Graphing Polynomial Functions Section 4.2 Graphing Polynomial Functions

Objectives Graph polynomial functions. Use the intermediate value theorem to determine whether a function has a real zero between two given real numbers.

Graphing Polynomial Functions If P(x) is a polynomial function of degree n, the graph of the function has:  at most n real zeros, and thus at most n x- intercepts;  at most n  1 turning points. (Turning points on a graph, also called relative maxima and minima, occur when the function changes from decreasing to increasing or from increasing to decreasing.)

To Graph a Polynomial Function 1. Use the leading-term test to determine the end behavior. 2. Find the zeros of the function by solving f (x) = 0. Any real zeros are the first coordinates of the x-intercepts. 3. Use the x-intercepts (zeros) to divide the x-axis into intervals and choose a test point in each interval to determine the sign of all function values in that interval. 4. Find f (0). This gives the y-intercept of the function.

To Graph a Polynomial Function 5. If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n x-intercepts and at most n  1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x-axis. We can also check the graph with a graphing calculator.

Example Graph the polynomial function h(x) = −2x4 + 3x3. Solution: 1. The leading term is −2x4 . The degree, 4, is even, and the coefficient, −2, is negative. Thus the end behavior of the graph can be sketched as followed. 2. To find the zeros, we solve h(x) = 0. Here we can use factoring by grouping.

Example continued Factor: The zeros are 0 and 3/2. The x-intercepts are (0,0) and (3/2, 0). 3. The zeros divide the x-axis into four intervals: (, 0), (0, 3/2), and (3/2, ). We choose a test value for x from each interval and find h(x).

Example continued

Example continued 4. We found the y-intercept in step (2) but if we hadn’t, we would find h(0) to determine it: The y-intercept is (0, 0).

Example continued 5. A few additional points are helpful when completing the graph. 6. The degree of h is 4. The graph of h can have at most 4 x-intercepts and at most 3 turning points. It has 2 x-intercepts and 1 turning point. The zeros, 0 and 3/2, each have odd multiplicities: 3 for 0 and 1 for 3/2. Since the multiplicities are odd, the graph crosses the x-axis at 0 and 3/2. The end behavior of the graph is what we described in step (1).

Intermediate Value Theorem For any polynomial function P(x) with real coefficients, suppose that for a  b, P(a) and P(b) are of opposite signs. Then the function has a real zero between a and b. The intermediate value theorem cannot be used to determine whether there is or is not a real zero between a and b when P(a) and P(b) have the same sign.

Example Using the intermediate value theorem, determine, if possible, whether the function has a real zero between a and b. a) f(x) = x3 + x2  6x; a = 4 b = 2 b) f(x) = x3 + x2  6x; a = −1 b = 3

Solution We find f(a) and f(b) and determine where they differ in sign. The graph of f(x) provides a visual check. f(4) = (4)3 + (4)2  6(4) = 24 f(2) = (2)3 + (2)2  6(2) = 8 By the intermediate value theorem, since f(4) and f(2) have opposite signs, then f(x) has a zero between 4 and 2. The graph above confirms this.

Solution b) f(−1) = (−1)3 + (−1)2  6(−1) = 6 Both f(-1) and f(3) are positive. Thus the intermediate value theorem does not allow us to determine whether there is a real zero between −1 and 3. Note that the graph of f(x) shows that there are two zeros between −1 and 3.