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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 4$ by $- 2$ .

$(- 2 x^{2} + 8 x - 2 \cdot 4) (x^{2} - 9)$

Multiply $- 2$ by $4$ .

$(- 2 x^{2} + 8 x - 8) (x^{2} - 9)$

Expand $(- 2 x^{2} + 8 x - 8) (x^{2} - 9)$ by multiplying each term in the first expression by each term in the second expression.

$- 2 x^{2} x^{2} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 2 (x^{2} x^{2}) - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

$- 2 x^{2 + 2} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Add $2$ and $2$ .

$- 2 x^{4} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Multiply $- 9$ by $- 2$ .

$- 2 x^{4} + 18 x^{2} + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

Move $x^{2}$ .

$- 2 x^{4} + 18 x^{2} + 8 (x^{2} x) + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

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

$- 2 x^{4} + 18 x^{2} + 8 (x^{2} x^{1}) + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

$- 2 x^{4} + 18 x^{2} + 8 x^{2 + 1} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Add $2$ and $1$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Multiply $- 9$ by $8$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} - 72 x - 8 x^{2} - 8 \cdot - 9$

Multiply $- 8$ by $- 9$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} - 72 x - 8 x^{2} + 72$

Subtract $8 x^{2}$ from $18 x^{2}$ .

$- 2 x^{4} + 10 x^{2} + 8 x^{3} - 72 x + 72$

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.

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Subtract $2 x$ from $- 2 x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $- 4$ by $- 2$ .

$(- 2 x^{2} + 8 x - 2 \cdot 4) (x^{2} - 9)$

Multiply $- 2$ by $4$ .

$(- 2 x^{2} + 8 x - 8) (x^{2} - 9)$

Expand $(- 2 x^{2} + 8 x - 8) (x^{2} - 9)$ by multiplying each term in the first expression by each term in the second expression.

$- 2 x^{2} x^{2} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 2 (x^{2} x^{2}) - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

$- 2 x^{2 + 2} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Add $2$ and $2$ .

$- 2 x^{4} - 2 x^{2} \cdot - 9 + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Multiply $- 9$ by $- 2$ .

$- 2 x^{4} + 18 x^{2} + 8 x \cdot x^{2} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

Move $x^{2}$ .

$- 2 x^{4} + 18 x^{2} + 8 (x^{2} x) + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

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

$- 2 x^{4} + 18 x^{2} + 8 (x^{2} x^{1}) + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

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

$- 2 x^{4} + 18 x^{2} + 8 x^{2 + 1} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Add $2$ and $1$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} + 8 x \cdot - 9 - 8 x^{2} - 8 \cdot - 9$

Multiply $- 9$ by $8$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} - 72 x - 8 x^{2} - 8 \cdot - 9$

Multiply $- 8$ by $- 9$ .

$- 2 x^{4} + 18 x^{2} + 8 x^{3} - 72 x - 8 x^{2} + 72$

Simplify by adding terms.

Subtract $8 x^{2}$ from $18 x^{2}$ .

$- 2 x^{4} + 10 x^{2} + 8 x^{3} - 72 x + 72$

Move $10 x^{2}$ .

$- 2 x^{4} + 8 x^{3} + 10 x^{2} - 72 x + 72$

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

$- 2 x^{4}$

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

$- 2$

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 f(x)=-2(x-2)^2(x^2-9)

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