = f(u + 1) so the point (u, f(u +1)) is on the graph of g(x).
- 58. Hence to obtain the graph of y = f(x + 1), shift the entire graph of y = f(x) left by 1 unit. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). u (u+1, f(u+1)) ht = f(u+1) u+1x+1 y = g(x) or y = f(x + 1) (x, f(x +1)) (u, f(u +1)) Likewise if the input is u, then y = g(u) = f(u + 1) so the point (u, f(u +1)) is on the graph of g(x). Shifting left by 1
- 59. Hence to obtain the graph of y = f(x + 1), shift the entire graph of y = f(x) left by 1 unit. Horizontal Translations x (x+1, f(x+1)) ht =f(x+1) y= f(x) x Let y = f(x) be as shown. Let’s define g(x) = f(x + 1), and the goal is to graph g(x). For the point x, the output is y = g(x) = f(x + 1), so the point (x, g(x) = f(x +1)) is on the graph of g(x). u (u+1, f(u+1)) ht = f(u+1) u+1x+1 y = g(x) or y = f(x + 1) (x, f(x +1)) (u, f(u +1)) Likewise if the input is u, then y = g(u) = f(u + 1) so the point (u, f(u +1)) is on the graph of g(x). Shifting left by 1 Similarly demonstrations show that to obtain the graph of y = f(x – 1), shift the graph of y = f(x) right by 1 unit.
- 60. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, Horizontal Translations
- 61. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, Horizontal Translations Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Horizontal Shifts
- 62. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: Horizontal Translations moves y = f(x) to the left for y = f(x + c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) b. h(x) = (x – 2)2 = f(x – 2) x y=x 2 Horizontal Shifts
- 63. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: Horizontal Translations moves y = f(x) to the left for y = f(x + c). b. h(x) = (x – 2)2 = f(x – 2) Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. x y=x 2 Horizontal Shifts
- 64. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: y=(x + 2)2 Horizontal Translations moves y = f(x) to the left for y = f(x + c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Horizontal Shifts
- 65. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: y=(x + 2)2 Horizontal Translations moves y = f(x) to the left for y = f(x + c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. The vertex of g(x) = (x + 2)2 is (–2, 0). x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Horizontal Shifts (–2, 0)
- 66. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: y=(x + 2)2 Horizontal Translations moves y = f(x) to the left for y = f(x + c). moves y = f(x) to the right for y = f(x – c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. The vertex of g(x) = (x + 2)2 is (–2, 0). x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Horizontal Shifts (–2, 0)
- 67. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: y=(x + 2)2 Horizontal Translations moves y = f(x) to the left for y = f(x + c). moves y = f(x) to the right for y = f(x – c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. The vertex of g(x) = (x + 2)2 is (–2, 0). x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Shift the graph of y = x2 right 2 units. The vertex of h(x) is (2, 0). Horizontal Shifts (–2, 0)
- 68. Horizontal Translations The graphs of y = f(x + c) are the horizontal shifts (or translations) of y = f(x) by c units, assuming c > 0: y=(x + 2)2 y=(x – 2)2 Horizontal Translations moves y = f(x) to the left for y = f(x + c). moves y = f(x) to the right for y = f(x – c). Example B. Graph the following functions by shifting the graph of y = f(x) = x2. Label their vertices. a. g(x) = (x + 2)2 = f(x + 2) Shift of the graph of y = x2 left 2 units. The vertex of g(x) = (x + 2)2 is (–2, 0). x y=x 2 b. h(x) = (x – 2)2 = f(x – 2) Shift the graph of y = x2 right 2 units. The vertex of h(x) is (2, 0). Horizontal Shifts (–2, 0) (2, 0)
- 69. Horizontal Stretches and Compressions x Let y = f(x) with its graph shown here and let g(x) = f(2x). 0 y= f(x) Horizontal Stretches and Compressions
- 70. Horizontal Stretches and Compressions x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) x0 y= f(x) Horizontal Stretches and Compressions x
- 71. Horizontal Stretches and Compressions x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). x0 y= f(x) Horizontal Stretches and Compressions x
