1. Table of Contents Functions: Transformations of Graphs -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Vertical Translation: The graph of f(x) + k appears as the graph of f(x) shifted up (k > 0) or down k (k < 0) units. 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.

2. Table of Contents Functions: Transformations of Graphs Slide 2 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Horizontal Translation: The graph of f(x + k) appears as the graph of f(x) shifted left (k > 0) or right k (k < 0) units. 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.

3. Table of Contents Functions: Transformations of Graphs Slide 3 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 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. 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

4. Table of Contents Functions: Transformations of Graphs Slide 4 -6 -5 -4 -3 -2 0 1 2 3 4 5 6 -5-4-3-2123456 Vertical stretches and compressions: The graph of k f(x) appears as the graph of f(x) vertically stretched (k > 1) or vertically compressed (0 < k < 1) by a factor of k. 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 compressed 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

5. Table of Contents Combinations of Transformations: When there are multiple transformations of a graph of a function, they should be done in this order:(1) Reflections, (2) vertical stretch and shrink, (3) shifts. Functions: Transformations of Graphs Slide 5 Example 5: Sketch the graph of f (x) = - 1/4  x + 1  + 2 by performing translations on the graph of another function. First, sketch the graph of y =  x  (see next slide).

6. Table of Contents Functions: Transformations of Graphs Slide 6 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 reflection across the x-axis: f (x) = - 1/4  x + 1  + 2. Next, do the vertical shrink: f (x) = - 1/4  x + 1  + 2. Last, do the horizontal and vertical shifts: f (x) = - 1/4  x + 1  + 2. Graph of f (x) = - 1/4  x + 1  + 2

7. Table of Contents Functions: Transformations of Graphs Slide 7 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

8. Table of Contents Functions: Transformations of Graphs