1. Copyright © Cengage Learning. All rights reserved. Functions and Graphs 3

2. Copyright © Cengage Learning. All rights reserved. 3.5 Graphs of Functions

3. 3 Copyright © Cengage Learning. All rights reserved. 3 Functions and Graphs 3.5 Graph of Function

4. 4 Graphs of Functions In this section we discuss aids for sketching graphs of certain types of functions. In particular, a function f is called even if f (–x) = f (x) for every x in its domain. In this case, the equation y = f (x) is not changed if –x is substituted for x, and hence, from symmetry test 1, the graph of an even function is symmetric with respect to the y-axis. A function f is called odd if f (–x) = –f (x) for every x in its domain. If we apply symmetry test 3 to the equation y = f (x), we see that the graph of an odd function is symmetric with respect to the origin.

5. 5 Graphs of Functions These facts are summarized in the first two columns of the next chart. Even and Odd Functions

6. 6 Example 1 – Determining whether a function is even or odd Determine whether f is even, odd, or neither even nor odd. (a) f (x) = 3x 4 – 2x 2 + 5 (b) f (x) = 2x 5 – 7x 3 + 4x (c) f (x) = x 3 + x 2 Solution: In each case the domain of f is. To determine whether f is even or odd, we begin by examining f (–x), where x is any real number.

7. 7 Example 1 – Solution (a) f (–x) = 3(–x) 4 – 2(–x) 2 + 5 = 3x 4 – 2x 2 + 5 = f (x) Since f (–x) = f (x), f is an even function. (b) f (–x) = 2(–x) 5 – 7(–x) 3 + 4(–x) = –2x 5 + 7x 3 – 4x definition of f simplify substitute –x for x in f (x) simplify substitute –x for x in f (x) cont’d

8. 8 Example 1 – Solution = –(2x 5 – 7x 3 + 4x) = –f (x) Since f (–x) = –f (x), f is an odd function. (c) f (–x) = (–x) 3 + (–x) 2 = –x 3 + x 2 Since f (–x) ≠ f (x), and f (–x) ≠ –f (x) (note that –f (x) = –x 3 – x 2 ), the function f is neither even nor odd. factor out –1 definition of f substitute –x for x in f(x) simplify cont’d

9. 9 Graphs of Functions In the next example we consider the absolute value function f, defined by f (x) = | x |.

10. 10 Example 2 – Sketching the graph of the absolute value function Let f (x) = | x |. (a) Determine whether f is even or odd. (b) Sketch the graph of f. (c) Find the intervals on which f is increasing or is decreasing.

11. 11 Example 2 – Solution (a) The domain of f is, because the absolute value of x exists for every real number x. If x is in, then f (–x) = | –x | = | x | = f (x). Thus, f is an even function, since f (–x) = f (x). (b) Since f is even, its graph is symmetric with respect to the y-axis. If x  0, then | x | = x, and therefore the first quadrant part of the graph coincides with the line y = x.

12. 12 Example 2 – Solution Sketching this half-line and using symmetry gives us Figure 1. Figure 1 cont’d

13. 13 Example 2 – Solution (c) Referring to the graph, we see that f is decreasing on (–, 0] and is increasing on [0, ). cont’d

14. 14 Graphs of Functions If we know the graph of y = f (x), it is easy to sketch the graphs of y = f (x) + c and y = f (x) – c for any positive real number c. As in the next chart, for y = f (x) + c, we add c to the y-coordinate of each point on the graph of y = f (x). This shifts the graph of f upward a distance c. For y = f (x) – c with c > 0, we subtract c from each y-coordinate, thereby shifting the graph of f a distance c downward. These are called vertical shifts of graphs.

15. 15 Graphs of Functions Vertically Shifting the Graph of y = f (x)

16. 16 Example 3 – Vertically shifting a graph Sketch the graph of f : (a) f (x) = x 2 (b) f (x) = x 2 + 4 (c) f (x) = x 2 – 4 Solution: We shall sketch all graphs on the same coordinate plane. (a) Since, f (–x) = (–x) 2 = x 2 = f (x), the function f is even, and hence its graph is symmetric with respect to the y-axis.

