Algebra Examples

$f (x) = - 4^{x - 4} + 5$

Step 1

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

$y = - 4^{(- 2) - 4} + 5$

Step 2

Simplify $- 4^{(- 2) - 4} + 5$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $4$ from $- 2$ .

$y = - 4^{- 6} + 5$

Step 2.1.2

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

$y = - \frac{1}{4^{6}} + 5$

Step 2.1.3

Raise $4$ to the power of $6$ .

$y = - \frac{1}{4096} + 5$

Step 2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{4096}{4096}$ .

$y = - \frac{1}{4096} + 5 \cdot \frac{4096}{4096}$

Step 2.3

Combine $5$ and $\frac{4096}{4096}$ .

$y = - \frac{1}{4096} + \frac{5 \cdot 4096}{4096}$

Step 2.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 4096}{4096}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $5$ by $4096$ .

$y = \frac{- 1 + 20480}{4096}$

Step 2.5.2

Add $- 1$ and $20480$ .

$y = \frac{20479}{4096}$

Step 3

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

$y = - 4^{(- 1) - 4} + 5$

Step 4

Simplify $- 4^{(- 1) - 4} + 5$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Subtract $4$ from $- 1$ .

$y = - 4^{- 5} + 5$

Step 4.1.2

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

$y = - \frac{1}{4^{5}} + 5$

Step 4.1.3

Raise $4$ to the power of $5$ .

$y = - \frac{1}{1024} + 5$

Step 4.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{1024}{1024}$ .

$y = - \frac{1}{1024} + 5 \cdot \frac{1024}{1024}$

Step 4.3

Combine $5$ and $\frac{1024}{1024}$ .

$y = - \frac{1}{1024} + \frac{5 \cdot 1024}{1024}$

Step 4.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 1024}{1024}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $5$ by $1024$ .

$y = \frac{- 1 + 5120}{1024}$

Step 4.5.2

Add $- 1$ and $5120$ .

$y = \frac{5119}{1024}$

Step 5

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

$y = - 4^{(0) - 4} + 5$

Step 6

Simplify $- 4^{(0) - 4} + 5$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Subtract $4$ from $0$ .

$y = - 4^{- 4} + 5$

Step 6.1.2

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

$y = - \frac{1}{4^{4}} + 5$

Step 6.1.3

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

$y = - \frac{1}{256} + 5$

Step 6.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$y = - \frac{1}{256} + 5 \cdot \frac{256}{256}$

Step 6.3

Combine $5$ and $\frac{256}{256}$ .

$y = - \frac{1}{256} + \frac{5 \cdot 256}{256}$

Step 6.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 256}{256}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $5$ by $256$ .

$y = \frac{- 1 + 1280}{256}$

Step 6.5.2

Add $- 1$ and $1280$ .

$y = \frac{1279}{256}$

Step 7

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

$y = - 4^{(1) - 4} + 5$

Step 8

Simplify $- 4^{(1) - 4} + 5$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Subtract $4$ from $1$ .

$y = - 4^{- 3} + 5$

Step 8.1.2

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

$y = - \frac{1}{4^{3}} + 5$

Step 8.1.3

Raise $4$ to the power of $3$ .

$y = - \frac{1}{64} + 5$

Step 8.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$y = - \frac{1}{64} + 5 \cdot \frac{64}{64}$

Step 8.3

Combine $5$ and $\frac{64}{64}$ .

$y = - \frac{1}{64} + \frac{5 \cdot 64}{64}$

Step 8.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 64}{64}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $5$ by $64$ .

$y = \frac{- 1 + 320}{64}$

Step 8.5.2

Add $- 1$ and $320$ .

$y = \frac{319}{64}$

Step 9

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

$y = - 4^{(2) - 4} + 5$

Step 10

Simplify $- 4^{(2) - 4} + 5$ .

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Step 10.1

Simplify each term.

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Step 10.1.1

Subtract $4$ from $2$ .

$y = - 4^{- 2} + 5$

Step 10.1.2

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

$y = - \frac{1}{4^{2}} + 5$

Step 10.1.3

Raise $4$ to the power of $2$ .

$y = - \frac{1}{16} + 5$

Step 10.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$y = - \frac{1}{16} + 5 \cdot \frac{16}{16}$

Step 10.3

Combine $5$ and $\frac{16}{16}$ .

$y = - \frac{1}{16} + \frac{5 \cdot 16}{16}$

Step 10.4

Combine the numerators over the common denominator.

$y = \frac{- 1 + 5 \cdot 16}{16}$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Multiply $5$ by $16$ .

$y = \frac{- 1 + 80}{16}$

Step 10.5.2

Add $- 1$ and $80$ .

$y = \frac{79}{16}$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & \frac{20479}{4096} \\ - 1 & \frac{5119}{1024} \\ 0 & \frac{1279}{256} \\ 1 & \frac{319}{64} \\ 2 & \frac{79}{16} \end{array}$

Step 12