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

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

Step 1

Identify the degree of the function.

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

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

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

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

$8 \to 0$

$- 5 x^{3} \to 3$

$- 19 x \to 1$

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

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

Simplify the expression.

Move $8$ .

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

Reorder $\frac{x^{2}}{2}$ and $- 5 x^{3}$ .

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

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

$- 5 x^{3}$

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

$- 5$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

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 falls to the right

Step 7