Find the Behavior (Leading Coefficient Test) -3x^5+6x^2-9x^3-x

$- 3 x^{5} + 6 x^{2} - 9 x^{3} - x$

Step 1

Write $- 3 x^{5} + 6 x^{2} - 9 x^{3} - x$ as a function.

$f (x) = - 3 x^{5} + 6 x^{2} - 9 x^{3} - x$

Step 2

Identify the degree of the function.

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

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

$6 x^{2} \to 2$

$- 9 x^{3} \to 3$

$- x \to 1$

The largest exponent is the degree of the polynomial.

$5$

Step 3

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Step 4

Identify the leading coefficient.

Move $6 x^{2}$ .

$- 3 x^{5} - 9 x^{3} + 6 x^{2} - x$

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

$- 3 x^{5}$

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

$- 3$

Step 5

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

Negative

Step 6

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 7

Determine the behavior.

Rises to the left and falls to the right

Step 8