Algebra Examples

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

Step 1

Find the point at $x = - 2$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $- 1$ by $- 8$ .

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

Step 1.2.3

Subtract $2$ from $8$ .

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

Step 1.2.4

Simplify each term.

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Step 1.2.4.1

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

$f (- 2) = - 1 \cdot 4 + 1 + \frac{6}{2}$

Step 1.2.4.2

Multiply $- 1$ by $4$ .

$f (- 2) = - 4 + 1 + \frac{6}{2}$

Step 1.2.4.3

Divide $6$ by $2$ .

$f (- 2) = - 4 + 1 + 3$

Step 1.2.5

Simplify by adding numbers.

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Step 1.2.5.1

Add $- 4$ and $1$ .

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

Step 1.2.5.2

Add $- 3$ and $3$ .

$f (- 2) = 0$

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.

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

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

Write $- {(0)}^{2}$ as a fraction with denominator $1$ .

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

Step 2.2.1.2

Multiply $\frac{- {(0)}^{2}}{1}$ by $\frac{2}{2}$ .

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

Step 2.2.1.3

Multiply $\frac{- {(0)}^{2}}{1}$ by $\frac{2}{2}$ .

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

Step 2.2.1.4

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

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

Step 2.2.1.5

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

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

Step 2.2.1.6

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

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Multiply $- 1$ by $0$ .

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Multiply $- 0 \cdot 2$ .

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Step 2.2.3.4.1

Multiply $- 1$ by $0$ .

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

Step 2.2.3.4.2

Multiply $0$ by $2$ .

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

Step 2.2.4

Simplify the expression.

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Step 2.2.4.1

Add $0$ and $0$ .

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

Step 2.2.4.2

Add $0$ and $2$ .

$f (0) = \frac{2}{2}$

Step 2.2.4.3

Divide $2$ by $2$ .

$f (0) = 1$

Step 2.2.5

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 1$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

One to any power is one.

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

Step 3.2.2.2

Multiply $- 1$ by $1$ .

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

Step 3.2.3

Add $- 1$ and $1$ .

$f (1) = - {(1)}^{2} + 1 + \frac{0}{2}$

Step 3.2.4

Find the common denominator.

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Step 3.2.4.1

Write $- {(1)}^{2}$ as a fraction with denominator $1$ .

$f (1) = \frac{- {(1)}^{2}}{1} + 1 + \frac{0}{2}$

Step 3.2.4.2

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

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

Step 3.2.4.3

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

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

Step 3.2.4.4

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

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

Step 3.2.4.5

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

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

Step 3.2.4.6

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

Simplify each term.

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Step 3.2.6.1

One to any power is one.

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

Step 3.2.6.2

Multiply $- 1 \cdot 1 \cdot 2$ .

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Step 3.2.6.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.6.2.2

Multiply $- 1$ by $2$ .

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

Step 3.2.7

Simplify the expression.

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Step 3.2.7.1

Add $- 2$ and $2$ .

$f (1) = \frac{0}{2}$

Step 3.2.7.2

Divide $0$ by $2$ .

$f (1) = 0$

Step 3.2.8

The final answer is $0$ .

$0$

Step 3.3

Convert $0$ to decimal.

$y = 0$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $- 1$ by $8$ .

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

Step 4.2.3

Add $- 8$ and $2$ .

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

$f (2) = - 1 \cdot 4 + 1 + \frac{- 6}{2}$

Step 4.2.4.2

Multiply $- 1$ by $4$ .

$f (2) = - 4 + 1 + \frac{- 6}{2}$

Step 4.2.4.3

Divide $- 6$ by $2$ .

$f (2) = - 4 + 1 - 3$

Step 4.2.5

Simplify by adding and subtracting.

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Step 4.2.5.1

Add $- 4$ and $1$ .

$f (2) = - 3 - 3$

Step 4.2.5.2

Subtract $3$ from $- 3$ .

$f (2) = - 6$

Step 4.2.6

The final answer is $- 6$ .

$- 6$

Step 4.3

Convert $- 6$ to decimal.

$y = - 6$

Step 5

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

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 2 & - 6 \end{array}$

Step 6

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

Rises to the left and falls to the right

$\begin{array}{c} x & y \\ - 2 & 0 \\ - 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 2 & - 6 \end{array}$

Step 7