Find the End Behavior f(x)=-3072+2432x+128x^4-576x^3+256x^2-8x^5

$f (x) = - 3072 + 2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5}$

Step 1

Identify the degree of the function.

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$- 3072 \to 0$

$2432 x \to 1$

$128 x^{4} \to 4$

$- 576 x^{3} \to 3$

$256 x^{2} \to 2$

$- 8 x^{5} \to 5$

The largest exponent is the degree of the polynomial.

$5$

Step 2

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Move $- 3072$ .

$2432 x + 128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} - 3072$

Move $2432 x$ .

$128 x^{4} - 576 x^{3} + 256 x^{2} - 8 x^{5} + 2432 x - 3072$

Move $256 x^{2}$ .

$128 x^{4} - 576 x^{3} - 8 x^{5} + 256 x^{2} + 2432 x - 3072$

Move $- 576 x^{3}$ .

$128 x^{4} - 8 x^{5} - 576 x^{3} + 256 x^{2} + 2432 x - 3072$

Reorder $128 x^{4}$ and $- 8 x^{5}$ .

$- 8 x^{5} + 128 x^{4} - 576 x^{3} + 256 x^{2} + 2432 x - 3072$

The leading term in a polynomial is the term with the highest degree.

$- 8 x^{5}$

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 8$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Rises to the left and falls to the right

Step 7