Find the End Behavior f(x)=460x-10x^4-40x^3-1680+310x^2

$f (x) = 460 x - 10 x^{4} - 40 x^{3} - 1680 + 310 x^{2}$

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.

$460 x \to 1$

$- 10 x^{4} \to 4$

$- 40 x^{3} \to 3$

$- 1680 \to 0$

$310 x^{2} \to 2$

The largest exponent is the degree of the polynomial.

$4$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

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

Move $- 1680$ .

$460 x - 10 x^{4} - 40 x^{3} + 310 x^{2} - 1680$

Move $460 x$ .

$- 10 x^{4} - 40 x^{3} + 310 x^{2} + 460 x - 1680$

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

$- 10 x^{4}$

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

$- 10$

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.

Falls to the left and falls to the right

Step 7