Algebra Examples

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

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

Step 1.2

Simplify the result.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Subtract $1$ from $1$ .

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

Step 1.2.3

Find the common denominator.

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Step 1.2.3.1

Write ${(- 1)}^{3}$ as a fraction with denominator $1$ .

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

Step 1.2.3.2

Multiply $\frac{{(- 1)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 1.2.3.3

Multiply $\frac{{(- 1)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 1.2.3.4

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

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

Step 1.2.3.5

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

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

Step 1.2.3.6

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify each term.

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Step 1.2.5.1

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

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

Step 1.2.5.2

Multiply $- 1$ by $2$ .

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

Step 1.2.5.3

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

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Step 1.2.5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.5.3.2

Multiply $2$ by $1$ .

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

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $- 2$ and $2$ .

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

Step 1.2.6.2

Divide $0$ by $2$ .

$f (- 1) = 0$

Step 1.2.7

The final answer is $0$ .

$0$

Step 1.3

Convert $0$ to decimal.

$y = 0$

Step 2

Find the point at $x = 3$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Combine the numerators over the common denominator.

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

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Subtract $1$ from $9$ .

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

Step 2.2.3

Find the common denominator.

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Step 2.2.3.1

Write ${(3)}^{3}$ as a fraction with denominator $1$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

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

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

Step 2.2.3.4

Write $- (3)$ as a fraction with denominator $1$ .

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

Step 2.2.3.5

Multiply $\frac{- (3)}{1}$ by $\frac{2}{2}$ .

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

Step 2.2.3.6

Multiply $\frac{- (3)}{1}$ by $\frac{2}{2}$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify each term.

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $27$ by $2$ .

$f (3) = \frac{54 - 3 \cdot 2 + 8}{2}$

Step 2.2.5.3

Multiply $- (3) \cdot 2$ .

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Step 2.2.5.3.1

Multiply $- 1$ by $3$ .

$f (3) = \frac{54 - 3 \cdot 2 + 8}{2}$

Step 2.2.5.3.2

Multiply $- 3$ by $2$ .

$f (3) = \frac{54 - 6 + 8}{2}$

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Subtract $6$ from $54$ .

$f (3) = \frac{48 + 8}{2}$

Step 2.2.6.2

Add $48$ and $8$ .

$f (3) = \frac{56}{2}$

Step 2.2.6.3

Divide $56$ by $2$ .

$f (3) = 28$

Step 2.2.7

The final answer is $28$ .

$28$

Step 2.3

Convert $28$ to decimal.

$y = 28$

Step 3

Find the point at $x = - 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Subtract $1$ from $4$ .

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

Step 3.2.3

Find the common denominator.

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Step 3.2.3.1

Write ${(- 2)}^{3}$ as a fraction with denominator $1$ .

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

Step 3.2.3.2

Multiply $\frac{{(- 2)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 3.2.3.3

Multiply $\frac{{(- 2)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 3.2.3.4

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

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

Step 3.2.3.5

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

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

Step 3.2.3.6

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

Step 3.2.5

Simplify each term.

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Step 3.2.5.1

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

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

Step 3.2.5.2

Multiply $- 8$ by $2$ .

$f (- 2) = \frac{- 16 + 2 \cdot 2 + 3}{2}$

Step 3.2.5.3

Multiply $- (- 2) \cdot 2$ .

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Step 3.2.5.3.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = \frac{- 16 + 2 \cdot 2 + 3}{2}$

Step 3.2.5.3.2

Multiply $2$ by $2$ .

$f (- 2) = \frac{- 16 + 4 + 3}{2}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $- 16$ and $4$ .

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

Step 3.2.6.2

Add $- 12$ and $3$ .

$f (- 2) = \frac{- 9}{2}$

Step 3.2.6.3

Move the negative in front of the fraction.

$f (- 2) = - \frac{9}{2}$

Step 3.2.7

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

Step 3.3

Convert $- \frac{9}{2}$ to decimal.

$y = - 4.5$

Step 4

Find the point at $x = 0$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Combine the numerators over the common denominator.

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

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Subtract $1$ from $0$ .

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

Step 4.2.3

Find the common denominator.

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Step 4.2.3.1

Write ${(0)}^{3}$ as a fraction with denominator $1$ .

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

Step 4.2.3.2

Multiply $\frac{{(0)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 4.2.3.3

Multiply $\frac{{(0)}^{3}}{1}$ by $\frac{2}{2}$ .

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

Step 4.2.3.4

Write $- (0)$ as a fraction with denominator $1$ .

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

Step 4.2.3.5

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

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

Step 4.2.3.6

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify each term.

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Step 4.2.5.1

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

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

Step 4.2.5.2

Multiply $0$ by $2$ .

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

Step 4.2.5.3

Multiply $- (0) \cdot 2$ .

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Step 4.2.5.3.1

Multiply $- 1$ by $0$ .

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

Step 4.2.5.3.2

Multiply $0$ by $2$ .

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

Step 4.2.6

Simplify the expression.

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Step 4.2.6.1

Add $0$ and $0$ .

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

Step 4.2.6.2

Subtract $1$ from $0$ .

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

Step 4.2.6.3

Move the negative in front of the fraction.

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

Step 4.2.7

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 4.3

Convert $- \frac{1}{2}$ to decimal.

$y = - 0.5$

Step 5

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

$\begin{array}{c} x & y \\ - 2 & - 4.5 \\ - 1 & 0 \\ 0 & - 0.5 \\ 1 & 0 \\ 3 & 28 \end{array}$

Step 6

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

Falls to the left and rises to the right

$\begin{array}{c} x & y \\ - 2 & - 4.5 \\ - 1 & 0 \\ 0 & - 0.5 \\ 1 & 0 \\ 3 & 28 \end{array}$

Step 7