Algebra Examples

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

Step 1

Substitute $- 2$ for $x$ and find the result for $y$ .

$y = \frac{1}{2} \cdot 2^{- 2} - 6$

Step 2

Simplify $\frac{1}{2} \cdot 2^{- 2} - 6$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{2} \cdot \frac{1}{2^{2}} - 6$

Step 2.1.2

Combine.

$y = \frac{1 \cdot 1}{2 \cdot 2^{2}} - 6$

Step 2.1.3

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 2.1.3.1

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

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Step 2.1.3.1.1

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

$y = \frac{1 \cdot 1}{2^{1} \cdot 2^{2}} - 6$

Step 2.1.3.1.2

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

$y = \frac{1 \cdot 1}{2^{1 + 2}} - 6$

Step 2.1.3.2

Add $1$ and $2$ .

$y = \frac{1 \cdot 1}{2^{3}} - 6$

Step 2.1.4

Multiply $1$ by $1$ .

$y = \frac{1}{2^{3}} - 6$

Step 2.1.5

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

$y = \frac{1}{8} - 6$

Step 2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{1}{8} - 6 \cdot \frac{8}{8}$

Step 2.3

Combine $- 6$ and $\frac{8}{8}$ .

$y = \frac{1}{8} + \frac{- 6 \cdot 8}{8}$

Step 2.4

Combine the numerators over the common denominator.

$y = \frac{1 - 6 \cdot 8}{8}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $- 6$ by $8$ .

$y = \frac{1 - 48}{8}$

Step 2.5.2

Subtract $48$ from $1$ .

$y = \frac{- 47}{8}$

Step 2.6

Move the negative in front of the fraction.

$y = - \frac{47}{8}$

Step 3

Substitute $- 1$ for $x$ and find the result for $y$ .

$y = \frac{1}{2} \cdot 2^{- 1} - 6$

Step 4

Simplify $\frac{1}{2} \cdot 2^{- 1} - 6$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = \frac{1}{2} \cdot \frac{1}{2} - 6$

Step 4.1.2

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

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Step 4.1.2.1

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

$y = \frac{1}{2 \cdot 2} - 6$

Step 4.1.2.2

Multiply $2$ by $2$ .

$y = \frac{1}{4} - 6$

Step 4.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{1}{4} - 6 \cdot \frac{4}{4}$

Step 4.3

Combine $- 6$ and $\frac{4}{4}$ .

$y = \frac{1}{4} + \frac{- 6 \cdot 4}{4}$

Step 4.4

Combine the numerators over the common denominator.

$y = \frac{1 - 6 \cdot 4}{4}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 6$ by $4$ .

$y = \frac{1 - 24}{4}$

Step 4.5.2

Subtract $24$ from $1$ .

$y = \frac{- 23}{4}$

Step 4.6

Move the negative in front of the fraction.

$y = - \frac{23}{4}$

Step 5

Substitute $0$ for $x$ and find the result for $y$ .

$y = \frac{1}{2} \cdot 2^{0} - 6$

Step 6

Simplify $\frac{1}{2} \cdot 2^{0} - 6$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Anything raised to $0$ is $1$ .

$y = \frac{1}{2} \cdot 1 - 6$

Step 6.1.2

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

$y = \frac{1}{2} - 6$

Step 6.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{1}{2} - 6 \cdot \frac{2}{2}$

Step 6.3

Combine $- 6$ and $\frac{2}{2}$ .

$y = \frac{1}{2} + \frac{- 6 \cdot 2}{2}$

Step 6.4

Combine the numerators over the common denominator.

$y = \frac{1 - 6 \cdot 2}{2}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $- 6$ by $2$ .

$y = \frac{1 - 12}{2}$

Step 6.5.2

Subtract $12$ from $1$ .

$y = \frac{- 11}{2}$

Step 6.6

Move the negative in front of the fraction.

$y = - \frac{11}{2}$

Step 7

Substitute $1$ for $x$ and find the result for $y$ .

$y = \frac{1}{2} \cdot 2^{1} - 6$

Step 8

Simplify $\frac{1}{2} \cdot 2^{1} - 6$ .

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $2^{1}$ .

$y = \frac{1}{2} \cdot (2 \cdot 1) - 6$

Step 8.1.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \cdot 1) - 6$

Step 8.1.3

Rewrite the expression.

$y = 1 - 6$

Step 8.2

Subtract $6$ from $1$ .

$y = - 5$

Step 9

Substitute $2$ for $x$ and find the result for $y$ .

$y = \frac{1}{2} \cdot 2^{2} - 6$

Step 10

Simplify $\frac{1}{2} \cdot 2^{2} - 6$ .

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Step 10.1

Cancel the common factor of $2$ .

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Step 10.1.1

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

$y = \frac{1}{2} \cdot (2 \cdot 2) - 6$

Step 10.1.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \cdot 2) - 6$

Step 10.1.3

Rewrite the expression.

$y = 2 - 6$

Step 10.2

Subtract $6$ from $2$ .

$y = - 4$

Step 11

This is a table of possible values to use when graphing the equation.

$\begin{array}{c} x & y \\ - 2 & - \frac{47}{8} \\ - 1 & - \frac{23}{4} \\ 0 & - \frac{11}{2} \\ 1 & - 5 \\ 2 & - 4 \end{array}$

Step 12