17. 17 Example 3 – Solution Several points on the graph of y = x 2 are (0, 0), (1, 1), (2, 4), and (3, 9). Drawing a smooth curve through these points and reflecting through the y-axis gives us the sketch in Figure 2. The graph is a parabola with vertex at the origin and opening upward. Figure 2 cont’d

18. 18 Example 3 – Solution (b) To sketch the graph of y = x 2 + 4, we add 4 to the y-coordinate of each point on the graph of y = x 2 ; that is, we shift the graph in part (a) upward 4 units, as shown in the figure. (c) To sketch the graph of y = x 2 – 4, we decrease the y-coordinates of y = x 2 by 4; that is, we shift the graph in part (a) downward 4 units. cont’d

19. 19 Graphs of Functions We can also consider horizontal shifts of graphs. Specifically, if c > 0, consider the graphs of y = f (x) and y = g(x) = f(x – c) sketched on the same coordinate plane, as illustrated in the next chart. Since g(a + c) = f([ a + c] – c) = f(a), we see that the point with x-coordinate a on the graph of y = f (x) has the same y-coordinate as the point with x-coordinate a + c on the graph of y = g (x) = f (x – c).

20. 20 Graphs of Functions This implies that the graph of y = g(x) = f (x – c) can be obtained by shifting the graph of y = f (x) to the right a distance c. Similarly, the graph of y = h(x) = f (x + c) can be obtained by shifting the graph of f to the left a distance c, as shown in the chart.

21. 21 Graphs of Functions Horizontally Shifting the Graph of y = f (x)

22. 22 Graphs of Functions Horizontal and vertical shifts are also referred to as translations.

23. 23 Example 4 – Horizontally shifting a graph Sketch the graph of f : (a) f (x) = (x – 4) 2 (b) f (x) = (x + 2) 2 Solution: The graph of y = x 2 is sketched in Figure 3. Figure 3

24. 24 Example 4 – Solution (a) Shifting the graph of y = x 2 to the right 4 units gives us the graph of y = (x – 4) 2, shown in the figure. (b) Shifting the graph of y = x 2 to the left 2 units leads us the graph of y = (x + 2) 2, shown in the figure. cont’d

25. 25 Graphs of Functions To obtain the graph of y = cf (x) for some real number c, we may multiply the y-coordinates of points on the graph of y = f (x) by c. For example, if y = 2f (x), we double the y-coordinates; or if y =, we multiply each y-coordinate by. This procedure is referred to as vertically stretching the graph of f (if c > 1) or vertically compressing the graph (if 0 < c < 1) and is summarized in the next chart.

26. 26 Graphs of Functions Vertically Stretching or Compressing the Graph of y = f (x)

27. 27 Example 5 – Vertically stretching or compressing a graph Sketch the graph of the equation: (a) y = 4x 2 (b) y = Solution: (a) To sketch the graph of y = 4x 2, we may refer to the graph of y = x 2 in Figure 4 and multiply the y-coordinate of each point by 4. Figure 4

28. 28 Example 5 – Solution This stretches the graph of y = x 2 vertically by a factor 4 and gives us a narrower parabola that is sharper at the vertex, as illustrated in the figure. (b) The graph of y = may be sketched by multiplying the y-coordinates of points on the graph of y = x 2 by. This compresses the graph of y = x 2 vertically by a factor = 4 and gives us a wider parabola that is flatter at the vertex, as shown in Figure 4. cont’d

29. 29 Graphs of Functions We may obtain the graph of y = –f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by –1. Thus, every point (a, b) on the graph of y = f (x) that lies above the x-axis determines a point (a, –b) on the graph of y = –f (x) that lies below the x-axis. Similarly, if (c, d ) lies below the x-axis (that is, d < 0), then (c, –d ) lies above the x-axis. The graph of y = –f (x) is a reflection of the graph of y = f (x) through the x-axis.

30. 30 Example 6 – Reflecting a graph through the x-axis Sketch the graph of y = –x 2. Solution: The graph may be found by plotting points; however, since the graph of y = x 2 is familiar to us, we sketch it as in Figure 5 and then multiply the y-coordinates of points by –1. This procedure gives us the reflection through the x-axis indicated in the figure. Figure 5

31. 31 Graphs of Functions Sometimes it is useful to compare the graphs of y = f (x) and y = f (cx) if c ≠ 0. In this case the function values f (x) for a  x  b are the same as the function values f (cx) for a  cx  b or, equivalently, This implies that the graph of f is horizontally compressed (if c > 1) or horizontally stretched (if 0 < c < 1), as summarized in the next chart.

