Algebra Examples

$f (x) = \frac{1}{8} (x + 4) (x + 2) (x - 2)$

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

$f (- 2) = \frac{{(- 2)}^{3}}{8} + \frac{{(- 2)}^{2}}{2} - \frac{- 2}{2} - 2$

Step 1.2

Simplify the result.

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Step 1.2.1

Combine fractions.

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Step 1.2.1.1

Combine the numerators over the common denominator.

$f (- 2) = - 2 + \frac{{(- 2)}^{3}}{8} + \frac{{(- 2)}^{2} + 2}{2}$

Step 1.2.1.2

Simplify the expression.

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Step 1.2.1.2.1

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

$f (- 2) = - 2 + \frac{{(- 2)}^{3}}{8} + \frac{4 + 2}{2}$

Step 1.2.1.2.2

Add $4$ and $2$ .

$f (- 2) = - 2 + \frac{{(- 2)}^{3}}{8} + \frac{6}{2}$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

$f (- 2) = - 2 + \frac{- 8}{8} + \frac{6}{2}$

Step 1.2.2.2

Divide $- 8$ by $8$ .

$f (- 2) = - 2 - 1 + \frac{6}{2}$

Step 1.2.2.3

Divide $6$ by $2$ .

$f (- 2) = - 2 - 1 + 3$

Step 1.2.3

Simplify by adding and subtracting.

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Step 1.2.3.1

Subtract $1$ from $- 2$ .

$f (- 2) = - 3 + 3$

Step 1.2.3.2

Add $- 3$ and $3$ .

$f (- 2) = 0$

Step 1.2.4

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 0$ .

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Step 2.1

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

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2}}{2} - \frac{0}{2} - 2$

Step 2.2

Simplify the result.

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Step 2.2.1

Find the common denominator.

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Step 2.2.1.1

Multiply $\frac{{(0)}^{2}}{2}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2}}{2} \cdot \frac{4}{4} - \frac{0}{2} - 2$

Step 2.2.1.2

Multiply $\frac{{(0)}^{2}}{2}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - \frac{0}{2} - 2$

Step 2.2.1.3

Multiply $\frac{0}{2}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - (\frac{0}{2} \cdot \frac{4}{4}) - 2$

Step 2.2.1.4

Multiply $\frac{0}{2}$ by $\frac{4}{4}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - \frac{0 \cdot 4}{2 \cdot 4} - 2$

Step 2.2.1.5

Write $- 2$ as a fraction with denominator $1$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - \frac{0 \cdot 4}{2 \cdot 4} + \frac{- 2}{1}$

Step 2.2.1.6

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - \frac{0 \cdot 4}{2 \cdot 4} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 2.2.1.7

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{2 \cdot 4} - \frac{0 \cdot 4}{2 \cdot 4} + \frac{- 2 \cdot 8}{8}$

Step 2.2.1.8

Multiply $2$ by $4$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{8} - \frac{0 \cdot 4}{2 \cdot 4} + \frac{- 2 \cdot 8}{8}$

Step 2.2.1.9

Multiply $2$ by $4$ .

$f (0) = \frac{{(0)}^{3}}{8} + \frac{{(0)}^{2} \cdot 4}{8} - \frac{0 \cdot 4}{8} + \frac{- 2 \cdot 8}{8}$

Step 2.2.2

Combine the numerators over the common denominator.

$f (0) = \frac{{(0)}^{3} + {(0)}^{2} \cdot 4 + 0 \cdot 4 - 2 \cdot 8}{8}$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

$f (0) = \frac{{(0)}^{3} + 0 \cdot 4 + 0 \cdot 4 - 2 \cdot 8}{8}$

Step 2.2.3.2

Multiply $0$ by $4$ .

$f (0) = \frac{{(0)}^{3} + 0 + 0 \cdot 4 - 2 \cdot 8}{8}$

Step 2.2.3.3

Multiply $0$ by $4$ .

$f (0) = \frac{{(0)}^{3} + 0 + 0 - 2 \cdot 8}{8}$

Step 2.2.3.4

Multiply $- 2$ by $8$ .

$f (0) = \frac{{(0)}^{3} + 0 + 0 - 16}{8}$

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add ${(0)}^{3}$ and $0$ .

$f (0) = \frac{{(0)}^{3} + 0 - 16}{8}$

Step 2.2.4.2

Add ${(0)}^{3}$ and $0$ .

