Find the End Behavior p(x)=-9x^9+6x^6-3x^3+1

$p (x) = - 9 x^{9} + 6 x^{6} - 3 x^{3} + 1$

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.

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

$6 x^{6} \to 6$

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

$1 \to 0$

The largest exponent is the degree of the polynomial.

$9$

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.

$- 9 x^{9}$

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

$- 9$

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