364 CHAPTER 3 Polynomial and Rational Functions Graphing Techniques We have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function ƒ1x2 will show the following characteristics. • all x-intercepts (indicating the real zeros) and the behavior of the graph at these zeros • the y-intercept • the sign of ƒ1x2 within the intervals formed by the x-intercepts • enough of the domain to show the end behavior In Examples 3 and 4, we sketch the graphs of two polynomial functions by hand. We use the following general guidelines. Graphing a Polynomial Function Let ƒ1x2 = anx n + a n-1x n-1 + g+ a 1x + a0, with an ≠0, be a polynomial function of degree n. To sketch its graph, follow these steps. Step 1 Find the real zeros of ƒ. Plot the corresponding x-intercepts. Step 2 Find ƒ102 = a0. Plot the corresponding y-intercept. Step 3 Use end behavior, whether the graph crosses, bounces on, or wiggles through the x-axis at the x-intercepts, and selected points as necessary to complete the graph. EXAMPLE 3 Graphing a Polynomial Function Graph ƒ1x2 = 2x3 + 5x2 - x - 6. SOLUTION Step 1 The possible rational zeros are {1, {2, {3, {6, { 1 2 , and { 3 2 . Use synthetic division to show that 1 is a zero. 1)2 5 -1 -6 2 7 6 2 7 6 0 ƒ112 = 0 We use the results of the synthetic division to factor as follows. ƒ1x2 = 1x - 1212x2 + 7x + 62 ƒ1x2 = 1x - 1212x + 321x + 22 Factor again. Set each linear factor equal to 0, and then solve for x to find zeros. The three zeros of ƒ are 1, - 3 2 , and -2. Plot the corresponding x-intercepts. See Figure 30. Step 2 ƒ102 = -6, so plot 10, -62. See Figure 30. Step 3 The dominating term of ƒ1x2 is 2x3, so the graph will have end behavior similar to that of ƒ1x2 = x3. It will rise to the right and fall to the left as . See Figure 30. Each zero of ƒ1x2 occurs with multiplicity 1, meaning that the graph of ƒ1x2 will cross the x-axis at each of its zeros. Because the graph of a polynomial function has no breaks, gaps, or sudden jumps, we now have sufficient information to sketch the graph of ƒ1x2. Rises to the right Falls to the left (–2, 0) (1, 0) – 3 2 x y 6 (0, –6) 0 Q , 0R Figure 30
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