Graphing Polynomial Functions

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.

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 $x=2$.