Algebra Examples

$f (x) = - 3^{x - 5} + 1$

Step 1

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

$y = - 3^{(- 2) - 5} + 1$

Step 2

Simplify $- 3^{(- 2) - 5} + 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract $5$ from $- 2$ .

$y = - 3^{- 7} + 1$

Step 2.1.2

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

$y = - \frac{1}{3^{7}} + 1$

Step 2.1.3

Raise $3$ to the power of $7$ .

$y = - \frac{1}{2187} + 1$

Step 2.2

Simplify the expression.

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Step 2.2.1

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

$y = - \frac{1}{2187} + \frac{2187}{2187}$

Step 2.2.2

Combine the numerators over the common denominator.

$y = \frac{- 1 + 2187}{2187}$

Step 2.2.3

Add $- 1$ and $2187$ .

$y = \frac{2186}{2187}$

Step 3

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

$y = - 3^{(- 1) - 5} + 1$

Step 4

Simplify $- 3^{(- 1) - 5} + 1$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Subtract $5$ from $- 1$ .

$y = - 3^{- 6} + 1$

Step 4.1.2

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

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

Step 4.1.3

Raise $3$ to the power of $6$ .

$y = - \frac{1}{729} + 1$

Step 4.2

Simplify the expression.

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Step 4.2.1

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

$y = - \frac{1}{729} + \frac{729}{729}$

Step 4.2.2

Combine the numerators over the common denominator.

$y = \frac{- 1 + 729}{729}$

Step 4.2.3

Add $- 1$ and $729$ .

$y = \frac{728}{729}$

Step 5

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

$y = - 3^{(0) - 5} + 1$

Step 6

Simplify $- 3^{(0) - 5} + 1$ .

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Step 6.1

Simplify each term.

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Step 6.1.1

Subtract $5$ from $0$ .

$y = - 3^{- 5} + 1$

Step 6.1.2

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

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

Step 6.1.3

Raise $3$ to the power of $5$ .

$y = - \frac{1}{243} + 1$

Step 6.2

Simplify the expression.

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Step 6.2.1

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

$y = - \frac{1}{243} + \frac{243}{243}$

Step 6.2.2

Combine the numerators over the common denominator.

$y = \frac{- 1 + 243}{243}$

Step 6.2.3

Add $- 1$ and $243$ .

$y = \frac{242}{243}$

Step 7

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

$y = - 3^{(1) - 5} + 1$

Step 8

Simplify $- 3^{(1) - 5} + 1$ .

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Step 8.1

Simplify each term.

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Step 8.1.1

Subtract $5$ from $1$ .

$y = - 3^{- 4} + 1$

Step 8.1.2

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

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

Step 8.1.3

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

$y = - \frac{1}{81} + 1$

Step 8.2

Simplify the expression.

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Step 8.2.1

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

$y = - \frac{1}{81} + \frac{81}{81}$

Step 8.2.2

Combine the numerators over the common denominator.

$y = \frac{- 1 + 81}{81}$

Step 8.2.3

Add $- 1$ and $81$ .

$y = \frac{80}{81}$

Step 9

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

$y = - 3^{(2) - 5} + 1$

Step 10

Simplify $- 3^{(2) - 5} + 1$ .

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Step 10.1

Simplify each term.

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Step 10.1.1

Subtract $5$ from $2$ .

$y = - 3^{- 3} + 1$

Step 10.1.2

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

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

Step 10.1.3

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

$y = - \frac{1}{27} + 1$

Step 10.2

Simplify the expression.

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Step 10.2.1

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

$y = - \frac{1}{27} + \frac{27}{27}$

Step 10.2.2

Combine the numerators over the common denominator.

$y = \frac{- 1 + 27}{27}$

Step 10.2.3

Add $- 1$ and $27$ .

$y = \frac{26}{27}$

Step 11

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

$\begin{array}{c} x & y \\ - 2 & \frac{2186}{2187} \\ - 1 & \frac{728}{729} \\ 0 & \frac{242}{243} \\ 1 & \frac{80}{81} \\ 2 & \frac{26}{27} \end{array}$

Step 12