1. 1 Transformations of Functions SECTION 2.7 1 2 3 4 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.

2. 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 transformation of the graph of f.

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

8. 8 EXAMPLE 1 Graphing Vertical Shifts 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.

9. 9 EXAMPLE 1 Graphing Vertical Shifts Solution Make a table of values.

10. 10 EXAMPLE 1 Graphing Vertical Shifts 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.

11. 11 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.

12. 12 EXAMPLE 2 Writing Functions for Horizontal Shifts Let f (x) = x 2, 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.

13. 13 EXAMPLE 2 Writing Functions for Horizontal Shifts

14. 14 EXAMPLE 2 Writing Functions for Horizontal Shifts

15. 15 EXAMPLE 2 Writing Functions for Horizontal Shifts All three functions are squaring functions. Solution The x-intercept of f is 0. The x-intercept of g is 2. a.g is obtained by replacing x with x – 2 in f. 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.

16. 16 EXAMPLE 2 Writing Functions for Horizontal Shifts Solution continued The x-intercept of f is 0. The x-intercept of h is –3. b.h is obtained by replacing x with x + 3 in f. 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.

17. 17 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.

19. 19 EXAMPLE 3 Sketch the graph of the function Solution Identify and graph the parent function Graphing Combined Vertical and Horizontal Shifts

20. 20 EXAMPLE 3 Solution continued Graphing Combined Vertical and Horizontal Shifts Translate 2 units to the left Translate 3 units down

21. 21 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).

22. 22 REFLECTION IN THE x -AXIS

23. 23 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).

24. 24 REFLECTION IN THE y -AXIS

25. 25 EXAMPLE 4 Combining Transformations Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|. Solution Step 1Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|.

26. 26 EXAMPLE 4 Combining Transformations Solution continued Step 2Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

27. 27 EXAMPLE 4 Combining Transformations Solution continued Step 3Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

28. 28 EXAMPLE 5 Stretching or Compressing a Function Vertically Solution 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. Let x–2–1012 f(x)f(x)21012 g(x)g(x)42024 h(x)h(x)11/20 1

29. 29 EXAMPLE 5 Stretching or Compressing a Function Vertically Solution continued

30. 30 EXAMPLE 5 Stretching or Compressing a Function Vertically 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.

31. 31 VERTICAL STRETCHING OR COMPRESSING The graph of y = a f (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 1.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.