- 72. Horizontal Stretches and Compressions x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions 2x ht =f(2x) (2x, f(2x)) x
- 73. Horizontal Stretches and Compressions 2x ht =f(2x) x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). x0 y= f(x) (2x, f(2x)) Horizontal Stretches and Compressions (x, g(x)=f(2x))
- 74. Horizontal Stretches and Compressions 2x ht =f(2x) x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). x0 y= f(x) (2x, f(2x)) Horizontal Stretches and Compressions x Likewise for the point u, the output is g(u) = f(2u) so the point (u, y = f(2u)) is on the graph of g(x). (x, g(x)=f(2x)) u
- 75. Horizontal Stretches and Compressions 2x ht =f(2x) x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). x0 y= f(x) (2x, f(2x)) Horizontal Stretches and Compressions x 2u (2u, f(2u)) ht = f(2u) Likewise for the point u, the output is g(u) = f(2u) so the point (u, y = f(2u)) is on the graph of g(x). (x, g(x)=f(2x)) u
- 76. Horizontal Stretches and Compressions 2x ht =f(2x) x Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). x0 y= f(x) (2x, f(2x)) Horizontal Stretches and Compressions x 2u (2u, f(2u)) ht = f(2u) Likewise for the point u, the output is g(u) = f(2u) so the point (u, y = f(2u)) is on the graph of g(x). (x, g(x)=f(2x)) u (u,g(u))=f(2u)
- 77. Horizontal Stretches and Compressions 2x (u,g(u))=f(2u) ht =f(2x) y=g(x)=f(2x) x 2u (2u, f(2u)) ht = f(2u) u Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). Likewise for the point u, the output is g(u) = f(2u) so the point (u, y = f(2u)) is on the graph of g(x). x0 y= f(x) Horizontal stretch by a factor of 2 Hence we see that the graph of y = f(2x) is the horizontal compression of the graph y = f(x) by a factor of ½ . (x, g(x)=f(2x)) (2x, f(2x)) Horizontal Stretches and Compressions
- 78. Horizontal Stretches and Compressions 2x (u,g(u))=f(2u) ht =f(2x) y=g(x)=f(2x) x 2u (2u, f(2u)) ht = f(2u) u Let y = f(x) with its graph shown here and let g(x) = f(2x). For the point x, the output is g(x) = f(2x) so the point (x, y = f(2x)) is on the graph of g(x). Likewise for the point u, the output is g(u) = f(2u) so the point (u, y = f(2u)) is on the graph of g(x). x0 y= f(x) Horizontal stretch by a factor of 2 Hence we see that the graph of y = f(2x) is the horizontal compression of the graph y = f(x) by a factor of ½ . Similarly, the graph of y = f(½ * x) is the horizontal stretch the graph of y = f(x) by a factor of 2. (Convince yourself of this fact.) (x, g(x)=f(2x)) (2x, f(2x)) Horizontal Stretches and Compressions
- 79. Horizontal Reflections Horizontal Stretches and Compressions Let y = f(x) with its graph shown here and let g(x) = f(–x). x y= f(x) y 0 Horizontal reflection
- 80. Horizontal Reflections Horizontal Stretches and Compressions Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) x y= f(x) x y 0 Horizontal reflection
- 81. Horizontal Reflections Horizontal Stretches and Compressions Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). x y= f(x) x y –x 0 Horizontal reflection
- 82. Horizontal Reflections Horizontal Stretches and Compressions Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). x y= f(x) x y –x (x, g(x)=f(–x)) 0 Horizontal reflection
- 83. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) u Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). y Horizontal reflection
- 84. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) –uu Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). y Horizontal reflection
- 85. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) –uu Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). (u, g(u)=f(–u)) y Horizontal reflection
- 86. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) –uu Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). (u, g(u)=f(–u)) y Horizontal reflection Hence we reflect the graph of y = f(x) horizontally across the y–axis to obtain the graph of y = f(–x).
- 87. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) –uu (u, g(x)=f(–x)) Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). (u, g(u)=f(–u)) y Horizontal reflection Hence we reflect the graph of y = f(x) horizontally across the y–axis to obtain the graph of y = f(–x).