$f (0) = \frac{{(0)}^{3} - 16}{8}$

Step 2.2.4.3

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

$f (0) = \frac{0 - 16}{8}$

Step 2.2.4.4

Subtract $16$ from $0$ .

$f (0) = \frac{- 16}{8}$

Step 2.2.4.5

Divide $- 16$ by $8$ .

$f (0) = - 2$

Step 2.2.5

The final answer is $- 2$ .

$- 2$

Step 2.3

Convert $- 2$ to decimal.

$y = - 2$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \frac{{(2)}^{3}}{8} + \frac{{(2)}^{2}}{2} - \frac{2}{2} - 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine fractions.

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Step 3.2.1.1

Combine the numerators over the common denominator.

$f (2) = - 2 + \frac{{(2)}^{3}}{8} + \frac{{(2)}^{2} - 2}{2}$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

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

$f (2) = - 2 + \frac{{(2)}^{3}}{8} + \frac{4 - 2}{2}$

Step 3.2.1.2.2

Subtract $2$ from $4$ .

$f (2) = - 2 + \frac{{(2)}^{3}}{8} + \frac{2}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (2) = - 2 + \frac{8}{8} + \frac{2}{2}$

Step 3.2.2.2

Divide $8$ by $8$ .

$f (2) = - 2 + 1 + \frac{2}{2}$

Step 3.2.2.3

Divide $2$ by $2$ .

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

Step 3.2.3

Simplify by adding numbers.

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Step 3.2.3.1

Add $- 2$ and $1$ .

$f (2) = - 1 + 1$

Step 3.2.3.2

Add $- 1$ and $1$ .

$f (2) = 0$

Step 3.2.4

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = - 3$ .

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Step 4.1

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

$f (- 3) = \frac{{(- 3)}^{3}}{8} + \frac{{(- 3)}^{2}}{2} - \frac{- 3}{2} - 2$

Step 4.2

Simplify the result.

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Step 4.2.1

Combine fractions.

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Step 4.2.1.1

Combine the numerators over the common denominator.

$f (- 3) = - 2 + \frac{{(- 3)}^{3}}{8} + \frac{{(- 3)}^{2} + 3}{2}$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

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

$f (- 3) = - 2 + \frac{{(- 3)}^{3}}{8} + \frac{9 + 3}{2}$

Step 4.2.1.2.2

Add $9$ and $3$ .

$f (- 3) = - 2 + \frac{{(- 3)}^{3}}{8} + \frac{12}{2}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

$f (- 3) = - 2 + \frac{- 27}{8} + \frac{12}{2}$

Step 4.2.2.2

Move the negative in front of the fraction.

$f (- 3) = - 2 - \frac{27}{8} + \frac{12}{2}$

Step 4.2.2.3

Divide $12$ by $2$ .

$f (- 3) = - 2 - \frac{27}{8} + 6$

Step 4.2.3

Find the common denominator.

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Step 4.2.3.1

Write $- 2$ as a fraction with denominator $1$ .

$f (- 3) = \frac{- 2}{1} - \frac{27}{8} + 6$

Step 4.2.3.2

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (- 3) = \frac{- 2}{1} \cdot \frac{8}{8} - \frac{27}{8} + 6$

Step 4.2.3.3

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (- 3) = \frac{- 2 \cdot 8}{8} - \frac{27}{8} + 6$

Step 4.2.3.4

Write $6$ as a fraction with denominator $1$ .

$f (- 3) = \frac{- 2 \cdot 8}{8} - \frac{27}{8} + \frac{6}{1}$

Step 4.2.3.5

Multiply $\frac{6}{1}$ by $\frac{8}{8}$ .

$f (- 3) = \frac{- 2 \cdot 8}{8} - \frac{27}{8} + \frac{6}{1} \cdot \frac{8}{8}$

Step 4.2.3.6

Multiply $\frac{6}{1}$ by $\frac{8}{8}$ .

$f (- 3) = \frac{- 2 \cdot 8}{8} - \frac{27}{8} + \frac{6 \cdot 8}{8}$

Step 4.2.4

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 2 \cdot 8 - 27 + 6 \cdot 8}{8}$

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

Multiply $- 2$ by $8$ .

$f (- 3) = \frac{- 16 - 27 + 6 \cdot 8}{8}$

Step 4.2.5.2

Multiply $6$ by $8$ .