32. 32 Graphs of Functions Horizontally Compressing or Stretching the Graph of y = f (x)

33. 33 Graphs of Functions If c < 0, then the graph of y = f (cx) may be obtained by reflecting the graph of y = f (| c |x) through the y-axis. For example, to sketch the graph of y = f (–2x), we reflect the graph of y = f (2x) through the y-axis. As a special case, the graph of y = f (–x), is a reflection of the graph of y = f (x), through the y-axis.

34. 34 Example 7 – Horizontally stretching or compressing a graph If f(x) = x 3 – 4x 2, sketch the graphs of y = f (x), y = f (2x), and y = Solution: We have the following: y = f (x) = x 3 – 4x 2 y = f (2x) = (2x) 3 – 4(2x) 2 = x 2 (x – 4) = 8x 3 –16x 2 = 8x 2 (x – 2)

35. 35 Example 7 – Solution Note that the x-intercepts of the graph of y = f (2x), are 0 and 2, which are the x-intercepts of 0 and 4 for y = f (x). This indicates a horizontal compression by a factor 2. The x-intercepts of the graph of are 0 and 8, which are 2 times the x-intercepts for y = f (x). This indicates a horizontal stretching by a factor cont’d

36. 36 Example 7 – Solution The graphs, obtained by using a graphing calculator with viewing rectangle [–6, 15] by [–10, 4], are shown in Figure 6. Figure 6 cont’d [–6, 15] by [–10, 4]

37. 37 Graphs of Functions Functions are sometimes described by more than one expression, as in the next examples. We call such functions piecewise-defined functions.

38. 38 Example 8 – Sketching the graph of a piecewise-defined function Sketch the graph of the function f if 2x + 5 if x  –1 f (x) = x 2 if | x | < 1 2 if x  1 Solution: If x  –1, then f (x) = 2x + 5 and the graph of f coincides with the line y = 2x + 5 and is represented by the portion of the graph to the left of the line x = –1 in Figure 7. Figure 7

39. 39 Example 8 – Solution The small dot indicates that the point (–1, 3) is on the graph. If | x | < 1 (or, equivalently, –1 < x < 1) we use x 2 to find values of f, and therefore this part of the graph of f coincides with the parabola y = x 2, as indicated in the figure. Note that the points (–1,1) and (1,1) are not on the graph. cont’d

40. 40 Example 8 – Solution Finally, if x  1, the values of f are always 2. Thus, the graph of f for x  1 is the horizontal half-line in Figure 7. Note: When you finish sketching the graph of a piecewise-defined function, check that it passes the vertical line test. Figure 7 cont’d

41. 41 Graphs of Functions If x is a real number, we define the symbol as follows: = n, where n is the greatest integer such that n  x If we identify with points on a coordinate line, then n is the first integer to the left of (or equal to) x.

42. 42 Graphs of Functions Illustration: The Symbol

43. 43 Graphs of Functions The greatest integer function f is defined by f (x) =.

44. 44 Example 11 – Sketching the graph of the greatest integer function Sketch the graph of the greatest integer function. Solution: The x- and y-coordinates of some points on the graph may be listed as follows:

45. 45 Example 11 – Solution Whenever x is between successive integers, the corresponding part of the graph is a segment of a horizontal line. Part of the graph is sketched in Figure 10. The graph continues indefinitely to the right and to the left. Figure 10 cont’d

46. 46 Graphs of Functions In general, if the graph of y = f (x) contains a point P(c, –d) with d positive, then the graph of y = | f (x) | contains the point Q(c, d) —that is, Q is the reflection of P through the x-axis. Points with nonnegative y-values are the same for the graphs of y = f (x) and y = | f (x) |. We used algebraic methods to solve inequalities involving absolute values of polynomials of degree 1, such as | 2x – 5 | < 7 and | 5x + 2 |  3.

47. 47 Graphs of Functions The processes of shifting, stretching, compressing, and reflecting a graph may be collectively termed transforming a graph, and the resulting graph is called a transformation of the original graph.