Find the End Behavior f(x)=2(x+1)^3-5

$f (x) = 2 {(x + 1)}^{3} - 5$

Identify the degree of the function.

Simplify and reorder the polynomial.

Simplify each term.

Use the Binomial Theorem.

$2 (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) - 5$

Simplify each term.

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) - 5$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) - 5$

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x + 1^{3}) - 5$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x + 1) - 5$

Apply the distributive property.

$2 x^{3} + 2 (3 x^{2}) + 2 (3 x) + 2 \cdot 1 - 5$

Simplify.

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 2 (3 x) + 2 \cdot 1 - 5$

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 6 x + 2 \cdot 1 - 5$

Multiply $2$ by $1$ .

$2 x^{3} + 6 x^{2} + 6 x + 2 - 5$

Subtract $5$ from $2$ .

$2 x^{3} + 6 x^{2} + 6 x - 3$

Identify the exponents on the variables in each term, and add them together to find the degree of each term.

$2 x^{3} \to 3$

$6 x^{2} \to 2$

$6 x \to 1$

$- 3 \to 0$

The largest exponent is the degree of the polynomial.

$3$

Since the degree is odd, the ends of the function will point in the opposite directions.

Odd

Identify the leading coefficient.

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

Simplify each term.

Use the Binomial Theorem.

$2 (x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3}) - 5$

Simplify each term.

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3}) - 5$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3}) - 5$

Multiply $3$ by $1$ .

$2 (x^{3} + 3 x^{2} + 3 x + 1^{3}) - 5$

One to any power is one.

$2 (x^{3} + 3 x^{2} + 3 x + 1) - 5$

Apply the distributive property.

$2 x^{3} + 2 (3 x^{2}) + 2 (3 x) + 2 \cdot 1 - 5$

Simplify.

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 2 (3 x) + 2 \cdot 1 - 5$

Multiply $3$ by $2$ .

$2 x^{3} + 6 x^{2} + 6 x + 2 \cdot 1 - 5$

Multiply $2$ by $1$ .

$2 x^{3} + 6 x^{2} + 6 x + 2 - 5$

Subtract $5$ from $2$ .

$2 x^{3} + 6 x^{2} + 6 x - 3$

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

$2 x^{3}$

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

$2$

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

Positive

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

Determine the behavior.

Falls to the left and rises to the right

Find the End Behavior f(x)=2(x+1)^3-5

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