Graphing Polynomial Functions

In this section, we will bring all we know about polynomial functions and use it to sketch a graph given an equation. You will also learn about the intermediate value theorem and how we can use it to analyze behaviors when we don't know exactly where the zeros of a polynomial are.

Using the Intermediate Value Theorem

In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the $x$ -axis, we can confirm that there is a zero between them. Consider a polynomial function $f$ whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers $a$ and $b$ in the domain of $f$ , if $a < b$ and $f(a) \neq f(b)$ , then the function $f$ takes on every value between $f(a)$ and $f(b)$ . (While the theorem is intuitive, the proof is actually quite complicated and requires higher mathematics.) We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous function $f$ at $x=a$ lies above the $x$ - axis and another point at $x=b$ lies below the $x$ - axis, there must exist a third point between $x=a$ and $x=b$ where the graph crosses the $x$ - axis. Call this point $(c, \quad f(c))$ . This means that we are assured there is a solution $c$ where $f(c)=0$ .

In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the $x$ - axis. Figure 17 shows that there is a zero between $a$ and $b$ .

Figure 17 Using the Intermediate Value Theorem to show there exists a zero.

INTERMEDIATE VALUE THEOREM

Let $f$ be a polynomial function. The Intermediate Value Theorem states that if $f(a)$ and $f(b)$ have opposite signs, then there exists at least one value $c$ between $a$ and $b$ for which $f(c)=0$ .

EXAMPLE 9

Using the Intermediate Value Theorem

Show that the function $f(x)=x^{3}-5 x^{2}+3 x+6$ has at least two real zeros between $x=1$ and $x=4$ .

Solution

As a start, evaluate $f(x)$ at the integer values $x=1,2,3$ , and $4$ . See Table 2.

$x$	$1$	$2$	$3$	$4$
$f(x)$	$5$	$0$	$- 3$	$2$

Table 2

We see that one zero occurs at $x=2$ . Also, since $f(3)$ is negative and $f(4)$ is positive, by the Intermediate Value Theorem, there must be at least one real zero between $3$ and $4$ .

We have shown that there are at least two real zeros between $x=1$ and $x=4$ .

Analysis

We can also see on the graph of the function in Figure 18 that there are two real zeros between $x=1$ and $x=4$ .

Figure 18

TRY IT #4

Show that the function $f(x)=7 x^{5}-9 x^{4}-x^{2}$ has at least one real zero between $x=1$ and $x=2$ .