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

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

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Use the Binomial Theorem.

$2 (x^{4} + 4 x^{3} \cdot - 4 + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Simplify terms.

Simplify each term.

Multiply $- 4$ by $4$ .

$2 (x^{4} - 16 x^{3} + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Raise $- 4$ to the power of $2$ .

$2 (x^{4} - 16 x^{3} + 6 x^{2} \cdot 16 + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Multiply $16$ by $6$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Raise $- 4$ to the power of $3$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} + 4 x \cdot - 64 + {(- 4)}^{4})$

Multiply $- 64$ by $4$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + {(- 4)}^{4})$

Raise $- 4$ to the power of $4$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256)$

Apply the distributive property.

$2 x^{4} + 2 (- 16 x^{3}) + 2 (96 x^{2}) + 2 (- 256 x) + 2 \cdot 256$

Simplify.

Multiply $- 16$ by $2$ .

$2 x^{4} - 32 x^{3} + 2 (96 x^{2}) + 2 (- 256 x) + 2 \cdot 256$

Multiply $96$ by $2$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} + 2 (- 256 x) + 2 \cdot 256$

Multiply $- 256$ by $2$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} - 512 x + 2 \cdot 256$

Multiply $2$ by $256$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} - 512 x + 512$

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.

Use the Binomial Theorem.

$2 (x^{4} + 4 x^{3} \cdot - 4 + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Simplify terms.

Simplify each term.

Multiply $- 4$ by $4$ .

$2 (x^{4} - 16 x^{3} + 6 x^{2} {(- 4)}^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Raise $- 4$ to the power of $2$ .

$2 (x^{4} - 16 x^{3} + 6 x^{2} \cdot 16 + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Multiply $16$ by $6$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} + 4 x {(- 4)}^{3} + {(- 4)}^{4})$

Raise $- 4$ to the power of $3$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} + 4 x \cdot - 64 + {(- 4)}^{4})$

Multiply $- 64$ by $4$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + {(- 4)}^{4})$

Raise $- 4$ to the power of $4$ .

$2 (x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256)$

Apply the distributive property.

$2 x^{4} + 2 (- 16 x^{3}) + 2 (96 x^{2}) + 2 (- 256 x) + 2 \cdot 256$

Simplify.

Multiply $- 16$ by $2$ .

$2 x^{4} - 32 x^{3} + 2 (96 x^{2}) + 2 (- 256 x) + 2 \cdot 256$

Multiply $96$ by $2$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} + 2 (- 256 x) + 2 \cdot 256$

Multiply $- 256$ by $2$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} - 512 x + 2 \cdot 256$

Multiply $2$ by $256$ .

$2 x^{4} - 32 x^{3} + 192 x^{2} - 512 x + 512$

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 positive, the graph rises to the right.

Positive

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 rises to the right

Step 7

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

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