Precalculus Examples

$f (x) = x (x - 2) (x + 2)$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$f (- 2) = - 8 - 4 \cdot - 2$

Step 1.2.1.2

Multiply $- 4$ by $- 2$ .

$f (- 2) = - 8 + 8$

Step 1.2.2

Add $- 8$ and $8$ .

$f (- 2) = 0$

Step 1.2.3

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = {(- 1)}^{3} - 4 \cdot - 1$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 1) = - 1 - 4 \cdot - 1$

Step 2.2.1.2

Multiply $- 4$ by $- 1$ .

$f (- 1) = - 1 + 4$

Step 2.2.2

Add $- 1$ and $4$ .

$f (- 1) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 2.3

Convert $3$ to decimal.

$y = 3$

Step 3

Find the point at $x = 0$ .

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Step 3.1

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

$f (0) = {(0)}^{3} - 4 \cdot 0$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (0) = 0 - 4 \cdot 0$

Step 3.2.1.2

Multiply $- 4$ by $0$ .

$f (0) = 0 + 0$

Step 3.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 1$ .

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Step 4.1

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

$f (1) = {(1)}^{3} - 4 \cdot 1$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

One to any power is one.

$f (1) = 1 - 4 \cdot 1$

Step 4.2.1.2

Multiply $- 4$ by $1$ .

$f (1) = 1 - 4$

Step 4.2.2

Subtract $4$ from $1$ .

$f (1) = - 3$

Step 4.2.3

The final answer is $- 3$ .

$- 3$

Step 4.3

Convert $- 3$ to decimal.

$y = - 3$

Step 5

Find the point at $x = 2$ .

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Step 5.1

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

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

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f (2) = 8 - 4 \cdot 2$

Step 5.2.1.2

Multiply $- 4$ by $2$ .

$f (2) = 8 - 8$

Step 5.2.2

Subtract $8$ from $8$ .

$f (2) = 0$

Step 5.2.3

The final answer is $0$ .

$0$

Step 5.3

Convert $0$ to decimal.

$y = 0$

Step 6

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

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

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

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

Step 8