- 88. Horizontal Reflections x Let y = f(x) with its graph shown here and let g(x) = f(–x). For the point x, the output is g(x) = f(–x) so the point (x, y = f(–x)) is on the graph of g(x). 0 y= f(x) Horizontal Stretches and Compressions x–x (x, g(x)=f(–x)) –uu (u, g(x)=f(–x)) Likewise for the point u, the output is g(u) = f(–u) so the point (u, y = f(–u)) is on the graph of g(x). (u, g(u)=f(–u)) y Horizontal reflection Hence we reflect the graph of y = f(x) horizontally across the y–axis to obtain the graph of y = f(–x). To graph y = f(–2x), we compress y = f(x) by a factor of ½ to obtain the graph of y = f(2x), then reflect the result to obtain the graph of y = f(–2x).
- 89. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions x y= f(x) y= f(x) x
- 90. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions x y= f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c x
- 91. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x) + 1 x y= f(x) + 2 y= f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c x
- 92. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c x
- 93. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c x c > 1, y = cf(x) stretches f vertically
- 94. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= 2f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c c > 1, y = cf(x) stretches f vertically x
- 95. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= 3f(x) y= 2f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c c > 1, y = cf(x) stretches f vertically x
- 96. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= 3f(x) y= f(x)/3 y= 2f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c c > 1, y = cf(x) stretches f vertically 0 < c < 1, y = cf(x) compresses f x
- 97. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= 3f(x) y= f(x)/3 y= 2f(x) y= –f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c c > 1, y = cf(x) stretches f vertically y = –f(x) reflects f vertically 0 < c < 1, y = cf(x) compresses f x
- 98. Summary of vertical transformations of graphs (c > 0). Transformations of Graphs Vertical Shifts Vertical Stretches and Compressions y= f(x)–1 y= f(x) + 1 x y= f(x) + 2 y= f(x)–2 y= f(x)–3 y= f(x) y= 3f(x) y= f(x)/3 y= 2f(x) y= –f(x) y= –2f(x) y= –3f(x) y= f(x) y = f(x) + c moves f up by c y = f(x) – c move f down by c c > 1, y = cf(x) stretches f vertically y = –f(x) reflects f vertically 0 < c < 1, y = cf(x) compresses f x
- 99. x –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x) –2–3 y Horizontal Stretches and Compressions Summary of horizontal transformations of graph (c > 0). x
- 100. x –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x) –2–3 y Horizontal Stretches and Compressions y = f(x + c) moves f left by c y = f(x – c) moves f right by c Summary of horizontal transformations of graph (c > 0). x
- 101. Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) Horizontal Stretches and Compressions y = f(x + c) moves f left by c y = f(x – c) moves f right by c Summary of horizontal transformations of graph (c > 0). x x –1 y=f(x) –2–3 y
- 102. Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) Horizontal Stretches and Compressions y = f(x + c) moves f left by c y = f(x – c) moves f right by c Summary of horizontal transformations of graph (c > 0). x x –1 y=f(x) –2–3 y
- 103. Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) Horizontal Stretches and Compressions y = f(x + c) moves f left by c y = f(x – c) moves f right by c Summary of horizontal transformations of graph (c > 0). x x –1 y=f(x) –2–3 y c > 1, y = f(cx) compresses f horizontally 0 < c < 1, y = f(cx) stretches f horizontally.
