Find the Behavior (Leading Coefficient Test) f(x)=(x+4)^6

$f (x) = {(x + 4)}^{6}$

Step 1

Identify the degree of the function.

Simplify and reorder the polynomial.

Use the Binomial Theorem.

$x^{6} + 6 x^{5} \cdot 4 + 15 x^{4} \cdot 4^{2} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Simplify each term.

Multiply $4$ by $6$ .

$x^{6} + 24 x^{5} + 15 x^{4} \cdot 4^{2} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 15 x^{4} \cdot 16 + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Multiply $16$ by $15$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 240 x^{4} + 20 x^{3} \cdot 64 + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Multiply $64$ by $20$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 15 x^{2} \cdot 256 + 6 x \cdot 4^{5} + 4^{6}$

Multiply $256$ by $15$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6 x \cdot 4^{5} + 4^{6}$

Raise $4$ to the power of $5$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6 x \cdot 1024 + 4^{6}$

Multiply $1024$ by $6$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6144 x + 4^{6}$

Raise $4$ to the power of $6$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6144 x + 4096$

The largest exponent is the degree of the polynomial.

$6$

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.

$x^{6} + 6 x^{5} \cdot 4 + 15 x^{4} \cdot 4^{2} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Simplify each term.

Multiply $4$ by $6$ .

$x^{6} + 24 x^{5} + 15 x^{4} \cdot 4^{2} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 15 x^{4} \cdot 16 + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Multiply $16$ by $15$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 20 x^{3} \cdot 4^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 240 x^{4} + 20 x^{3} \cdot 64 + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

Multiply $64$ by $20$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 15 x^{2} \cdot 4^{4} + 6 x \cdot 4^{5} + 4^{6}$

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

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 15 x^{2} \cdot 256 + 6 x \cdot 4^{5} + 4^{6}$

Multiply $256$ by $15$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6 x \cdot 4^{5} + 4^{6}$

Raise $4$ to the power of $5$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6 x \cdot 1024 + 4^{6}$

Multiply $1024$ by $6$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6144 x + 4^{6}$

Raise $4$ to the power of $6$ .

$x^{6} + 24 x^{5} + 240 x^{4} + 1280 x^{3} + 3840 x^{2} + 6144 x + 4096$

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

$x^{6}$

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

$1$

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 Behavior (Leading Coefficient Test) f(x)=(x+4)^6

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