1. Vertical and horizontal shifts If f is the function y = f(x) = x 2, then we can plot points and draw its graph as: If we add 1 (outside change) to f(x), we have y = f(x) + 1 = x 2 + 1. We simply take the graph above and move it up 1 unit to get the new graph. x y x y (2,5)

2. If we replace x by x+2 (inside change) to form the function y = f(x+2) = (x+2) 2, then the corresponding graph is obtained from the graph of y = x 2 by moving it 2 units to the left along the x-axis. If we replace x by x–1 (inside change) to form the function y = f(x–1) = (x–1) 2, then the corresponding graph is obtained from the graph of y = x 2 by moving it 1 unit to the right along the x-axis. x y x y (-2,0) (1,0)

3. If y = g(x) is a function and k is a constant, then the graph of: y = g(x) + k is the graph of y = g(x) shifted vertically by |k| units. If k > 0, the shift is up, and if k < 0, the shift is down. y = g(x+k) is the graph of y = g(x) shifted horizontally by |k| units. If k > 0, the shift is left, and if k < 0, the shift is right. Horizontal and vertical shifts of the graph of a function are called translations.

4. An example which combines horizontal and vertical shifts Problem. Use the graph of y = f(x) = x 2 to sketch the graph of g(x) = f(x–2) – 1 = (x–2) 2 – 1. Solution. The graph of g is the graph of f shifted to the right by 2 units and down 1 unit as shown below. y (2,-1) x

5. Reflections and symmetry Suppose that we are given the function y = f(x) as shown. If we define y = g(x) = –f(x), then the graph of g may be obtained by reflecting the graph of f vertically across the x-axis as shown next. x y y x

6. If we define y = h(x) = f(–x), then the graph of h is obtained by reflecting the graph of f horizontally across the y-axis as shown next. Next, we define y = p(x) = –f(–x). The graph of p is obtained by reflecting the graph of f about the origin as shown next. x y x y Continuation of example from previous slide

7. For any function f: The graph of y = –f(x) is the reflection of the graph of y = f(x) across the x-axis. The graph of y = f(–x) is the reflection of the graph of y = f(x) across the y-axis. The graph of y = –f(–x) is the reflection of the graph of y = f(x) about the origin. Note that this reflection can be obtained by applying the two previous reflections in sequence.

8. Symmetries of graphs A function is called an even function if, for all values of x in the domain of f, The graph of an even function is symmetric across the y-axis. Examples of even functions are power functions with even exponents, such as y = x 2, y = x 4, y = x 6,... A function is called an odd function if, for all values of x in the domain of f, The graph of an odd function is symmetric about the origin. Examples of odd functions are power functions with odd exponents, such as y = x 1, y = x 3, y = x 5,...

9. Problem. Is the function f(x) = x 3 +x even, odd, or neither? Solution. Since –2 = f(–1) is not equal to f(1) = 2, it follows that f is not even. Since f(–x) = = –f(x), it follows that f is odd. y = x 3 +x Note the symmetry about the origin.

10. Problem. Is the function f(x) = |x| even, odd, or neither? Solution. Since f(–x) = |x| = f(x), it follows that f is even. Since 1 = f(–1) is not equal to –f(1) = –1, it follows that f is not odd. Question. Is it possible for a function to be both even and odd? y = |x| Note the symmetry about the y-axis.

11. Combining shifts and reflections--an example In an earlier example, we discussed an investment of $10000 in the latest dotcom venture. This investment had a value of 10000(0.95) t dollars after t years. Suppose that we want to graph the amount of the loss after t years for this investment. The formula for the loss is: 10000 – 10000(0.95) t The loss is graphed on the next slide using Maple. Shift UpwardsReflect across t-axis

12. Use of Maple to graph loss on dotcom investment > plot({10000,10000-10000*(0.95)^t},t=0..80, color=black,labels=["t","L"]); The graph of the loss has a horizontal asymptote, L = 10000.

