Graphing Polynomial Functions

Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have an $x$-intercept where each factor is equal to zero, we can form a function that will pass through a set of $x$-intercepts by introducing a corresponding set of factors.

FACTORED FORM OF POLYNOMIALS

If a polynomial of lowest degree p has horizontal intercepts at $x=x_{1}, x_{2}, \ldots, x_{n}$, then the polynomial can be written in the factored form: $f(x)=a\left(x-x_{1}\right)^{p_{1}}\left(x-x_{2}\right)^{p_{2}} \cdots\left(x-x_{n}\right)^{p_{n}}$ where the powers $p_{i}$ on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the $x$-intercept.

HOW TO

Given a graph of a polynomial function, write a formula for the function.

EXAMPLE 10

Writing a Formula for a Polynomial Function from the Graph

Write a formula for the polynomial function shown in Figure 19.

Figure 19

Solution

This graph has three x-intercepts: $x=−3,2$, and $5$. The $y$-intercept is located at $(0,2)$. At $x=−3$ and $x=5$, the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At $x=2$, the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us

$f(x)=a(x+3)(x-2)^{2}(x-5)$

To determine the stretch factor, we utilize another point on the graph. We will use the $y$- intercept $(0,–2)$, to solve for $a$.

$\begin{aligned} f(0) &=a(0+3)(0-2)^{2}(0-5) \\ -2 &=a(0+3)(0-2)^{2}(0-5) \\ -2 &=-60 a \\ a &=\frac{1}{30} \end{aligned}$

The graphed polynomial appears to represent the function $f(x)=\frac{1}{30}(x+3)(x-2)^{2}(x-5)$.

TRY IT #5

Given the graph shown in Figure 20, write a formula for the function shown.

Figure 20