Find the End Behavior f(x)=-832x^2+12x^4+4096+128x^3-2x^5

$f (x) = - 832 x^{2} + 12 x^{4} + 4096 + 128 x^{3} - 2 x^{5}$

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.

$- 832 x^{2} \to 2$

$12 x^{4} \to 4$

$4096 \to 0$

$128 x^{3} \to 3$

$- 2 x^{5} \to 5$

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 $4096$ .

$- 832 x^{2} + 12 x^{4} + 128 x^{3} - 2 x^{5} + 4096$

Move $- 832 x^{2}$ .

$12 x^{4} + 128 x^{3} - 2 x^{5} - 832 x^{2} + 4096$

Move $128 x^{3}$ .

$12 x^{4} - 2 x^{5} + 128 x^{3} - 832 x^{2} + 4096$

Reorder $12 x^{4}$ and $- 2 x^{5}$ .

$- 2 x^{5} + 12 x^{4} + 128 x^{3} - 832 x^{2} + 4096$

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

$- 2 x^{5}$

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

$- 2$

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