Algebra Examples

$f (x) = x^{3}$ $g (x) = x^{3} - 4$

Step 1

Graph $f (x) = x^{3}$ .

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Step 1.1

Find the point at $x = - 2$ .

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Step 1.1.1

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

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

Step 1.1.2

Simplify the result.

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Step 1.1.2.1

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

$f (- 2) = - 8$

Step 1.1.2.2

The final answer is $- 8$ .

$- 8$

Step 1.1.3

Convert $- 8$ to decimal.

$y = - 8$

Step 1.2

Find the point at $x = - 1$ .

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Step 1.2.1

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

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

Step 1.2.2

Simplify the result.

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Step 1.2.2.1

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

$f (- 1) = - 1$

Step 1.2.2.2

The final answer is $- 1$ .

$- 1$

Step 1.2.3

Convert $- 1$ to decimal.

$y = - 1$

Step 1.3

Find the point at $x = 0$ .

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Step 1.3.1

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

$f (0) = {(0)}^{3}$

Step 1.3.2

Simplify the result.

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Step 1.3.2.1

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

$f (0) = 0$

Step 1.3.2.2

The final answer is $0$ .

$0$

Step 1.3.3

Convert $0$ to decimal.

$y = 0$

Step 1.4

Find the point at $x = 1$ .

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Step 1.4.1

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

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

Step 1.4.2

Simplify the result.

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Step 1.4.2.1

One to any power is one.

$f (1) = 1$

Step 1.4.2.2

The final answer is $1$ .

$1$

Step 1.4.3

Convert $1$ to decimal.

$y = 1$

Step 1.5

Find the point at $x = 2$ .

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Step 1.5.1

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

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

Step 1.5.2

Simplify the result.

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Step 1.5.2.1

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

$f (2) = 8$

Step 1.5.2.2

The final answer is $8$ .

$8$

Step 1.5.3

Convert $8$ to decimal.

$y = 8$

Step 1.6

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

$\begin{array}{c} x & y \\ - 2 & - 8 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 8 \end{array}$

Step 1.7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 8 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 8 \end{array}$

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 8 \\ - 1 & - 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 8 \end{array}$

Step 2

Graph $g (x) = x^{3} - 4$ .

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Step 2.1

Find the point at $x = - 2$ .

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Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Subtract $4$ from $- 8$ .

$f (- 2) = - 12$

Step 2.1.2.3

The final answer is $- 12$ .

$- 12$

Step 2.1.3

Convert $- 12$ to decimal.

$y = - 12$

Step 2.2

Find the point at $x = - 1$ .

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Step 2.2.1

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

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

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Subtract $4$ from $- 1$ .

$f (- 1) = - 5$

Step 2.2.2.3

The final answer is $- 5$ .

$- 5$

Step 2.2.3

Convert $- 5$ to decimal.

$y = - 5$

Step 2.3

Find the point at $x = 0$ .

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Step 2.3.1

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

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

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

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

$f (0) = 0 - 4$

Step 2.3.2.2

Subtract $4$ from $0$ .

$f (0) = - 4$

Step 2.3.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3.3

Convert $- 4$ to decimal.

$y = - 4$

Step 2.4

Find the point at $x = 1$ .

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Step 2.4.1

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

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

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

One to any power is one.

$f (1) = 1 - 4$

Step 2.4.2.2

Subtract $4$ from $1$ .

$f (1) = - 3$

Step 2.4.2.3

The final answer is $- 3$ .

$- 3$

Step 2.4.3

Convert $- 3$ to decimal.

$y = - 3$

Step 2.5

Find the point at $x = 2$ .

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Step 2.5.1

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

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

Step 2.5.2

Simplify the result.

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Step 2.5.2.1

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

$f (2) = 8 - 4$

Step 2.5.2.2

Subtract $4$ from $8$ .

$f (2) = 4$

Step 2.5.2.3

The final answer is $4$ .

$4$

Step 2.5.3

Convert $4$ to decimal.

$y = 4$

Step 2.6

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

$\begin{array}{c} x & y \\ - 2 & - 12 \\ - 1 & - 5 \\ 0 & - 4 \\ 1 & - 3 \\ 2 & 4 \end{array}$

Step 2.7

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 12 \\ - 1 & - 5 \\ 0 & - 4 \\ 1 & - 3 \\ 2 & 4 \end{array}$

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 12 \\ - 1 & - 5 \\ 0 & - 4 \\ 1 & - 3 \\ 2 & 4 \end{array}$

Step 3

Plot each graph on the same coordinate system.

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

$g (x) = x^{3} - 4$

Step 4