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

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

Step 1

Write $x (x^{2} - 9)$ as a function.

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

Step 2

Identify the degree of the function.

Simplify and reorder the polynomial.

Apply the distributive property.

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

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

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$x^{1} x^{2} + x \cdot - 9$

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

$x^{1 + 2} + x \cdot - 9$

Add $1$ and $2$ .

$x^{3} + x \cdot - 9$

Move $- 9$ to the left of $x$ .

$x^{3} - 9 x$

The largest exponent is the degree of the polynomial.

$3$

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.

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

Apply the distributive property.

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

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

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$x^{1} x^{2} + x \cdot - 9$

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

$x^{1 + 2} + x \cdot - 9$

Add $1$ and $2$ .

$x^{3} + x \cdot - 9$

Move $- 9$ to the left of $x$ .

$x^{3} - 9 x$

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

$x^{3}$

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

$1$

Step 5

Since the leading coefficient is positive, the graph rises to the right.

Positive

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.

Falls to the left and rises to the right

Step 8

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

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