Find the Behavior (Leading Coefficient Test) f(x)=1/3x^3+8x^4-5x-19x^2

$f (x) = \frac{1}{3} x^{3} + 8 x^{4} - 5 x - 19 x^{2}$

Step 1

Identify the degree of the function.

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

$\frac{x^{3}}{3} + 8 x^{4} - 5 x - 19 x^{2}$

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

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

$8 x^{4} \to 4$

$- 5 x \to 1$

$- 19 x^{2} \to 2$

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.

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

$\frac{x^{3}}{3} + 8 x^{4} - 5 x - 19 x^{2}$

Simplify the expression.

Move $- 5 x$ .

$\frac{x^{3}}{3} + 8 x^{4} - 19 x^{2} - 5 x$

Reorder $\frac{x^{3}}{3}$ and $8 x^{4}$ .

$8 x^{4} + \frac{x^{3}}{3} - 19 x^{2} - 5 x$

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

$8 x^{4}$

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

$8$

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