QuestionAnalyze the function $f(x)=x^{3}-16 x$ .
(a) Describe the end behavior. (b) Find the $x$ - and $y$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is $f(x)=x^{3}-16 x$ . We are asked to analyze the function, which includes determining the end behavior, and finding the x- and y-intercepts.

STEP 2

First, we need to factor the polynomial function.
$f(x)=x^{}-16 x = x(x^{2}-16)$

STEP 3

Further factor the polynomial function by using the difference of squares formula, $a^{2}-b^{2}=(a-b)(a+b)$ .
$f(x)=x(x^{2}-16) = x(x-)(x+)$

STEP 4

(a) To determine the end behavior of the graph of the function, we look at the highest power of $x$ , which is $x^{3}$ . Since the coefficient of $x^{3}$ is positive, as $x$ approaches positive infinity, $f(x)$ approaches positive infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity.

STEP 5

(b) To find the x-intercepts of the graph of the function, we set $f(x)$ equal to zero and solve for $x$ .
$0 = x(x-4)(x+4)$

STEP 6

Setting each factor equal to zero gives the solutions $x=0$ , $x=4$ , and $x=-4$ . These are the x-intercepts of the graph.

STEP 7

To find the y-intercept of the graph of the function, we set $x=0$ in the function $f(x)$ .
$f(0) =0(0-4)(0+4)$

STEP 8

Calculate the y-intercept.
$f(0) =0(0-4)(0+4) =0$ The y-intercept of the graph is $0$ .

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