Precalculus Examples

$\begin{array}{c} x & - 3 \\ - 1 & \frac{1}{2} \\ \frac{2}{3} & y \\ 4 & 12 \\ - 24 & - 18 \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 1, \frac{1}{2}), (- 24, - 18)$

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.

$m = \frac{- 18 - (\frac{1}{2})}{- 24 - (- 1)}$

Step 6

Simplify.

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Step 6.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 6.1.1

Multiply $\frac{- 18 - \frac{1}{2}}{- 24 - (- 1)}$ by $\frac{2}{2}$ .

$m = \frac{2}{2} \cdot \frac{- 18 - \frac{1}{2}}{- 24 - (- 1)}$

Step 6.1.2

Combine.

$m = \frac{2 (- 18 - \frac{1}{2})}{2 (- 24 - (- 1))}$

Step 6.2

Apply the distributive property.

$m = \frac{2 \cdot - 18 + 2 (- \frac{1}{2})}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.3

Cancel the common factor of $2$ .

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Step 6.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$m = \frac{2 \cdot - 18 + 2 (\frac{- 1}{2})}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.3.2

Cancel the common factor.

$m = \frac{2 \cdot - 18 + 2 (\frac{- 1}{2})}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.3.3

Rewrite the expression.

$m = \frac{2 \cdot - 18 - 1}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $2$ by $- 18$ .

$m = \frac{- 36 - 1}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.4.2

Subtract $1$ from $- 36$ .

$m = \frac{- 37}{2 \cdot - 24 + 2 (- (- 1))}$

Step 6.5

Simplify the denominator.

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Step 6.5.1

Multiply $2$ by $- 24$ .

$m = \frac{- 37}{- 48 + 2 (- (- 1))}$

Step 6.5.2

Multiply $2 (- (- 1))$ .

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Step 6.5.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 37}{- 48 + 2 \cdot 1}$

Step 6.5.2.2

Multiply $2$ by $1$ .

$m = \frac{- 37}{- 48 + 2}$

Step 6.5.3

Add $- 48$ and $2$ .

$m = \frac{- 37}{- 46}$

Step 6.6

Dividing two negative values results in a positive value.

$m = \frac{37}{46}$

Step 7