1. SECTION 1.5 GRAPHING TECHNIQUES; GRAPHING TECHNIQUES; TRANSFORMATIONS TRANSFORMATIONS

2. TRANSFORMATIONS Recall our “library” of functions. Here we will learn techniques for graphing a function which is “related” to one we already know how to graph.

3. HORIZONTAL SHIFTS On the same screen, graph each of the following functions: Y 1 = x 2 Y 2 = (x - 1) 2 Y 3 = (x - 3) 2 Y 4 = (x + 2) 2

4. COMPARING y = x 2 and y = (x - 2) 2 If we named the first function f(x), we could denote the second one by f(x - 2). In general, we can refer to any horizontal shift of a function f(x) by using the notation f(x - h)

5. y = f(x - 2)y = f(x + 3) When h is positive (that is, when there is a value being subtracted from x) the shift is to the right. When h is positive (that is, when there is a value being subtracted from x) the shift is to the right. When h is negative (that is, when there is a value being added to x) the shift is to the left. When h is negative (that is, when there is a value being added to x) the shift is to the left.

6. VERTICAL SHIFTS In general, we can refer to any vertical shift of a function f(x) by using the notation: f(x) + k

7. y = f(x) + 4y = f(x) - 1 When k is positive, the shift is upward. When k is negative, the shift is downward.

8. EXAMPLE: The figure shows the graph of f(x). Sketch the graphs of f(x + 1) and f(x) - 1. - 2 - 1 1 2

9. y = f(x + 1) - 3 - 2 - 1 1 2

10. y = f(x) - 1

11. VERTICAL STRETCHES When we compare the graph of y = x 2 to the graph of y = 2x 2, we find the second one is more narrow than the first. When we compare the graph of y = x 2 to the graph of y = 2x 2, we find the second one is more narrow than the first. This is called a vertical stretch. All the y-values are being doubled. This is called a vertical stretch. All the y-values are being doubled.

12. VERTICAL SHRINKS When we compare the graph of y = x 2 to the graph of y =.5x 2, we find the second one is wider than the first. When we compare the graph of y = x 2 to the graph of y =.5x 2, we find the second one is wider than the first. This is called a vertical shrink. All the y-values are being halved. This is called a vertical shrink. All the y-values are being halved.

13. In general, we can denote vertical stretches and shrinks to a function f(x) in the following way: For  a  > 1, stretch y = af(x) For 0 <  a  < 1, shrink

14. EXAMPLE: Sketch the graphs of y = 3f(x), y =.5f(x), and y = -.5f(x) Sketch the graphs of y = 3f(x), y =.5f(x), and y = -.5f(x)

15. y = 3f (x) - 2 - 1 12

16. y =.5f (x)

17. y = -.5f (x)

18. HORIZONTAL STRETCHES AND SHRINKS In general, we can denote horizontal stretches and shrinks to a function f(x) in the following way: For c > 1, shrink y = f(cx) For 0 < c < 1, stretch

19. EXAMPLE: Sketch the graphs of y = f(2x) and y = f(.5x) Sketch the graphs of y = f(2x) and y = f(.5x)

20. y = f (2x)

21. y = f (.5x)

22. EXAMPLE: Given f(x) = x 3 - 4x, explain the transformations that will occur to the graph of the function for f(2x) + 3 The graph will be compressed horizontally and shifted 3 units up. Graph it!

23. CONCLUSION OF SECTION 1.5 CONCLUSION OF SECTION 1.5