Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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Step 1.2.2.1.1

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

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Step 1.2.2.1.1.1

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

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

Step 1.2.2.1.1.2

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

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

Step 1.2.2.1.2

Add $1$ and $3$ .

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

Step 1.2.2.2

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

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

Step 1.2.3

Subtract $1$ from $1$ .

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

Step 1.2.4

Find the common denominator.

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

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

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

Step 1.2.4.4

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

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

Step 1.2.4.5

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 1.2.4.6

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify each term.

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Step 1.2.6.1

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

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

Step 1.2.6.2

Multiply $2$ by $1$ .

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

Step 1.2.6.3

Multiply $- 1$ by $2$ .

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

Step 1.2.7

Simplify the expression.

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Step 1.2.7.1

Subtract $2$ from $2$ .

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

Step 1.2.7.2

Divide $0$ by $2$ .

$f (- 1) = 0$

Step 1.2.8

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 $\frac{2}{2}$ .

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Multiply $- 1$ by $0$ .

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Multiply $0$ by $2$ .

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

Step 2.2.3.5

Multiply $- 1$ 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

Subtract $2$ from $0$ .

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

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

Simplify each term.

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Step 3.2.4.1

One to any power is one.

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

Step 3.2.4.2

Divide $0$ by $2$ .

$f (1) = 1 - 1 + 0$

Step 3.2.5

Simplify by adding and subtracting.

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Step 3.2.5.1

Subtract $1$ from $1$ .

$f (1) = 0 + 0$

Step 3.2.5.2

Add $0$ and $0$ .

$f (1) = 0$

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.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Multiply $- 1$ by $27$ .

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

Step 4.2.3

Add $- 27$ and $3$ .

$f (3) = {(3)}^{2} - 1 + \frac{- 24}{2}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

$f (3) = 9 - 1 + \frac{- 24}{2}$

Step 4.2.4.2

Divide $- 24$ by $2$ .

$f (3) = 9 - 1 - 12$

Step 4.2.5

Simplify by subtracting numbers.

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Step 4.2.5.1

Subtract $1$ from $9$ .

$f (3) = 8 - 12$

Step 4.2.5.2

Subtract $12$ from $8$ .

$f (3) = - 4$

Step 4.2.6

The final answer is $- 4$ .

$- 4$

Step 4.3

Convert $- 4$ to decimal.

$y = - 4$

Step 5

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

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

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

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

Step 7