Graph h(x)=2(x+3)(x^2-1)

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

Step 1

Find the point at $x = - 3$ .

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = 2 {(- 3)}^{3} - 2 \cdot - 3 + 6 {(- 3)}^{2} - 6$

Simplify the result.

Simplify each term.

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

$f (- 3) = 2 \cdot - 27 - 2 \cdot - 3 + 6 {(- 3)}^{2} - 6$

Multiply $2$ by $- 27$ .

$f (- 3) = - 54 - 2 \cdot - 3 + 6 {(- 3)}^{2} - 6$

Multiply $- 2$ by $- 3$ .

$f (- 3) = - 54 + 6 + 6 {(- 3)}^{2} - 6$

Raise $- 3$ to the power of $2$ .

$f (- 3) = - 54 + 6 + 6 \cdot 9 - 6$

Multiply $6$ by $9$ .

$f (- 3) = - 54 + 6 + 54 - 6$

Simplify by adding and subtracting.

Add $- 54$ and $6$ .

$f (- 3) = - 48 + 54 - 6$

Add $- 48$ and $54$ .

$f (- 3) = 6 - 6$

Subtract $6$ from $6$ .

$f (- 3) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = - 2$ .

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = 2 {(- 2)}^{3} - 2 \cdot - 2 + 6 {(- 2)}^{2} - 6$

Simplify the result.

Simplify each term.

Raise $- 2$ to the power of $3$ .

$f (- 2) = 2 \cdot - 8 - 2 \cdot - 2 + 6 {(- 2)}^{2} - 6$

Multiply $2$ by $- 8$ .

$f (- 2) = - 16 - 2 \cdot - 2 + 6 {(- 2)}^{2} - 6$

Multiply $- 2$ by $- 2$ .

$f (- 2) = - 16 + 4 + 6 {(- 2)}^{2} - 6$

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

$f (- 2) = - 16 + 4 + 6 \cdot 4 - 6$

Multiply $6$ by $4$ .

$f (- 2) = - 16 + 4 + 24 - 6$

Simplify by adding and subtracting.

Add $- 16$ and $4$ .

$f (- 2) = - 12 + 24 - 6$

Add $- 12$ and $24$ .

$f (- 2) = 12 - 6$

Subtract $6$ from $12$ .

$f (- 2) = 6$

The final answer is $6$ .

$6$

Convert $6$ to decimal.

$y = 6$

Step 3

Find the point at $x = - 1$ .

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = 2 {(- 1)}^{3} - 2 \cdot - 1 + 6 {(- 1)}^{2} - 6$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $3$ .

$f (- 1) = 2 \cdot - 1 - 2 \cdot - 1 + 6 {(- 1)}^{2} - 6$

Multiply $2$ by $- 1$ .

$f (- 1) = - 2 - 2 \cdot - 1 + 6 {(- 1)}^{2} - 6$

Multiply $- 2$ by $- 1$ .

$f (- 1) = - 2 + 2 + 6 {(- 1)}^{2} - 6$

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 2 + 2 + 6 \cdot 1 - 6$

Multiply $6$ by $1$ .

$f (- 1) = - 2 + 2 + 6 - 6$

Simplify by adding and subtracting.

Add $- 2$ and $2$ .

$f (- 1) = 0 + 6 - 6$

Add $0$ and $6$ .

$f (- 1) = 6 - 6$

Subtract $6$ from $6$ .

$f (- 1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = 2 {(0)}^{3} - 2 \cdot 0 + 6 {(0)}^{2} - 6$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = 2 \cdot 0 - 2 \cdot 0 + 6 {(0)}^{2} - 6$

Multiply $2$ by $0$ .

$f (0) = 0 - 2 \cdot 0 + 6 {(0)}^{2} - 6$

Multiply $- 2$ by $0$ .

$f (0) = 0 + 0 + 6 {(0)}^{2} - 6$

Raising $0$ to any positive power yields $0$ .

$f (0) = 0 + 0 + 6 \cdot 0 - 6$

Multiply $6$ by $0$ .

$f (0) = 0 + 0 + 0 - 6$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 + 0 - 6$

Add $0$ and $0$ .

$f (0) = 0 - 6$

Subtract $6$ from $0$ .

$f (0) = - 6$

The final answer is $- 6$ .

$- 6$

Convert $- 6$ to decimal.

$y = - 6$

Step 5

Find the point at $x = 1$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = 2 {(1)}^{3} - 2 \cdot 1 + 6 {(1)}^{2} - 6$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 2 \cdot 1 - 2 \cdot 1 + 6 {(1)}^{2} - 6$

Multiply $2$ by $1$ .

$f (1) = 2 - 2 \cdot 1 + 6 {(1)}^{2} - 6$

Multiply $- 2$ by $1$ .

$f (1) = 2 - 2 + 6 {(1)}^{2} - 6$

One to any power is one.

$f (1) = 2 - 2 + 6 \cdot 1 - 6$

Multiply $6$ by $1$ .

$f (1) = 2 - 2 + 6 - 6$

Simplify by adding and subtracting.

Subtract $2$ from $2$ .

$f (1) = 0 + 6 - 6$

Add $0$ and $6$ .

$f (1) = 6 - 6$

Subtract $6$ from $6$ .

$f (1) = 0$

The final answer is $0$ .

$0$

Convert $0$ to decimal.

$y = 0$

Step 6

The cubic function can be graphed using the function behavior and the points.

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 6 \\ - 1 & 0 \\ 0 & - 6 \\ 1 & 0 \end{array}$

Step 7

The cubic function can be graphed using the function behavior and the selected points.

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & 6 \\ - 1 & 0 \\ 0 & - 6 \\ 1 & 0 \end{array}$

Step 8

Graph h(x)=2(x+3)(x^2-1)

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