Find the End Behavior f(x)=2520+3x^3-291x-219x^2+3x^4

$f (x) = 2520 + 3 x^{3} - 291 x - 219 x^{2} + 3 x^{4}$

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.

$2520 \to 0$

$3 x^{3} \to 3$

$- 291 x \to 1$

$- 219 x^{2} \to 2$

$3 x^{4} \to 4$

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 $2520$ .

$3 x^{3} - 291 x - 219 x^{2} + 3 x^{4} + 2520$

Move $- 291 x$ .

$3 x^{3} - 219 x^{2} + 3 x^{4} - 291 x + 2520$

Move $- 219 x^{2}$ .

$3 x^{3} + 3 x^{4} - 219 x^{2} - 291 x + 2520$

Reorder $3 x^{3}$ and $3 x^{4}$ .

$3 x^{4} + 3 x^{3} - 219 x^{2} - 291 x + 2520$

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

$3 x^{4}$

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

$3$

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.

Rises to the left and rises to the right

Step 7