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Exercises Polynomials and factoring

Example 12 from Algebra and Trigonometry, 5.2 Power Functions and Polynomial Functions

Drawing Conclusions about a Polynomial Function from the Factors

Given the function $f (x) = - 4 x (x + 3) (x - 4),$ determine the local behavior.

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
The y-intercept is found by evaluating $f (0) .$
$\begin{matrix} f (0) & = & - 4 (0) (0 + 3) (0 - 4 \\ = & 0 \end{matrix}$
The y-intercept is $(0, 0) .$
The x-intercepts are found by determining the zeros of the function.

$0 = −4 x (x + 3) (x - 4)$

$\begin{matrix} x & = & 0 & or & x + 3 & = & 0 & or & x - 4 & = & 0 \\ x & = & 0 & or & x & = & - 3 & or & x & = & 4 \end{matrix}$
The x-intercepts are $(0, 0), (–3, 0),$ and $(4, 0) .$
The degree is 3 so the graph has at most 2 turning points.

Your turn

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

Given the function $f (x) = 0.2 (x - 2) (x + 1) (x - 5),$ determine the local behavior.

How did it go?

PreviousDrawing Conclusions about a Polynomial Function from the Graph Next Recognizing Polynomial Functions

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.2, 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.