Find the End Behavior f(x)=-4(x-3)^2(x^2-4)

$f (x) = - 4 {(x - 3)}^{2} (x^{2} - 4)$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$- 4 ((x - 3) (x - 3)) (x^{2} - 4)$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$- 4 (x (x - 3) - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$- 4 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$- 4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Move $- 3$ to the left of $x$ .

$- 4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Multiply $- 3$ by $- 3$ .

$- 4 (x^{2} - 3 x - 3 x + 9) (x^{2} - 4)$

Subtract $3 x$ from $- 3 x$ .

$- 4 (x^{2} - 6 x + 9) (x^{2} - 4)$

Apply the distributive property.

$(- 4 x^{2} - 4 (- 6 x) - 4 \cdot 9) (x^{2} - 4)$

Simplify.

Multiply $- 6$ by $- 4$ .

$(- 4 x^{2} + 24 x - 4 \cdot 9) (x^{2} - 4)$

Multiply $- 4$ by $9$ .

$(- 4 x^{2} + 24 x - 36) (x^{2} - 4)$

Expand $(- 4 x^{2} + 24 x - 36) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$- 4 x^{2} x^{2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$- 4 x^{2 + 2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $2$ .

$- 4 x^{4} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$- 4 x^{4} + 16 x^{2} + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

Move $x^{2}$ .

$- 4 x^{4} + 16 x^{2} + 24 (x^{2} x) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

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

$- 4 x^{4} + 16 x^{2} + 24 (x^{2} x^{1}) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$- 4 x^{4} + 16 x^{2} + 24 x^{2 + 1} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $1$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $24$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} - 36 \cdot - 4$

Multiply $- 36$ by $- 4$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} + 144$

Subtract $36 x^{2}$ from $16 x^{2}$ .

$- 4 x^{4} - 20 x^{2} + 24 x^{3} - 96 x + 144$

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 ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$- 4 ((x - 3) (x - 3)) (x^{2} - 4)$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$- 4 (x (x - 3) - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$- 4 (x \cdot x + x \cdot - 3 - 3 (x - 3)) (x^{2} - 4)$

Apply the distributive property.

$- 4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Move $- 3$ to the left of $x$ .

$- 4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) (x^{2} - 4)$

Multiply $- 3$ by $- 3$ .

$- 4 (x^{2} - 3 x - 3 x + 9) (x^{2} - 4)$

Subtract $3 x$ from $- 3 x$ .

$- 4 (x^{2} - 6 x + 9) (x^{2} - 4)$

Apply the distributive property.

$(- 4 x^{2} - 4 (- 6 x) - 4 \cdot 9) (x^{2} - 4)$

Simplify.

Multiply $- 6$ by $- 4$ .

$(- 4 x^{2} + 24 x - 4 \cdot 9) (x^{2} - 4)$

Multiply $- 4$ by $9$ .

$(- 4 x^{2} + 24 x - 36) (x^{2} - 4)$

Expand $(- 4 x^{2} + 24 x - 36) (x^{2} - 4)$ by multiplying each term in the first expression by each term in the second expression.

$- 4 x^{2} x^{2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Simplify terms.

Simplify each term.

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

Move $x^{2}$ .

$- 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$- 4 x^{2 + 2} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $2$ .

$- 4 x^{4} - 4 x^{2} \cdot - 4 + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $- 4$ .

$- 4 x^{4} + 16 x^{2} + 24 x \cdot x^{2} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

Move $x^{2}$ .

$- 4 x^{4} + 16 x^{2} + 24 (x^{2} x) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

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

$- 4 x^{4} + 16 x^{2} + 24 (x^{2} x^{1}) + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

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

$- 4 x^{4} + 16 x^{2} + 24 x^{2 + 1} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Add $2$ and $1$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} + 24 x \cdot - 4 - 36 x^{2} - 36 \cdot - 4$

Multiply $- 4$ by $24$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} - 36 \cdot - 4$

Multiply $- 36$ by $- 4$ .

$- 4 x^{4} + 16 x^{2} + 24 x^{3} - 96 x - 36 x^{2} + 144$

Simplify by adding terms.

Subtract $36 x^{2}$ from $16 x^{2}$ .

$- 4 x^{4} - 20 x^{2} + 24 x^{3} - 96 x + 144$

Move $- 20 x^{2}$ .

$- 4 x^{4} + 24 x^{3} - 20 x^{2} - 96 x + 144$

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

$- 4 x^{4}$

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

$- 4$

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)=-4(x-3)^2(x^2-4)

Related Questions

Support LCMGCF.com by disabling your adblocker.