Algebra Examples

$g (x) = \frac{1}{4} x^{3} + \frac{1}{2} x^{2} + \frac{1}{4} x$

Step 1

Find the point at $x = - 1$ .

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Step 1.1

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

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

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.

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

Step 1.2.1.2

Simplify the expression.

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Step 1.2.1.2.1

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

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

Step 1.2.1.2.2

Subtract $1$ from $- 1$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Cancel the common factor of $- 2$ and $4$ .

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Step 1.2.2.1.1

Factor $2$ out of $- 2$ .

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

Step 1.2.2.1.2

Cancel the common factors.

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Step 1.2.2.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.2.1.2.2

Cancel the common factor.

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

Step 1.2.2.1.2.3

Rewrite the expression.

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

Step 1.2.2.2

Move the negative in front of the fraction.

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

Step 1.2.2.3

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

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

Step 1.2.3

Combine fractions.

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Step 1.2.3.1

Combine the numerators over the common denominator.

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

Step 1.2.3.2

Simplify the expression.

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Step 1.2.3.2.1

Add $- 1$ and $1$ .

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

Step 1.2.3.2.2

Divide $0$ by $2$ .

$f (- 1) = 0$

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 $x = 1$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Combine fractions.

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Step 2.2.1.1

Combine the numerators over the common denominator.

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

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

One to any power is one.

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

Step 2.2.1.2.2

Add $1$ and $1$ .

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Cancel the common factor of $2$ and $4$ .

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Step 2.2.2.1.1

Factor $2$ out of $2$ .

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

Step 2.2.2.1.2

Cancel the common factors.

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Step 2.2.2.1.2.1

Factor $2$ out of $4$ .

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

Step 2.2.2.1.2.2

Cancel the common factor.

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

Step 2.2.2.1.2.3

Rewrite the expression.

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

Step 2.2.2.2

One to any power is one.

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

Step 2.2.3

Combine fractions.

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Step 2.2.3.1

Combine the numerators over the common denominator.

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

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $1$ and $1$ .

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

Step 2.2.3.2.2

Divide $2$ by $2$ .

$f (1) = 1$

Step 2.2.4

The final answer is $1$ .

$1$

Step 2.3

Convert $1$ to decimal.

$y = 1$

Step 3

Find the point at $x = 3$ .

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Step 3.1

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

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

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.

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

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

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

$f (3) = \frac{27 + 3}{4} + \frac{{(3)}^{2}}{2}$

Step 3.2.1.2.2

Add $27$ and $3$ .

$f (3) = \frac{30}{4} + \frac{{(3)}^{2}}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Cancel the common factor of $30$ and $4$ .

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Step 3.2.2.1.1

Factor $2$ out of $30$ .

$f (3) = \frac{2 (15)}{4} + \frac{{(3)}^{2}}{2}$

Step 3.2.2.1.2

Cancel the common factors.

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Step 3.2.2.1.2.1

Factor $2$ out of $4$ .

$f (3) = \frac{2 \cdot 15}{2 \cdot 2} + \frac{{(3)}^{2}}{2}$

Step 3.2.2.1.2.2

Cancel the common factor.

$f (3) = \frac{2 \cdot 15}{2 \cdot 2} + \frac{{(3)}^{2}}{2}$

Step 3.2.2.1.2.3

Rewrite the expression.

$f (3) = \frac{15}{2} + \frac{{(3)}^{2}}{2}$

Step 3.2.2.2

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

$f (3) = \frac{15}{2} + \frac{9}{2}$

Step 3.2.3

Combine fractions.

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Step 3.2.3.1

Combine the numerators over the common denominator.

$f (3) = \frac{15 + 9}{2}$

Step 3.2.3.2

Simplify the expression.

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Step 3.2.3.2.1

Add $15$ and $9$ .

$f (3) = \frac{24}{2}$

Step 3.2.3.2.2

Divide $24$ by $2$ .

$f (3) = 12$

Step 3.2.4

The final answer is $12$ .

$12$

Step 3.3

Convert $12$ to decimal.

$y = 12$

Step 4

Find the point at $x = - 2$ .

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Step 4.1

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

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

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.

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

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Subtract $2$ from $- 8$ .

$f (- 2) = \frac{- 10}{4} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Cancel the common factor of $- 10$ and $4$ .

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Step 4.2.2.1.1

Factor $2$ out of $- 10$ .

$f (- 2) = \frac{2 (- 5)}{4} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2.1.2

Cancel the common factors.

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Step 4.2.2.1.2.1

Factor $2$ out of $4$ .

$f (- 2) = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2.1.2.2

Cancel the common factor.

$f (- 2) = \frac{2 \cdot - 5}{2 \cdot 2} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2.1.2.3

Rewrite the expression.

$f (- 2) = \frac{- 5}{2} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2.2

Move the negative in front of the fraction.

$f (- 2) = - \frac{5}{2} + \frac{{(- 2)}^{2}}{2}$

Step 4.2.2.3

Cancel the common factor of ${(- 2)}^{2}$ and $2$ .

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Step 4.2.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$f (- 2) = - \frac{5}{2} + \frac{{(- 1 \cdot 2)}^{2}}{2}$

Step 4.2.2.3.2

Apply the product rule to $- 1 (2)$ .

$f (- 2) = - \frac{5}{2} + \frac{{(- 1)}^{2} \cdot 2^{2}}{2}$

Step 4.2.2.3.3

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

$f (- 2) = - \frac{5}{2} + \frac{1 \cdot 2^{2}}{2}$

Step 4.2.2.3.4

Multiply $2^{2}$ by $1$ .

$f (- 2) = - \frac{5}{2} + \frac{2^{2}}{2}$

Step 4.2.2.3.5

Factor $2$ out of $2^{2}$ .

$f (- 2) = - \frac{5}{2} + \frac{2 \cdot 2}{2}$

Step 4.2.2.3.6

Cancel the common factors.

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Step 4.2.2.3.6.1

Factor $2$ out of $2$ .

$f (- 2) = - \frac{5}{2} + \frac{2 \cdot 2}{2 (1)}$

Step 4.2.2.3.6.2

Cancel the common factor.

$f (- 2) = - \frac{5}{2} + \frac{2 \cdot 2}{2 \cdot 1}$

Step 4.2.2.3.6.3

Rewrite the expression.

$f (- 2) = - \frac{5}{2} + \frac{2}{1}$

Step 4.2.2.3.6.4

Divide $2$ by $1$ .

$f (- 2) = - \frac{5}{2} + 2$

Step 4.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - \frac{5}{2} + 2 \cdot \frac{2}{2}$

Step 4.2.4

Combine $2$ and $\frac{2}{2}$ .

$f (- 2) = - \frac{5}{2} + \frac{2 \cdot 2}{2}$

Step 4.2.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 5 + 2 \cdot 2}{2}$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $2$ by $2$ .

$f (- 2) = \frac{- 5 + 4}{2}$

Step 4.2.6.2

Add $- 5$ and $4$ .

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

Step 4.2.7

Move the negative in front of the fraction.

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

Step 4.2.8

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.

$\begin{array}{c} x & y \\ - 2 & - 0.5 \\ - 1 & 0 \\ 0 & 0 \\ 1 & 1 \\ 3 & 12 \end{array}$

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

$\begin{array}{c} x & y \\ - 2 & - 0.5 \\ - 1 & 0 \\ 0 & 0 \\ 1 & 1 \\ 3 & 12 \end{array}$

Step 7