1. Graphing Techniques: Transformations We will be looking at functions from our library of functions and seeing how various modifications to the functions transform them.

2. VERTICAL TRANSLATIONS Above is the graph of What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them). What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them). As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function.

3. VERTICAL TRANSLATIONS Above is the graph of What would f(x) + 2 look like? So the graph f(x) + k, where k is any real number is the graph of f(x) but vertically shifted by k. If k is positive it will shift up. If k is negative it will shift down What would f(x) - 4 look like?

4. Above is the graph of What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function). As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but opposite way of the sign of the number. HORIZONTAL TRANSLATIONS What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).

5. HORIZONTAL TRANSLATIONS Above is the graph of What would f(x+1) look like? So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). What would f(x-3) look like? So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). So shift along the x-axis by 3 shift right 3

6. We could have a function that is transformed or translated both vertically AND horizontally. Above is the graph of What would the graph of look like? up 3 left 2

7. and If we multiply a function by a non-zero real number it has the affect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number. Let's try some functions from our library of functions multiplied by non-zero real numbers to see this. DILATION

8. Above is the graph of So the graph a f(x), where a is any real number GREATER THAN 1, is the graph of f(x) but vertically stretched or dilated by a factor of a. What would 2f(x) look like? What would 4f(x) look like? Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.

9. Above is the graph of So the graph a f(x), where a is 0 < a < 1, is the graph of f(x) but vertically compressed or dilated by a factor of a. Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value. What if the value of a was positive but less than 1? What would 1/4 f(x) look like? What would 1/2 f(x) look like?

10. Above is the graph of So the graph - f(x) is a reflection about the x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the x-axis) What if the value of a was negative? What would - f(x) look like? Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.

11. 11 Reflecting Graphs in Summary: Suppose we know the graph of y = f (x). How do we use it to obtain the graphs of y = –f (x)? The y-coordinate of each point on the graph of y = –f (x) is simply the negative of the y-coordinate of the corresponding point on the graph of y = f (x). So the desired graph is the reflection of the graph of y = f (x) in the x-axis.

12. Above is the graph of There is one last transformation we want to look at. Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value. What would f(-x) look like? (This means we are going to take the negative of x before putting in the function) So the graph f(-x) is a reflection about the y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the y-axis)

13. 13 Reflecting Graphs Comparing the graphs of - f(x) and f( - x ):

14. 14 Example 4 – Reflecting Graphs Sketch the graph of each function. (a) f (x) = –x 2 (b) g (x) = Solution: (a) We start with the graph of y = x 2. The graph of f (x) = –x 2 is the graph of y = x 2 reflected in the x-axis (see Figure 4). Figure 4

15. 15 Example 4 – Solution (b) We start with the graph of y =. The graph of g (x) = is the graph of y = reflected in the y-axis (see Figure 5). Note that the domain of the function g (x) = is {x | x  0}. cont’d Figure 5

16. Summary of Transformations So Far horizontal translation of h (opposite sign of number with the x) If a > 1, then vertical dilation or stretch by a factor of a vertical translation of k If 0 < a < 1, then vertical dilation or compression by a factor of a f(-x) reflection about y-axis Do reflections BEFORE vertical and horizontal translations If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a)

17. Graph using transformations We know what the graph would look like if it was from our library of functions. moves up 1 moves right 2 reflects about the x -axis

18. 18 Horizontal Stretching and Shrinking

19. HORIZONTAL STRETCHING & SHRINKING

20. 20 Horizontal Stretching and Shrinking To change the graph of y = f (x) to the graph of y = f (cx), we must shrink (or stretch) the graph horizontally by a factor of 1/c, as summarized in the following box.

21. 21 Example 7 – Horizontal Stretching and Shrinking of Graphs The graph of y = f (x) is shown in Figure 8. Sketch the graph of each function. (a) y = f (2x) (b) Figure 8 y = f (x)

22. 22 Example 7 – Solution Using the principles described in the preceding box, we obtain the graphs shown in Figures 9 and 10. Figure 9 y = f(2x) Figure 10

23. 23 Even and Odd Functions

24. 24 Even and Odd Functions If a function f satisfies f (–x) = f (x) for every number x in its domain, then f is called an even function. For instance, the function f (x) = x 2 is even because f (–x) = (–x) 2 = (–1) 2 x 2 = x 2 = f (x) The graph of an even function is symmetric with respect to the y-axis (see Figure 11). Figure 11

25. 25 Even and Odd Functions This means that if we have plotted the graph of f for x  0, then we can obtain the entire graph simply by reflecting this portion in the y-axis. If f satisfies f (–x) = –f (x) for every number x in its domain, then f is called an odd function. For example, the function f (x) = x 3 is odd because f (–x) = (–x) 3 = (–1) 3 x 3 = –x 3 = –f (x)

26. 26 Even and Odd Functions

27. 27 Example 8 – Even and Odd Functions Determine whether the functions are even, odd, or neither even nor odd. (a) f (x) = x 5 + x (b) g (x) = 1 – x 4 (c) h (x) = 2x – x 2 Solution: (a) f ( – x) = ( – x 5 ) + ( – x) = – x 5 – x

28. 28 Example 8 – Solution = –( x 5 + x) = –f (x) Therefore, f is an odd function. (b) g ( – x) = 1 – ( – x) 4 = 1 – x 4 = g (x) So g is even. cont’d

29. 29 Example 8 – Solution (c) h ( – x) = 2( – x) – ( – x) 2 = –2 x – x 2 Since h ( – x)  h (x) and h ( – x)  –h (x), we conclude that h is neither even nor odd. cont’d

30. 30 Even and Odd Functions The graphs of the functions in Example 8 are shown in Figure 13. The graph of f is symmetric about the origin, and the graph of g is symmetric about the y-axis. The graph of h is not symmetric either about the y-axis or the origin. Figure 13(b) Figure 13(a) Figure 13(c)