Find the End Behavior f(x)=-3x^5+x^3-6x^2+9

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

Step 1

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.

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

$x^{3} \to 3$

$- 6 x^{2} \to 2$

$9 \to 0$

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.

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 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