13. Vertical Stretches and Compressions If f(x) = x 2 and g(x) = 5x 2, then the graph of g is obtained from the graph of f by stretching it vertically by a factor of 5 as the following Maple plot shows:

14. If f(x) = x 2 and g(x) = -5x 2, then the graph of g is obtained from the graph of f by stretching it vertically by a factor of 5 and then reflecting it across the x-axis as the following Maple plot shows:

15. If we compare the graphs of f(x) = x 2 and g(x) = (1/2)x 2, we notice that the graph of g can be found by vertically compressing the graph of f by a factor of 1/2. Generalizing the above examples yields the following: If f is a function and k is a constant, then the graph of y = kf(x) is the graph of y = f(x) Vertically stretched by a factor of k, if k > 1. Vertically compressed by a factor of k, if 0 < k < 1. Vertically stretched or compressed by a factor |k| and reflected across the x-axis, if k < 0.

16. Vertical Stretch Factors and Average Rates of Change If f(x) = x 2 and g(x) = 5x 2, we compute the average rates of change of the two functions on the interval [1,3] as follows: The above computation illustrates a general fact:

17. If f(x) = 4–x 2 and g(x) = 4 – (2x) 2, then the graph of g is obtained from the graph of f by compressing it horizontally by a factor of 1/2 as the following Maple plot shows:

18. If f(x) = 4–x 2 and g(x) = 4 – (0.5x) 2, then the graph of g is obtained from the graph of f by stretching it horizontally by a factor of 2 as the following Maple plot shows:

19. Generalizing the two previous examples yields the following results for horizontal stretch or compression. If f is a function and k is a positive constant, then the graph of y = f(kx) is the graph of f Horizontally compressed by a factor of 1/k if k > 1. Horizontally stretched by a factor of 1/k if k < 1. If k < 0, then the graph of y = f(kx) also involves a horizontal reflection about the y-axis.

20. Combining transformations For nonzero constants A, B, h and k, the graph of the function is obtained by applying the transformations to the graph of f in the following order: Horizontal stretch/compression by factor of 1/|B| Horizontal shift by h units Vertical stretch/compression by factor of |A| Vertical shift by k units If A<0, follow the vertical stretch/compression by a reflection about the x-axis. If B<0, follow the horizontal stretch/compression by a reflection about the y-axis.

21. Example for combining transformations Let y = 5(x+2) 2 +7. Use the method of the preceding slide to graph this function. First, we let f(x) = x 2 so that A = 5, B = 1, h = –2 and k = 7. Based on these values, we carry out these steps Horizontal shift 2 to the left of the graph of f(x) = x 2. Vertical stretch of the resulting graph by factor of 5. Vertical shift of the resulting graph up by 7. Since B = 1, there is no horizontal compression, stretch, or reflection. You should compare this result with the use of the vertex form of a quadratic function from Chapter 3. See the next slide for the graph.

22. Example for combining transformations

23. Another example Let f(x) = |x|. Use the method previously described to analyze y = |3(x–1)|. Here, A = 1, B = 3, h = 1, k = 0. We have a horizontal compression by a factor of 1/3 followed by a horizontal shift of 1 unit to the right. f(x)f(3x)f(3(x–1))

24. Summary for Transformation of Functions and their Graphs If y = g(x) is a function and k is a constant, then the graph of y = g(x) + k is the graph of y = g(x) shifted vertically by |k| units. If y = g(x) is a function and k is a constant, then the graph of y = g(x+k) is the graph of y = g(x) shifted horizontally by |k| units. A function is called an even function if, for all values of x in the domain of f, f(–x) = f(x). The graph of an even function is symmetric across the y-axis. A function is called an odd function if, for all values of x in the domain of f, f(–x) = –f(x). The graph of an odd function is symmetric about the origin.

25. Summary for Transformation of Fcts and their Graphs, cont’d When a function f(x) is replaced by kf(x), the graph is vertically stretched or compressed and the average rate of change on any interval is also multiplied by k. If k is negative, a vertical reflection about the x-axis is also involved. When a function f(x) is replaced by f(kx), the graph is horizontally stretched or compressed by a factor of 1/|k| and, if k < 0, reflected horizontally about the y-axis. The graph of the function is obtained by sequentially applying transformations to the graph of f.