Find the End Behavior f(x)=8x^5+1008x^2+1944x-288x^3-6480-32x^4

$f (x) = 8 x^{5} + 1008 x^{2} + 1944 x - 288 x^{3} - 6480 - 32 x^{4}$

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.

$8 x^{5} \to 5$

$1008 x^{2} \to 2$

$1944 x \to 1$

$- 288 x^{3} \to 3$

$- 6480 \to 0$

$- 32 x^{4} \to 4$

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 $- 6480$ .

$8 x^{5} + 1008 x^{2} + 1944 x - 288 x^{3} - 32 x^{4} - 6480$

Move $1944 x$ .

$8 x^{5} + 1008 x^{2} - 288 x^{3} - 32 x^{4} + 1944 x - 6480$

Move $1008 x^{2}$ .

$8 x^{5} - 288 x^{3} - 32 x^{4} + 1008 x^{2} + 1944 x - 6480$

Move $- 288 x^{3}$ .

$8 x^{5} - 32 x^{4} - 288 x^{3} + 1008 x^{2} + 1944 x - 6480$

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 positive, the graph rises to the right.

Positive

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.

Falls to the left and rises to the right

Step 7