- 104. –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) y=f(x) y=f(2x) –2–3 y Horizontal Stretches and Compressions (0,f(0)) y = f(x + c) moves f left by c y = f(x – c) moves f right by c c > 1, y = f(cx) compresses f horizontally 0 < c < 1, y = f(cx) stretches f horizontally. Summary of horizontal transformations of graph (c > 0). x
- 105. x –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) y=f(x) y=f(2x) –2–3 y=f(3x) y Horizontal Stretches and Compressions (0,f(0)) y = f(x + c) moves f left by c y = f(x – c) moves f right by c c > 1, y = f(cx) compresses f horizontally 0 < c < 1, y = f(cx) stretches f horizontally. Summary of horizontal transformations of graph (c > 0). x
- 106. x –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) y=f(x)y=f(x/2)y=f(x/3) y=f(2x) –2–3 y=f(3x) y Horizontal Stretches and Compressions (0,f(0)) y = f(x + c) moves f left by c y = f(x – c) moves f right by c c > 1, y = f(cx) compresses f horizontally 0 < c < 1, y = f(cx) stretches f horizontally. Summary of horizontal transformations of graph (c > 0). x
- 107. x –1 Transformations of Graphs Horizontal Shifts y=f(x) y=f(x+2) y=f(x+1) y=f(x–2) y=f(x–1) y=f(x)y=f(x/2)y=f(x/3) y=f(2x) –2–3 y=f(3x) y Horizontal Stretches and Compressions y = f(–x) reflect f horizontally y=f(–x/3) (0,f(0)) y = f(x + c) moves f left by c y = f(x – c) moves f right by c c > 1, y = f(cx) compresses f horizontally 0 < c < 1, y = f(cx) stretches f horizontally. Summary of horizontal transformations of graph (c > 0). x
- 108. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. y 1 x 2 3½ y = f(x)
- 109. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). y 1 x 2 3½ y = f(x) y=g(x)=f(½ * x)
- 110. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). The domain of y = g(x) = f(½ * x) correspondingly is the stretch of the domain of y = f(x), from [0, 1] to [0, 2]. y 1 x 2 3½ y = f(x) y=g(x)=f(½ * x)
- 111. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. y y = f(x) 1 x 2 3 The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). The domain of y = g(x) = f(½ * x) correspondingly is the stretch of the domain of y = f(x), from [0, 1] to [0, 2]. y=g(x)=f(½ * x) ½ y = f(x/3)
- 112. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). The domain of y = g(x) = f(½ * x) correspondingly is the stretch of the domain of y = f(x), from [0, 1] to [0, 2]. Similarly the domain of y = h(x) = f(2x) is the compression from [0, 1] to [0, ½ ]. y 1 x 2 3½ y = f(x) y=g(x)=f(½ * x)y = f(2x) y = f(x/3)
- 113. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. y y = f(x) 1 x 2 3 The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). The domain of y = g(x) = f(½ * x) correspondingly is the stretch of the domain of y = f(x), from [0, 1] to [0, 2]. Similarly the domain of y = h(x) = f(2x) is the compression from [0, 1] to [0, ½ ]. y = f(x/3)y=g(x)=f(½ * x)y = f(2x) y = f(3x) ½
- 114. Horizontal Translations Let y = f(x) be a function with the interval [0, 1] as its domain as shown. y y = f(x) 1 x 2 3 The graph of y = g(x) = f(½ * x) is the horizontal stretch, by a factor of 2, of the graph of y = f(x). The domain of y = g(x) = f(½ * x) correspondingly is the stretch of the domain of y = f(x), from [0, 1] to [0, 2]. Similarly the domain of y = h(x) = f(2x) is the compression from [0, 1] to [0, ½ ]. y=g(x)=f(½ * x) The Domain of y = f(cx), c > 0 If the domain of y = f(x) is [0, a], then the domain of y = f(cx) is [0, a/c]. y = f(2x) ½ y = f(x/3) y = f(3x)
- 115. Horizontal Translations Example C. a. Given the graph of the function y = f(x) with the domain [–2, 2], graph y = (x – 3)2 – 1 by applying the transformation rules. Give the new domain and label the end points of the graph. f(x)=x2 x (2,4)(–2,4) 2–2 (0,0)
- 116. Horizontal Translations Example C. a. Given the graph of the function y = f(x) with the domain [–2, 2], graph y = (x – 3)2 – 1 by applying the transformation rules. Give the new domain and label the end points of the graph. f(x)=x2 x (2,4)(–2,4) 2–2 i. Shift right 3 units the graph of f(x) = x2 to obtain the graph of y = (x – 3)2. (0,0) 2–2 Shift right 3 units
- 117. Horizontal Translations Example C. a. Given the graph of the function y = f(x) with the domain [–2, 2], graph y = (x – 3)2 – 1 by applying the transformation rules. Give the new domain and label the end points of the graph. f(x)=x2 x (2,4)(–2,4) 2–2 i. Shift right 3 units the graph of f(x) = x2 to obtain the graph of y = (x – 3)2. (0,0) ii. Lower the graph of y = (x – 3)2 by 1 unit for the graph of y = (x – 3)2 – 1. 2–2 (5,3)(1,3) (3,–1) Shift right 3 units Lower by 1 unit
- 118. Horizontal Translations Example C. a. Given the graph of the function y = f(x) with the domain [–2, 2], graph y = (x – 3)2 – 1 by applying the transformation rules. Give the new domain and label the end points of the graph. f(x)=x2 x (2,4)(–2,4) 2–2 i. Shift right 3 units the graph of f(x) = x2 to obtain the graph of y = (x – 3)2. (0,0) ii. Lower the graph of y = (x – 3)2 by 1 unit for the graph of y = (x – 3)2 – 1. 2–2 (5,3)(1,3) (3,–1) Shift right 3 units Lower by 1 unit The new domain is [–2 + 3, 2 + 3] = [1, 5]. The new vertex is (3, –1) and end points (1, 3) and (5, 3).
