Precalculus Examples

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

Step 1

Find the point at $x = - 3$ .

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Step 1.1

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

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

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

Add $- 3$ and $3$ .

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

Step 1.2.1.2

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

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

Step 1.2.1.3

Multiply $- 3$ by $0$ .

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

Step 1.2.1.4

Add $- 3$ and $3$ .

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

Step 1.2.1.5

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

$f (- 3) = 0 - 0$

Step 1.2.1.6

Multiply $- 1$ by $0$ .

$f (- 3) = 0 + 0$

Step 1.2.2

Add $0$ and $0$ .

$f (- 3) = 0$

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 $x = - 2$ .

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Step 2.1

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

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

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

Add $- 2$ and $3$ .

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

Step 2.2.1.2

One to any power is one.

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

Step 2.2.1.3

Multiply $- 2$ by $1$ .

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

Step 2.2.1.4

Add $- 2$ and $3$ .

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

Step 2.2.1.5

One to any power is one.

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

Step 2.2.1.6

Multiply $- 1$ by $1$ .

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

Step 2.2.2

Subtract $1$ from $- 2$ .

$f (- 2) = - 3$

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 $x = - 1$ .

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Step 3.1

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

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

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

Add $- 1$ and $3$ .

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

Step 3.2.1.2

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

$f (- 1) = - 1 \cdot 4 - {((- 1) + 3)}^{2}$

Step 3.2.1.3

Multiply $- 1$ by $4$ .

$f (- 1) = - 4 - {((- 1) + 3)}^{2}$

Step 3.2.1.4

Add $- 1$ and $3$ .

$f (- 1) = - 4 - 2^{2}$

Step 3.2.1.5

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

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

Step 3.2.1.6

Multiply $- 1$ by $4$ .

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

Step 3.2.2

Subtract $4$ from $- 4$ .

$f (- 1) = - 8$

Step 3.2.3

The final answer is $- 8$ .

$- 8$

Step 3.3

Convert $- 8$ to decimal.

$y = - 8$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

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

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

Add $0$ and $3$ .

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

Step 4.2.1.2

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

$f (0) = 0 \cdot 9 - {((0) + 3)}^{2}$

Step 4.2.1.3

Multiply $0$ by $9$ .

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

Step 4.2.1.4

Add $0$ and $3$ .

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

Step 4.2.1.5

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

$f (0) = 0 - 1 \cdot 9$

Step 4.2.1.6

Multiply $- 1$ by $9$ .

$f (0) = 0 - 9$

Step 4.2.2

Subtract $9$ from $0$ .

$f (0) = - 9$

Step 4.2.3

The final answer is $- 9$ .

$- 9$

Step 4.3

Convert $- 9$ to decimal.

$y = - 9$

Step 5

Find the point at $x = 1$ .

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Step 5.1

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

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

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

Multiply ${((1) + 3)}^{2}$ by $1$ .

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

Step 5.2.1.2

Add $1$ and $3$ .

$f (1) = 4^{2} - {((1) + 3)}^{2}$

Step 5.2.1.3

Raise $4$ to the power of $2$ .

$f (1) = 16 - {((1) + 3)}^{2}$

Step 5.2.1.4

Add $1$ and $3$ .

$f (1) = 16 - 4^{2}$

Step 5.2.1.5

Raise $4$ to the power of $2$ .

$f (1) = 16 - 1 \cdot 16$

Step 5.2.1.6

Multiply $- 1$ by $16$ .

$f (1) = 16 - 16$

Step 5.2.2

Subtract $16$ from $16$ .

$f (1) = 0$

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.

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & - 3 \\ - 1 & - 8 \\ 0 & - 9 \\ 1 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 0 \\ - 2 & - 3 \\ - 1 & - 8 \\ 0 & - 9 \\ 1 & 0 \end{array}$

Step 8