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

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

Step 1

Identify the degree of the function.

Combine $\frac{1}{4}$ and $x^{5}$ .

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

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

$\frac{x^{5}}{4} \to 5$

$8 x \to 1$

$- 5 x^{4} \to 4$

$- 19 \to 0$

The largest exponent is the degree of the polynomial.

$5$

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 $\frac{1}{4}$ and $x^{5}$ .

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

Move $8 x$ .

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

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

$\frac{x^{5}}{4}$

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

$\frac{1}{4}$

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