Algebra Examples

$F (x) = {(x - 2)}^{3} - 3$

Step 1

Find the point at $x = 0$ .

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Step 1.1

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

$f (0) = {((0) - 2)}^{3} - 3$

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

Subtract $2$ from $0$ .

$f (0) = {(- 2)}^{3} - 3$

Step 1.2.1.2

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

$f (0) = - 8 - 3$

Step 1.2.2

Subtract $3$ from $- 8$ .

$f (0) = - 11$

Step 1.2.3

The final answer is $- 11$ .

$- 11$

Step 1.3

Convert $- 11$ to decimal.

$y = - 11$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

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

Subtract $2$ from $1$ .

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

Step 2.2.1.2

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

$f (1) = - 1 - 3$

Step 2.2.2

Subtract $3$ from $- 1$ .

$f (1) = - 4$

Step 2.2.3

The final answer is $- 4$ .

$- 4$

Step 2.3

Convert $- 4$ to decimal.

$y = - 4$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

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

Subtract $2$ from $2$ .

$f (2) = 0^{3} - 3$

Step 3.2.1.2

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

$f (2) = 0 - 3$

Step 3.2.2

Subtract $3$ from $0$ .

$f (2) = - 3$

Step 3.2.3

The final answer is $- 3$ .

$- 3$

Step 3.3

Convert $- 3$ to decimal.

$y = - 3$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

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

Subtract $2$ from $3$ .

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

Step 4.2.1.2

One to any power is one.

$f (3) = 1 - 3$

Step 4.2.2

Subtract $3$ from $1$ .

$f (3) = - 2$

Step 4.2.3

The final answer is $- 2$ .

$- 2$

Step 4.3

Convert $- 2$ to decimal.

$y = - 2$

Step 5

Find the point at $x = 4$ .

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Step 5.1

Replace the variable $x$ with $4$ in the expression.

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

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

Subtract $2$ from $4$ .

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

Step 5.2.1.2

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

$f (4) = 8 - 3$

Step 5.2.2

Subtract $3$ from $8$ .

$f (4) = 5$

Step 5.2.3

The final answer is $5$ .

$5$

Step 5.3

Convert $5$ to decimal.

$y = 5$

Step 6

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

$\begin{array}{c} x & y \\ 0 & - 11 \\ 1 & - 4 \\ 2 & - 3 \\ 3 & - 2 \\ 4 & 5 \end{array}$

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

$\begin{array}{c} x & y \\ 0 & - 11 \\ 1 & - 4 \\ 2 & - 3 \\ 3 & - 2 \\ 4 & 5 \end{array}$

Step 8