Algebra Examples

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

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

Step 1

Substitute

- 2

$- 2$ for

x

$x$ and find the result for

y

$y$ .

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

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

Step 2

Simplify

- 3^{(- 2) - 5} + 1

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Subtract

5

$5$ from

- 2

$- 2$ .

y = - 3^{- 7} + 1

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

Step 2.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 2.1.3

Raise

3

$3$ to the power of

7

$7$ .

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

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

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

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 2.2.2

Combine the numerators over the common denominator.

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

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

Step 2.2.3

Add

- 1

$- 1$ and

2187

$2187$ .

y = \frac{2186}{2187}

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

y = \frac{2186}{2187}

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

y = \frac{2186}{2187}

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

Step 3

Substitute

- 1

$- 1$ for

x

$x$ and find the result for

y

$y$ .

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

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

Step 4

Simplify

- 3^{(- 1) - 5} + 1

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

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Step 4.1

Simplify each term.

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Step 4.1.1

Subtract

5

$5$ from

- 1

$- 1$ .

y = - 3^{- 6} + 1

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

Step 4.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 4.1.3

Raise

3

$3$ to the power of

6

$6$ .

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

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

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

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

Step 4.2

Simplify the expression.

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Step 4.2.1

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 4.2.2

Combine the numerators over the common denominator.

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

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

Step 4.2.3

Add

- 1

$- 1$ and

729

$729$ .

y = \frac{728}{729}

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

y = \frac{728}{729}

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

y = \frac{728}{729}

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

Step 5

Substitute

0

$0$ for

x

$x$ and find the result for

y

$y$ .

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

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

Step 6

Simplify

- 3^{(0) - 5} + 1

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

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Step 6.1

Simplify each term.

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Step 6.1.1

Subtract

5

$5$ from

0

$0$ .

y = - 3^{- 5} + 1

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

Step 6.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 6.1.3

Raise

3

$3$ to the power of

5

$5$ .

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

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

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

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 6.2.2

Combine the numerators over the common denominator.

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

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

Step 6.2.3

Add

- 1

$- 1$ and

243

$243$ .

y = \frac{242}{243}

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

y = \frac{242}{243}

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

y = \frac{242}{243}

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

Step 7

Substitute

1

$1$ for

x

$x$ and find the result for

y

$y$ .

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

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

Step 8

Simplify

- 3^{(1) - 5} + 1

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

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Step 8.1

Simplify each term.

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Step 8.1.1

Subtract

5

$5$ from

1

$1$ .

y = - 3^{- 4} + 1

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

Step 8.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 8.1.3

Raise

3

$3$ to the power of

4

$4$ .

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

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

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

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

Step 8.2

Simplify the expression.

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Step 8.2.1

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 8.2.2

Combine the numerators over the common denominator.

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

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

Step 8.2.3

Add

- 1

$- 1$ and

81

$81$ .

y = \frac{80}{81}

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

y = \frac{80}{81}

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

y = \frac{80}{81}

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

Step 9

Substitute

2

$2$ for

x

$x$ and find the result for

y

$y$ .

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

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

Step 10

Simplify

- 3^{(2) - 5} + 1

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

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Step 10.1

Simplify each term.

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Step 10.1.1

Subtract

5

$5$ from

2

$2$ .

y = - 3^{- 3} + 1

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

Step 10.1.2

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 10.1.3

Raise

3

$3$ to the power of

3

$3$ .

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

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

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

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

Step 10.2

Simplify the expression.

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Step 10.2.1

Write

1

$1$ as a fraction with a common denominator.

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

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

Step 10.2.2

Combine the numerators over the common denominator.

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

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

Step 10.2.3

Add

- 1

$- 1$ and

27

$27$ .

y = \frac{26}{27}

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

y = \frac{26}{27}

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

y = \frac{26}{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}

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

Step 12