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Exercises Polynomials and factoring

Example 10 from Algebra and Trigonometry, 5.3 Graphs of Polynomial Functions

Writing a Formula for a Polynomial Function from the Graph

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

Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.

Figure 19

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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 $a .$

$\begin{matrix} f (0) & = & a (0 + 3) {(0 - 2)}^{2} (0 - 5) \\ −2 & = & a (0 + 3) {(0 - 2)}^{2} (0 - 5) \\ −2 & = & - 60 a \\ a & = & \frac{1}{30} \end{matrix}$
The graphed polynomial appears to represent the function $f (x) = \frac{1}{30} (x + 3) {(x - 2)}^{2} (x - 5) .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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

Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.

Figure 20

How did it go?

PreviousUsing the Intermediate Value Theorem Next Using Local Extrema to Solve Applications

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.