Transformations of Functions

1. Transformations of Functions Learn the meaning of transformations. Use vertical or horizontal shifts to graph functions. Use reflections to graph functions. Use stretching or compressing to graph functions. SECTION 2.7 1 2 3 4

2. TRANSFORMATIONS If a new function is formed by performing certain operations on a given function f , then the graph of the new function is called a transformationof the graph of f.

5. Example: y = |x| + 2 The transformation of the parent function is shown in blue. It is a shift up (or vertical translation up) of 2 units.) Parent function (y = |x|) shown on graph in red.

6. Example: y = x - 1 The transformation of the parent function is shown in blue. It is a shift down (or vertical translation down) of 1 unit. Parent function (y = x) shown on graph in red.

7. Parent Functions – The simplest function of its kind. All other functions of its kind are Transformations of the parent.

10. Translation (Shift) A vertical translation is made on a function by adding or subtracting a number to the function. Example: y = x + 3 (translation up) Example: y = x² - 5 (translation down) A translation up is also called a vertical shift up. A translation down is also called a vertical shift down.

11. Vertical Shifting Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted upb units; the graph of y = f(x) b is the graph of y = f(x) shifted downb units.

12. The graph of Is like the graph of SHIFTED 3 units up

13. The graph of Is like the graph of SHIFTED 2 units down

14. Graphing Vertical Shifts EXAMPLE 1 Let Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.

15. Make a table of values. Solution Graphing Vertical Shifts EXAMPLE 1

16. Solution continued Graph the equations. The graph of y = |x| + 2 is the graph of y = |x| shifted two units up. The graph of y = |x| – 3 is the graph of y = |x| shifted three units down. Graphing Vertical Shifts EXAMPLE 1

17. VERTICAL SHIFT Let d > 0. The graph of y = f(x) + d is the graph of y = f(x) shifted d units up, and the graph of y = f(x) – d is the graph of y = f(x) shifted d units down.

18. The effect of the transformation on the graph Replacing function with function – numberSHIFTS the basic graph number units down Replacing function with function + numberSHIFTS the basic graph number units up

19. EXAMPLE 2 Writing Functions for Horizontal Shifts Let f(x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2. A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide. Describe how the graphs of g and h relate to the graph of f.

20. EXAMPLE 2 Writing Functions for Horizontal Shifts

21. EXAMPLE 2 Writing Functions for Horizontal Shifts

22. EXAMPLE 2 Writing Functions for Horizontal Shifts Solution All three functions are squaring functions. a. g is obtained by replacing x with x – 2 in f . The x-intercept of f is 0. The x-intercept of g is 2. For each point (x, y) on the graph of f , there will be a corresponding point (x + 2, y) on the graph of g. The graph of g is the graph of f shifted 2 units to the right.

23. EXAMPLE 2 Writing Functions for Horizontal Shifts Solution continued b. h is obtained by replacing x with x + 3 in f . The x-intercept of f is 0. The x-intercept of h is –3. For each point (x, y) on the graph of f , there will be a corresponding point (x – 3, y) on the graph of h. The graph of h is the graph of f shifted 3 units to the left. The tables confirm both these considerations.

24. HORIZONTAL SHIFT The graph of y = f(x – c) is the graph of y = f(x) shifted |c| units to the right, if c > 0, to the left if c < 0.

25. Horizontal Shifting Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted rightd units; the graph of y = f(x + d) is the graph of y = f(x) shifted leftd units.

26. The effect of the transformation on the graph Replacing x with x – numberSHIFTS the basic graph number units to the right Replacing x with x + numberSHIFTS the basic graph number units to the left

28. The graph of Is like the graph of SHIFTED 2 units to the right

29. Vertical shifts • Moves the graph up or down • Impacts only the “y” values of the function • No changes are made to the “x” values • Horizontal shifts • Moves the graph left or right • Impacts only the “x” values of the function • No changes are made to the “y” values

30. The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function.Numbers added or subtractedinside translate left or right, while numbers added or subtractedoutside translate up or down.

31. Recognizing the shift from the equation, examples of shifting the function f(x) = • Vertical shift of 3 units up • Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

32. Points represented by (x , y) on the graph of f(x) become If the point (6, -3) is on the graph of f(x), find the corresponding point on the graph of f(x+3) + 2

33. Combining a vertical & horizontal shift • Example of function that is shifted down 4 units and right 6 units from the original function.

34. Graphing Combined Vertical and Horizontal Shifts EXAMPLE 3 Sketch the graph of the function Solution Identify and graph the parent function

35. The graph of Is like the graph of SHIFTED 3 units to the left

36. Graphing Combined Vertical and Horizontal Shifts EXAMPLE 3 Solution continued Translate 2 units to the left Translate 3 units down

37. REFLECTION IN THE x-AXIS The graph of y = – f(x) is a reflection of the graph of y = f(x) in the x-axis. If a point (x, y) is on the graph of y = f(x), then the point (x, –y) is on the graph of y = – f(x).

38. REFLECTION IN THE x-AXIS

39. REFLECTION IN THE y-AXIS The graph of y = f(–x) is a reflection of the graph of y = f(x) in the y-axis. If a point (x, y) is on the graph of y = f(x), then the point (–x, y) is on the graph of y = f(–x).

40. REFLECTION IN THE y-AXIS

41. Solution Step 1 Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|. EXAMPLE 4 Combining Transformations Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|.

42. EXAMPLE 4 Combining Transformations Solution continued Step 2 Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

43. EXAMPLE 4 Combining Transformations Solution continued Step 3 Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

44. Stretching or Compressing a Function Vertically EXAMPLE 5 Let Sketch the graphs of f, g, and h on the same coordinate plane, and describe how the graphs of g and h are related to the graph of f. Solution

45. Stretching or Compressing a Function Vertically EXAMPLE 5 Solution continued

46. Stretching or Compressing a Function Vertically EXAMPLE 5 Solution continued The graph of y = 2|x| is the graph of y = |x| vertically stretched (expanded) by multiplying each of its y–coordinates by 2. The graph of |x| is the graph of y = |x| vertically compressed (shrunk) by multiplying each of its y–coordinates by .

47. VERTICAL STRETCHING OR COMPRESSING The graph of y = af(x) is obtained from the graph of y = f(x) by multiplying the y-coordinate of each point on the graph of y = f(x) by a and leaving the x-coordinate unchanged. The result is • A vertical stretch away from the x-axis if a > 1; 2. A vertical compression toward the x-axis if 0 < a < 1. If a < 0, the graph of f is first reflected in the x-axis, then vertically stretched or compressed.

48. g(x) = f(x-2) g(x)= 4f(x) g(x) = f(½x) g(x) = -f(x) (-10, 4) (-12, 16) (-24, 4) (-12, -4) The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x).