Transformation of Functions

1. Transformation of Functions • Recognize graphs of common functions • Use shifts to graph functions • Use reflections to graph functions • Use stretching & shrinking to graph functions • Graph functions w/ sequence of transformations

2. The following basic graphs will be used extensively in this section. It is important to be able to sketch these from memory.

3. The identity functionf(x) = x

4. The squaring function

5. The square root function

6. The absolute value function

7. The cubing function

8. The cube root function

9. We will now see how certain transformations (operations) of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) stretching.

10. 3 types of transformations: • Translation – shifting a graph horizontally or vertically • Reflection – rotating a graph across the x- or y- axis • Dilation – compressing or stretching a graph; makes a graph look narrow or wide

11. Translations

12. Vertical Translation 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.

13. Horizontal Translation 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.

14. 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

15. Use the basic graph to sketch the following:

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

17. Reflections • The graph of f(x) is the reflection of the graph of f(x) across the x-axis. • The graph of f(x) is the reflection of the graph of f(x) across the y-axis. • If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and • (x, y) is on the graph of f(x).

18. Reflecting • Across x-axis (y becomes negative, -f(x)) • Across y-axis (x becomes negative, f(-x))

19. Use the basic graph to sketch the following:

20. Vertical Stretching and Shrinking The graph of af(x) can be obtained from the graph of f(x) by stretching vertically for |a| > 1, or shrinking vertically for 0 < |a| < 1. For a < 0, the graph is also reflected across the x-axis. (The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

21. VERTICAL STRETCH (SHRINK) • y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high or low!)

22. Horizontal Stretching or Shrinking The graph of y = f(cx) can be obtained from the graph of y = f(x) by shrinking horizontally for |c| > 1, or stretching horizontally for 0 < |c| < 1. For c < 0, the graph is also reflected across the y-axis. (The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)

23. Horizontal stretch & shrink • We’re MULTIPLYING by an integer (not 1 or 0). • x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)