Precalculus Examples

$f (x) = x - x^{3}$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = (- 2) - {(- 2)}^{3}$

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 $3$ .

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

Step 1.2.1.2

Multiply $- 1$ by $- 8$ .

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

Step 1.2.2

Add $- 2$ and $8$ .

$f (- 2) = 6$

Step 1.2.3

The final answer is $6$ .

$6$

Step 1.3

Convert $6$ to decimal.

$y = 6$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

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

$f (- 1) = (- 1) - {(- 1)}^{3}$

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

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 2.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{3}$ .

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Step 2.2.1.1.1.1

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

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

Step 2.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.1.1.2

Add $1$ and $3$ .

$f (- 1) = - 1 + {(- 1)}^{4}$

Step 2.2.1.2

Raise $- 1$ to the power of $4$ .

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

Step 2.2.2

Add $- 1$ and $1$ .

$f (- 1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = (2) - {(2)}^{3}$

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

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

$f (2) = 2 - 1 \cdot 8$

Step 3.2.1.2

Multiply $- 1$ by $8$ .

$f (2) = 2 - 8$

Step 3.2.2

Subtract $8$ from $2$ .

$f (2) = - 6$

Step 3.2.3

The final answer is $- 6$ .

$- 6$

Step 3.3

Convert $- 6$ to decimal.

$y = - 6$

Step 4

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

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

Step 5

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

Rises to the left and falls to the right

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

Step 6