Graphing Polynomial Functions
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 -axis, we can confirm that there is a zero between them. Consider a polynomial function whose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbers and in the domain of , if and , then the function takes on every value between and . (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 at lies above the - axis and another point at lies below the - axis, there must exist a third point between and where the graph crosses the - axis. Call this point . This means that we are assured there is a solution where .
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 - axis. Figure 17 shows that there is a zero between and .
Figure 17 Using the Intermediate Value Theorem to show there exists a zero.
INTERMEDIATE VALUE THEOREM
Let be a polynomial function. The Intermediate Value Theorem states that if and have opposite signs, then there exists at least one value between and for which .
EXAMPLE 9
Using the Intermediate Value Theorem
Show that the function has at least two real zeros between and .
Solution
As a start, evaluate at the integer values , and . See Table 2.
Table 2
We see that one zero occurs at . Also, since is negative and is positive, by the Intermediate Value Theorem, there must be at least one real zero between and .
We have shown that there are at least two real zeros between and .
Analysis
We can also see on the graph of the function in Figure 18 that there are two real zeros between and .
Figure 18
TRY IT #4
Show that the function has at least one real zero between and .