1. Functions: Transformations of Graphs Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted up k units (k > 0) or down k units (k < 0).

2. Functions: Transformations of Graphs -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Example 1: Sketch the graphs of and on the same rectangular coordinate plane. Note g(x) = f(x) + 3 so the graph of g(x) is the graph of f(x) shifted up 3 units. Note h(x) = f(x) – 4 so the graph of h(x) is the graph of f(x) shifted down 4 units. The graph of f(x) is shown.

3. Functions: Transformations of Graphs Horizontal Translation: The graph of f(x + k) appears as the graph of f(x) shifted left k units (k > 0) or right k units (k < 0).

4. Functions: Transformations of Graphs -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Note g(x) = f(x + 4) so the graph of g(x) is the graph of f(x) shifted left 4 units. Note h(x) = f(x – 2) so the graph of h(x) is the graph of f(x) shifted right 2 units. Example 2: Sketch the graphs of and on the same rectangular coordinate plane. The graph of f(x) is shown.

5. Functions: Transformations of Graphs Reflections across the axes: The graph of - f(x) appears as the graph of f(x) reflected across the x-axis. The graph of f(- x) appears as the graph of f(x) reflected across the y-axis.

6. Functions: Transformations of Graphs -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Note g(x) = - f(x) so the graph of g(x) is the graph of f(x) reflected across the x-axis. Note h(x) = f(- x) so the graph of h(x) is the graph of f(x) reflected across the y-axis. The graph of f(x) is shown. Example 3: Sketch the graphs of on the same rectangular coordinate plane. and

7. Functions: Transformations of Graphs Vertical stretches and shrinks: The graph of k f(x) appears as the graph of f(x) vertically stretched (k > 1) or vertically shrunk (0 < k < 1) by a factor of k.

8. Functions: Transformations of Graphs -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Note g(x) = 2f(x) so the graph of g(x) is the graph of f(x) vertically stretched by a factor of 2. Note h(x) = 1/2 f(x) so the graph of h(x) is the graph of f(x) vertically shrunk by a factor of one-half. The graph of f(x) is shown. Example 4: Sketch the graphs of on the same rectangular coordinate plane. and

9. Combinations of Transformations: When there are multiple transformations of a graph of a function, they should be done in this order: (1) Reflections, vertical stretches and shrinks (2) Horizontal and vertical shifts (translations) Functions: Transformations of Graphs

10. Example 5: Sketch the graph of f (x) = - 1/4  x + 1  + 2 using transformations First, sketch the basic function y =  x . -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456

11. Functions: Transformations of Graphs f (x) = - 1/4  x + 1  + 2 Next, do the reflection across the x-axis: -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456

12. Functions: Transformations of Graphs f (x) = - 1/4  x + 1  + 2 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Next, do the vertical shrink:

13. Functions: Transformations of Graphs f (x) = - 1/4  x + 1  + 2 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Last, do the horizontal and vertical shifts: Graph of f (x) = - 1/4  x + 1  + 2

14. Functions: Transformations of Graphs Now try: Sketch the graph of f (x) = 2(x – 3) 2 – 4 by performing transformations on the graph of another function. -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456

15. Functions: Transformations of Graphs