## Graphing Polynomial Functions

Polynomial functions of the form f ( x) = x n (where n is a positive integer) form one of two basic graphs, shown in Figure 1.

Figure 1. Graphs of polynomials

Graphs of polynomials.

Each graph has the origin as its only x‐intercept and y‐intercept. Each graph contains the ordered pair (1,1). If a polynomial function can be factored, its x‐intercepts can be immediately found. Then a study is made as to what happens between these intercepts, to the left of the far left intercept and to the right of the far right intercept.

##### Example 1

Graph f ( x) = x 4 – 10 x 2 + 9.

The zeros of this function are –1, 1, –3, and 3. That is, –1, 1, –3, and 3 are the x‐intercepts of this function.

When x < –3, say, x = –4, then

So for x < –3, f ( x) > 0.

When –1 < x < 1, say, x = 0, then

So for –1 < x < 1, f ( x) > 0.

In a similar way, it can be seen that

• when x > 3, f ( x) > 0

• when –3 < x < –1, f ( x) < 0

• when 1 < x < 3, f ( x) < 0

The graph then has points in the shaded regions as shown in Figure 2.

The y‐intercept of this function is found by finding f (0).

f (0) = 9

so (0, 9) is a point on the graph. To complete the graph, find and plot several points. Evaluate f ( x) for several integer replacements; then connect these points to form a smooth curve (see Figure 3).

Notice that f ( x) = x 4 – 10 x 2 + 9 has a leading term with an even exponent. The far right and far left sides of the graph will go in the same direction. Because the leading coefficient is positive, the two sides will go up. If the leading coefficient were negative, the two sides would go down.