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 the numerators over the common denominator.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $- 1$ by $4$ .

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

Step 1.2.3

Subtract $2$ from $- 4$ .

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

Step 1.2.4

Simplify each term.

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Step 1.2.4.1

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

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

Step 1.2.4.2

Multiply $- 1$ by $- 8$ .

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

Step 1.2.4.3

Divide $8$ by $8$ .

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

Step 1.2.4.4

Divide $- 6$ by $2$ .

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

Step 1.2.5

Simplify by adding and subtracting.

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Step 1.2.5.1

Add $2$ and $1$ .

$f (- 2) = 3 - 3$

Step 1.2.5.2

Subtract $3$ from $3$ .

$f (- 2) = 0$

Step 1.2.6

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 $0$ .

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

Step 2.2.3.2

Multiply $- 1$ by $0$ .

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Multiply $- 0 \cdot 4$ .

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Step 2.2.3.4.1

Multiply $- 1$ by $0$ .

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

Step 2.2.3.4.2

Multiply $0$ by $4$ .

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

Step 2.2.3.5

Multiply $0$ by $4$ .

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

Step 2.2.3.6

Multiply $2$ by $8$ .

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

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $0$ and $0$ .

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

Step 2.2.4.2

Add $0$ and $0$ .

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

Step 2.2.4.3

Add $0$ and $16$ .

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

Step 2.2.4.4

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 the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $- 1$ by $4$ .

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

Step 3.2.3

Add $- 4$ and $2$ .

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

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

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

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

Step 3.2.4.2

Multiply $- 1$ by $8$ .

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

Step 3.2.4.3

Divide $- 8$ by $8$ .

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

Step 3.2.4.4

Divide $- 2$ by $2$ .

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

Step 3.2.5

Simplify by subtracting numbers.

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Step 3.2.5.1

Subtract $1$ from $2$ .

$f (2) = 1 - 1$

Step 3.2.5.2

Subtract $1$ from $1$ .

$f (2) = 0$

Step 3.2.6

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 the numerators over the common denominator.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

$f (- 3) = 2 + \frac{- {(- 3)}^{3}}{8} + \frac{- 1 \cdot 9 - 3}{2}$

Step 4.2.2.2

Multiply $- 1$ by $9$ .

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

Step 4.2.3

Subtract $3$ from $- 9$ .

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

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

Step 4.2.4.2

Multiply $- 1$ by $- 27$ .

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

Step 4.2.4.3

Divide $- 12$ by $2$ .

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

Step 4.2.5

Find the common denominator.

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Step 4.2.5.1

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

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

Step 4.2.5.2

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

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

Step 4.2.5.3

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

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

Step 4.2.5.4

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

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

Step 4.2.5.5

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 4.2.5.6

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

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

Step 4.2.6

Combine the numerators over the common denominator.

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

Step 4.2.7

Simplify each term.

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Step 4.2.7.1

Multiply $2$ by $8$ .

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

Step 4.2.7.2

Multiply $- 6$ by $8$ .

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

Step 4.2.8

Simplify the expression.

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Step 4.2.8.1

Add $16$ and $27$ .

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

Step 4.2.8.2

Subtract $48$ from $43$ .

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

Step 4.2.8.3

Move the negative in front of the fraction.

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

Step 4.2.9

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 the numerators over the common denominator.

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

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 5.2.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 5.2.2.1.1.1

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

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

Step 5.2.2.1.1.2

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

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

Step 5.2.2.1.2

Add $1$ and $2$ .

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

Step 5.2.2.2

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

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

Step 5.2.3

Subtract $1$ from $- 1$ .

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

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

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Step 5.2.4.1.1.1

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

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

Step 5.2.4.1.1.2

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

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

Step 5.2.4.1.2

Add $1$ and $3$ .

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

Step 5.2.4.2

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

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

Step 5.2.4.3

Divide $- 2$ by $2$ .

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

Step 5.2.5

Find the common denominator.

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Step 5.2.5.1

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

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

Step 5.2.5.2

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

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

Step 5.2.5.3

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

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

Step 5.2.5.4

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

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

Step 5.2.5.5

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 5.2.5.6

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

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

Step 5.2.6

Combine the numerators over the common denominator.

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

Step 5.2.7

Simplify each term.

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Step 5.2.7.1

Multiply $2$ by $8$ .

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

Step 5.2.7.2

Multiply $- 1$ by $8$ .

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

Step 5.2.8

Simplify by adding and subtracting.

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Step 5.2.8.1

Add $16$ and $1$ .

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

Step 5.2.8.2

Subtract $8$ from $17$ .

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

Step 5.2.9

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.

Rises to the left and falls to the right

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

Step 8