Precalculus Examples

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

Step 1

Choose the starting and ending values from the table.

$(\frac{1}{4}, - 1), (0, 2)$

Step 2

To find the average rate of change, find the slope.

Step 3

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 4

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 5

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{2 - (- 1)}{0 - (\frac{1}{4})}$

Step 6

Simplify.

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Step 6.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 6.1.1

Multiply $\frac{2 - (- 1)}{0 - \frac{1}{4}}$ by $\frac{4}{4}$ .

$m = \frac{4}{4} \cdot \frac{2 - (- 1)}{0 - \frac{1}{4}}$

Step 6.1.2

Combine.

$m = \frac{4 (2 - (- 1))}{4 (0 - \frac{1}{4})}$

Step 6.2

Apply the distributive property.

$m = \frac{4 \cdot 2 + 4 (- (- 1))}{4 \cdot 0 + 4 (- \frac{1}{4})}$

Step 6.3

Cancel the common factor of $4$ .

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Step 6.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$m = \frac{4 \cdot 2 + 4 (- (- 1))}{4 \cdot 0 + 4 (\frac{- 1}{4})}$

Step 6.3.2

Cancel the common factor.

$m = \frac{4 \cdot 2 + 4 (- (- 1))}{4 \cdot 0 + 4 (\frac{- 1}{4})}$

Step 6.3.3

Rewrite the expression.

$m = \frac{4 \cdot 2 + 4 (- (- 1))}{4 \cdot 0 - 1}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $4$ by $2$ .

$m = \frac{8 + 4 (- (- 1))}{4 \cdot 0 - 1}$

Step 6.4.2

Multiply $4 (- (- 1))$ .

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Step 6.4.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{8 + 4 \cdot 1}{4 \cdot 0 - 1}$

Step 6.4.2.2

Multiply $4$ by $1$ .

$m = \frac{8 + 4}{4 \cdot 0 - 1}$

Step 6.4.3

Add $8$ and $4$ .

$m = \frac{12}{4 \cdot 0 - 1}$

Step 6.5

Simplify the denominator.

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Step 6.5.1

Multiply $4$ by $0$ .

$m = \frac{12}{0 - 1}$

Step 6.5.2

Subtract $1$ from $0$ .

$m = \frac{12}{- 1}$

Step 6.6

Divide $12$ by $- 1$ .

$m = - 12$

Step 7