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# Sketching Graphs of Polynomials Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

## Introduction Sketching Graphs of Polynomials

To sketch the graph of most polynomial functions accurately, we need to use calculus (don’t let that scare you—the calculus part is easier than the algebra part!) We can still get a pretty good graph using algebra alone. The general method is to plot x -intercepts (if there are any), a point to the left of the smallest x -intercept, a point between any two x -intercepts, and a point to the right of the largest x -intercept. Because y -intercepts are easy to find, it wouldn’t hurt to plot these, too.

#### Examples

• f(x) = −(2 x − 1)( x + 2)( x − 3)

The x -intercepts are −2, 3, and (from 2 x − 1 = 0). In addition to the x -intercepts, we will plot the points for x = −2.5 (to the left of x = −2), x = − 1 (between x = −2 and x = ), x = 2 (between x = and x = 3), and x = 3.5 (to the right of x = 3).

Table 7.1

 x f ( x ) −2.5 16.5 −2 0 −1 −12 0 −6 0 2 12 3 0 3.5 −16.5

The reason we used x = −2.5 instead of x = − 3 and x = 3.5 instead of x = 4 is that their y -values were too large for our graph.

Fig. 7.8

## Sketching Graphs of Polynomials Practice Problems

#### Practice

Match the graph of the given function with one of the graphs in Figures 7.1-7.4.

1. f(x) = −8 x 3 + 4 x 2 − 9 x + 3
2. f(x) = 4 x 5 + 10 x 4 − 3 x 3 + x 2
3. P(x) = − x 2 + x − 6
4. g(x) = 1 + x + x 2 + x 3

Identify the x -intercepts and factors for the polynomial function whose graphs are given.

1.

Fig. 7.9

2.

Fig. 7.10

3.

Fig. 7.11

Match the polynomial function with one of the graphs in Figures 7.9 through 7.11 .

#### Solutions

1. Figure 7.4
2. Figure 7.3
3. Figure 7.2
4. Figure 7.3
5. The x -intercepts are −2, 0, and 1, so x + 2, x, and x − 1 are factors of the polynomial.
6. The x -intercepts are −3, 2, and 4, so x + 3, x − 2, and x − 4 are factors of the polynomial.
7. The x -intercepts are −4, −2, 2 and 4, so x + 4, x + 2, x − 2 and x − 4 are factors of the polynomial.
8. Figure 7.11
9. Figure 7.9
10. Figure 7.10
11.

Fig. 7.12

12.

Fig. 7.13

Practice problems for this concept can be found at: Polynomial Functions Practice Test.

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