$f (- 3) = \frac{- 16 - 27 + 48}{8}$

Step 4.2.6

Simplify by adding and subtracting.

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Step 4.2.6.1

Subtract $27$ from $- 16$ .

$f (- 3) = \frac{- 43 + 48}{8}$

Step 4.2.6.2

Add $- 43$ and $48$ .

$f (- 3) = \frac{5}{8}$

Step 4.2.7

The final answer is $\frac{5}{8}$ .

$\frac{5}{8}$

Step 4.3

Convert $\frac{5}{8}$ to decimal.

$y = 0.625$

Step 5

Find the point at $x = - 1$ .

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Step 5.1

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

$f (- 1) = \frac{{(- 1)}^{3}}{8} + \frac{{(- 1)}^{2}}{2} - \frac{- 1}{2} - 2$

Step 5.2

Simplify the result.

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Step 5.2.1

Combine fractions.

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Step 5.2.1.1

Combine the numerators over the common denominator.

$f (- 1) = - 2 + \frac{{(- 1)}^{3}}{8} + \frac{{(- 1)}^{2} + 1}{2}$

Step 5.2.1.2

Simplify the expression.

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Step 5.2.1.2.1

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 2 + \frac{{(- 1)}^{3}}{8} + \frac{1 + 1}{2}$

Step 5.2.1.2.2

Add $1$ and $1$ .

$f (- 1) = - 2 + \frac{{(- 1)}^{3}}{8} + \frac{2}{2}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

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

$f (- 1) = - 2 + \frac{- 1}{8} + \frac{2}{2}$

Step 5.2.2.2

Move the negative in front of the fraction.

$f (- 1) = - 2 - \frac{1}{8} + \frac{2}{2}$

Step 5.2.2.3

Divide $2$ by $2$ .

$f (- 1) = - 2 - \frac{1}{8} + 1$

Step 5.2.3

Find the common denominator.

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Step 5.2.3.1

Write $- 2$ as a fraction with denominator $1$ .

$f (- 1) = \frac{- 2}{1} - \frac{1}{8} + 1$

Step 5.2.3.2

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (- 1) = \frac{- 2}{1} \cdot \frac{8}{8} - \frac{1}{8} + 1$

Step 5.2.3.3

Multiply $\frac{- 2}{1}$ by $\frac{8}{8}$ .

$f (- 1) = \frac{- 2 \cdot 8}{8} - \frac{1}{8} + 1$

Step 5.2.3.4

Write $1$ as a fraction with denominator $1$ .

$f (- 1) = \frac{- 2 \cdot 8}{8} - \frac{1}{8} + \frac{1}{1}$

Step 5.2.3.5

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (- 1) = \frac{- 2 \cdot 8}{8} - \frac{1}{8} + \frac{1}{1} \cdot \frac{8}{8}$

Step 5.2.3.6

Multiply $\frac{1}{1}$ by $\frac{8}{8}$ .

$f (- 1) = \frac{- 2 \cdot 8}{8} - \frac{1}{8} + \frac{8}{8}$

Step 5.2.4

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 2 \cdot 8 - 1 + 8}{8}$

Step 5.2.5

Simplify the expression.

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Step 5.2.5.1

Multiply $- 2$ by $8$ .

$f (- 1) = \frac{- 16 - 1 + 8}{8}$

Step 5.2.5.2

Subtract $1$ from $- 16$ .

$f (- 1) = \frac{- 17 + 8}{8}$

Step 5.2.5.3

Add $- 17$ and $8$ .

$f (- 1) = \frac{- 9}{8}$

Step 5.2.5.4

Move the negative in front of the fraction.

$f (- 1) = - \frac{9}{8}$

Step 5.2.6

The final answer is $- \frac{9}{8}$ .

$- \frac{9}{8}$

Step 5.3

Convert $- \frac{9}{8}$ to decimal.

$y = - 1.125$

Step 6

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

$\begin{array}{c} x & y \\ - 3 & 0.625 \\ - 2 & 0 \\ - 1 & - 1.125 \\ 0 & - 2 \\ 2 & 0 \end{array}$

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

$\begin{array}{c} x & y \\ - 3 & 0.625 \\ - 2 & 0 \\ - 1 & - 1.125 \\ 0 & - 2 \\ 2 & 0 \end{array}$

Step 8