- 119. Horizontal Translations x (4,2) (0,0) y=g(x)=√x 4 b. Given the graphs of the function y = g(x) = √x and y = G(x) a transformation of y = g(x) as shown, express G(x) using g(x). cy=G(x) (6,2) 62
- 120. Horizontal Translations The graph of y = G(x) is obtained by horizontally compressing the graph of x (4,2) (0,0) y=g(x)=√x 4 b. Given the graphs of the function y = g(x) = √x and y = G(x) a transformation of y = g(x) as shown, express G(x) using g(x). cy=G(x) (6,2) 62 y = g(x) by a factor of ½, which gives the graph of h(x) = g(2x) = √2x as shown, x (4,2) (0,0) y=g(x)=√x 4 62 (2,2) c y=h(x)=√2x horizontal compression
- 121. Horizontal Translations The graph of y = G(x) is obtained by horizontally compressing the graph of x (4,2) (0,0) y=g(x)=√x 4 b. Given the graphs of the function y = g(x) = √x and y = G(x) a transformation of y = g(x) as shown, express G(x) using g(x). cy=G(x) (6,2) 62 y = g(x) by a factor of ½, which gives the graph of h(x) = g(2x) = √2x as shown, x (4,2) (0,0) y=g(x)=√x 4 62 (2,2) c then moving h(x) to the right by 4 units. y=h(x)=√2x x (0,0) 4 cy=G(x) (6,2) 62 (2,2) c y=h(x)=√2x horizontal compression horizontal shift
- 122. Horizontal Translations The graph of y = G(x) is obtained by horizontally compressing the graph of x (4,2) (0,0) y=g(x)=√x 4 b. Given the graphs of the function y = g(x) = √x and y = G(x) a transformation of y = g(x) as shown, express G(x) using g(x). cy=G(x) (6,2) 62 y = g(x) by a factor of ½, which gives the graph of h(x) = g(2x) = √2x as shown, x (4,2) (0,0) y=g(x)=√x 4 62 (2,2) c then moving h(x) to the right by 4 units. Hence G(x) = h(x – 4) = √2(x – 4) or that G(x) = √2x – 8. y=h(x)=√2x x (0,0) 4 cy=G(x) (6,2) 62 (2,2) c y=h(x)=√2x horizontal compression horizontal shift
- 123. Absolute-Value Flip
- 124. Absolute-Value Flip y = f(x) = x x -2 -1 0 1 y -2 -1 0 1
- 125. Absolute-Value Flip y = f(x) = x x -2 -1 0 1 y -2 -1 0 1
- 126. Absolute-Value Flip y = f(x) = x y = |f(x)| = |x| x -2 -1 0 1 y -2 -1 0 1 x -2 -1 0 1 y 2 1 0 1
- 127. Absolute-Value Flip y = f(x) = x y = |f(x)| = |x| x -2 -1 0 1 y -2 -1 0 1 x -2 -1 0 1 y 2 1 0 1
- 128. Absolute-Value Flip y = f(x) = x The graph of y = |f(x)| is obtained by reflecting the portion of the graph below the x-axis to above the x-axis. y = |f(x)| = |x| x -2 -1 0 1 y -2 -1 0 1 x -2 -1 0 1 y 2 1 0 1
- 129. Absolute-Value Flip Another example, y = x2 – 1 (0,–1)
- 130. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| (0,–1)
- 131. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| (0,–1) (1,0)
- 132. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| y = |x2 – 1| – 1 (0,–1) (1,0)
- 133. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| y = |x2 – 1| – 1 (0,–1) (0,0) (1,0)
- 134. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| y = |x2 – 1| – 1 y = 2(|x2 – 1| – 1) (0,–1) (0,0) (1,0)
- 135. Absolute-Value Flip Another example, y = x2 – 1 y = |x2 – 1| y = |x2 – 1| – 1 y = 2(|x2 – 1| – 1) (0,–1) (0,0) (0,0) (1,0)
- 136. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis.
