Find the End Behavior f(x)=-1/2x^2+6x^3-x

$f (x) = - \frac{1}{2} x^{2} + 6 x^{3} - x$

Step 1

Identify the degree of the function.

Combine $x^{2}$ and $\frac{1}{2}$ .

$- \frac{x^{2}}{2} + 6 x^{3} - x$

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

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

$6 x^{3} \to 3$

$- x \to 1$

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.

Combine $x^{2}$ and $\frac{1}{2}$ .

$- \frac{x^{2}}{2} + 6 x^{3} - x$

Reorder $- \frac{x^{2}}{2}$ and $6 x^{3}$ .

$6 x^{3} - \frac{x^{2}}{2} - x$

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

$6 x^{3}$

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

$6$

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