Math Snap

STEP 1

Assumptions1. The polynomial function is $f(x)=(x+1)(x-4)(x+6)$. 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.
$$f(x) = (x+1)(x-)(x+6)$$

STEP 4

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

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 $x$ approaches negative infinity ($x \rightarrow -\infty$), $f(x)$ approaches positive infinity ($f(x) \rightarrow \infty$)
- As $x$ approaches positive infinity ($x \rightarrow \infty$), $f(x)$ approaches negative infinity ($f(x) \rightarrow -\infty$)

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