Precalculus Examples

$\begin{array}{c} x & y \\ - 1 & \frac{8}{3} \\ 0 & 2 \\ 1 & \frac{3}{2} \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 1, \frac{8}{3}), (1, \frac{3}{2})$

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.

$m = \frac{\frac{3}{2} - (\frac{8}{3})}{1 - (- 1)}$

Step 6

Simplify.

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Step 6.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 6.1.1

Multiply $\frac{\frac{3}{2} - \frac{8}{3}}{1 - (- 1)}$ by $\frac{6}{6}$ .

$m = \frac{6}{6} \cdot \frac{\frac{3}{2} - \frac{8}{3}}{1 - (- 1)}$

Step 6.1.2

Combine.

$m = \frac{6 (\frac{3}{2} - \frac{8}{3})}{6 (1 - (- 1))}$

Step 6.2

Apply the distributive property.

$m = \frac{6 (\frac{3}{2}) + 6 (- \frac{8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3

Simplify by cancelling.

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Step 6.3.1

Cancel the common factor of $2$ .

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Step 6.3.1.1

Factor $2$ out of $6$ .

$m = \frac{2 (3) (\frac{3}{2}) + 6 (- \frac{8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.1.2

Cancel the common factor.

$m = \frac{2 \cdot (3 (\frac{3}{2})) + 6 (- \frac{8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.1.3

Rewrite the expression.

$m = \frac{3 \cdot 3 + 6 (- \frac{8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.2

Multiply $3$ by $3$ .

$m = \frac{9 + 6 (- \frac{8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.3

Cancel the common factor of $3$ .

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Step 6.3.3.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$m = \frac{9 + 6 (\frac{- 8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.3.2

Factor $3$ out of $6$ .

$m = \frac{9 + 3 (2) (\frac{- 8}{3})}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.3.3

Cancel the common factor.

$m = \frac{9 + 3 \cdot (2 (\frac{- 8}{3}))}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.3.4

Rewrite the expression.

$m = \frac{9 + 2 \cdot - 8}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.3.4

Multiply $2$ by $- 8$ .

$m = \frac{9 - 16}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.4

Subtract $16$ from $9$ .

$m = \frac{- 7}{6 \cdot 1 + 6 (- (- 1))}$

Step 6.5

Simplify the denominator.

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Step 6.5.1

Multiply $6$ by $1$ .

$m = \frac{- 7}{6 + 6 (- (- 1))}$

Step 6.5.2

Multiply $6 (- (- 1))$ .

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Step 6.5.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 7}{6 + 6 \cdot 1}$

Step 6.5.2.2

Multiply $6$ by $1$ .

$m = \frac{- 7}{6 + 6}$

Step 6.5.3

Add $6$ and $6$ .

$m = \frac{- 7}{12}$

Step 6.6

Move the negative in front of the fraction.

$m = - \frac{7}{12}$

Step 7