Find the End Behavior f(x)=2+1/2x^4-1/2x^3

$f (x) = 2 + \frac{1}{2} x^{4} - \frac{1}{2} x^{3}$

Step 1

Identify the degree of the function.

Simplify each term.

Combine $\frac{1}{2}$ and $x^{4}$ .

$2 + \frac{x^{4}}{2} - \frac{1}{2} x^{3}$

Combine $x^{3}$ and $\frac{1}{2}$ .

$2 + \frac{x^{4}}{2} - \frac{x^{3}}{2}$

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

$2 \to 0$

$\frac{x^{4}}{2} \to 4$

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

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.

Simplify each term.

Combine $\frac{1}{2}$ and $x^{4}$ .

$2 + \frac{x^{4}}{2} - \frac{1}{2} x^{3}$

Combine $x^{3}$ and $\frac{1}{2}$ .

$2 + \frac{x^{4}}{2} - \frac{x^{3}}{2}$

Move $2$ .

$\frac{x^{4}}{2} - \frac{x^{3}}{2} + 2$

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

$\frac{x^{4}}{2}$

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

$\frac{1}{2}$

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.

Rises to the left and rises to the right

Step 7