Math

QuestionAnalyze the function f(x)=(x+1)(x4)(x+6)f(x)=(x+1)(x-4)(x+6). Find its end behavior: ff behaves like y=y=\square as x|x| increases.

Studdy Solution

STEP 1

Assumptions1. The polynomial function is f(x)=(x+1)(x4)(x+6)f(x)=(x+1)(x-4)(x+6). We need to analyze the end behavior of the graph of this function

STEP 2

The end behavior of a polynomial function can be determined by the degree and the leading coefficient of the polynomial.

STEP 3

First, we need to expand the polynomial to find its degree and leading coefficient.
f(x)=(x+1)(x)(x+6)f(x) = (x+1)(x-)(x+6)

STEP 4

Expand the polynomial.
f(x)=x3+3x210x24f(x) = x^3 +3x^2 -10x -24

STEP 5

Now that we have the expanded form of the polynomial, we can see that the degree of the polynomial is3 and the leading coefficient is1.

STEP 6

The end behavior of a polynomial function is determined by the sign of the leading coefficient and whether the degree is even or odd.

STEP 7

Since the degree of the polynomial is odd and the leading coefficient is positive, the end behavior of the function is- As xx approaches negative infinity (xx \rightarrow -\infty), f(x)f(x) approaches positive infinity (f(x)f(x) \rightarrow \infty) - As xx approaches positive infinity (xx \rightarrow \infty), f(x)f(x) approaches negative infinity (f(x)f(x) \rightarrow -\infty)

STEP 8

So, the graph of ff behaves like y=x3y = -x^3 for large values of x|x|.

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