Find the End Behavior f(x)=-8x^5-1408x-512x^3+104x^4+640+1216x^2

$f (x) = - 8 x^{5} - 1408 x - 512 x^{3} + 104 x^{4} + 640 + 1216 x^{2}$

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.

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

$- 1408 x \to 1$

$- 512 x^{3} \to 3$

$104 x^{4} \to 4$

$640 \to 0$

$1216 x^{2} \to 2$

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 $640$ .

$- 8 x^{5} - 1408 x - 512 x^{3} + 104 x^{4} + 1216 x^{2} + 640$

Move $- 1408 x$ .

$- 8 x^{5} - 512 x^{3} + 104 x^{4} + 1216 x^{2} - 1408 x + 640$

Move $- 512 x^{3}$ .

$- 8 x^{5} + 104 x^{4} - 512 x^{3} + 1216 x^{2} - 1408 x + 640$

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