- 137. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2
- 138. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2
- 139. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2 y = f(-x) = – x3 – x2
- 140. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2 y = f(-x) = – x3 – x2
- 141. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2 y = f(-x) = – x3 – x2 A function is said to be even if f(x) = f(– x). Graphs of even functions are symmetric to the y-axis.
- 142. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2 y = f(-x) = – x3 – x2 A function is said to be even if f(x) = f(– x). Graphs of even functions are symmetric to the y-axis. Graph of an even function x (x, f(x)) –x (–x, f(–x))
- 143. Horizontal Flip The graph of y = f(–x) is the horizontal reflection of the graph of y = f(x) across the y axis. y = f(x) = x3 – x2 y = f(-x) = (-x)3 – (-x)2 y = f(-x) = – x3 – x2 A function is said to be even if f(x) = f(– x). Graphs of even functions are symmetric to the y-axis. Graph of an even function x (x, f(x)) –x (–x, f(–x))
- 144. Horizontal Flip Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2
- 145. Horizontal Flip A function is said to be odd iff f(–x) = – f(x). Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2
- 146. Horizontal Flip y = x4 – 4x2 A function is said to be odd iff f(–x) = – f(x). Graphs of odd functions are symmetric to the origin, Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2.
- 147. Horizontal Flip A function is said to be odd iff f(–x) = – f(x). Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis. Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2
- 148. Horizontal Flip Graph of an odd function A function is said to be odd iff f(–x) = – f(x). Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis. x–x 0 (x, f(x)) (–x, –f(x)) Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2
- 149. Horizontal Flip Graph of an odd function A function is said to be odd iff f(–x) = – f(x). Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis. x–x 0 (x, f(x)) (–x, –f(x)) u (u, f(u)) (–u, –f(u)) –u Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2
- 150. Horizontal Flip Polynomial-functions whose terms are all even powers are even. The graph on the right is the even polynomial– function y = x4 – 4x2. y = x4 – 4x2 Graph of an odd function A function is said to be odd iff f(–x) = – f(x). Graphs of odd functions are symmetric to the origin, that is, they're the same as reflecting across the y-axis followed by reflecting across the x-axis. x–x 0 (x, f(x)) (–x, –f(x)) u (u, f(u)) (–u, –f(u)) –u
- 151. Horizontal Flip Polynomial-functions whose terms are all odd powers are odd.
- 152. Horizontal Flip y = x3 – 4x Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 153. Horizontal Flip Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x. y = x3 – 4x Theorem (even and odd):
- 154. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 155. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 156. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 157. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. The product of an even function with an odd function is odd. Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 158. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. The product of an even function with an odd function is odd. (The same hold for quotient.) Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 159. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. The product of an even function with an odd function is odd. (The same hold for quotient.) is odd,x x4 + 1 Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 160. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. The product of an even function with an odd function is odd. (The same hold for quotient.) is odd, is even,x x4 + 1 x2 x4 + 1 Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 161. Horizontal Flip y = x3 – 4x Theorem (even and odd): I. The sum of even functions is even. The sum of odd functions is odd. II. The product of even functions is even. The product of odd functions is even. The product of an even function with an odd function is odd. (The same hold for quotient.) is odd, is even, x + 1 is neither.x x4 + 1 x2 x4 + 1 Polynomial-functions whose terms are all odd powers are odd. The graph on the right is the odd polynomial–function y = x3 – 4x.
- 162. HW Use the graph of y=x2, sketch the graphs of: 1. y = 3x2 2. y = -2x2 3. y = -0.5x2 4. y=x2 – 1 5. y=2x2 – 1 6. y= -x2 – 2 7. y=(x+1)2 8. y=(x–3)2
- 163. Transformations of Graphs
- 164. Transformations of Graphs
- 165. Transformations of Graphs
- 166. Transformations of Graphs
- 167. Transformations of Graphs

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