Find the End Behavior q(x)=-x^2(x^2-9)

$q (x) = - x^{2} (x^{2} - 9)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Apply the distributive property.

$- x^{2} x^{2} - x^{2} \cdot - 9$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$- (x^{2} x^{2}) - x^{2} \cdot - 9$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2 + 2} - x^{2} \cdot - 9$

Add $2$ and $2$ .

$- x^{4} - x^{2} \cdot - 9$

Multiply $- 9$ by $- 1$ .

$- x^{4} + 9 x^{2}$

The largest exponent is the degree of the polynomial.

$4$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

Apply the distributive property.

$- x^{2} x^{2} - x^{2} \cdot - 9$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$- (x^{2} x^{2}) - x^{2} \cdot - 9$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{2 + 2} - x^{2} \cdot - 9$

Add $2$ and $2$ .

$- x^{4} - x^{2} \cdot - 9$

Multiply $- 9$ by $- 1$ .

$- x^{4} + 9 x^{2}$

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

$- x^{4}$

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

$- 1$

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.

Falls to the left and falls to the right

Step 7

Find the End Behavior q(x)=-x^2(x^2-9)

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