Find the End Behavior f(x)=12x^2-18x+2x^3-108

$f (x) = 12 x^{2} - 18 x + 2 x^{3} - 108$

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.

$12 x^{2} \to 2$

$- 18 x \to 1$

$2 x^{3} \to 3$

$- 108 \to 0$

The largest exponent is the degree of the polynomial.

$3$

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 $- 18 x$ .

$12 x^{2} + 2 x^{3} - 18 x - 108$

Reorder $12 x^{2}$ and $2 x^{3}$ .

$2 x^{3} + 12 x^{2} - 18 x - 108$

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

$2 x^{3}$

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

$2$

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