Precalculus Examples

$\begin{array}{c} x & 0 \\ 1 & 2 \\ 3 & 4 \\ y & 40 \\ 10 & \frac{5}{2} \\ \frac{5}{8} & \frac{5}{32} \end{array}$

Step 1

Choose the starting and ending values from the table.

$(1, 2), (\frac{5}{8}, \frac{5}{32})$

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.

$m = \frac{\frac{5}{32} - (2)}{\frac{5}{8} - (1)}$

Step 6

Simplify.

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Step 6.1

Multiply the numerator and denominator of the fraction by $32$ .

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Step 6.1.1

Multiply $\frac{\frac{5}{32} - (2)}{\frac{5}{8} - (1)}$ by $\frac{32}{32}$ .

$m = \frac{32}{32} \cdot \frac{\frac{5}{32} - (2)}{\frac{5}{8} - (1)}$

Step 6.1.2

Combine.

$m = \frac{32 (\frac{5}{32} - (2))}{32 (\frac{5}{8} - (1))}$

Step 6.2

Apply the distributive property.

$m = \frac{32 (\frac{5}{32}) + 32 (- (2))}{32 (\frac{5}{8}) + 32 (- (1))}$

Step 6.3

Simplify by cancelling.

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Step 6.3.1

Cancel the common factor of $32$ .

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Step 6.3.1.1

Cancel the common factor.

$m = \frac{32 (\frac{5}{32}) + 32 (- (2))}{32 (\frac{5}{8}) + 32 (- (1))}$

Step 6.3.1.2

Rewrite the expression.

$m = \frac{5 + 32 (- (2))}{32 (\frac{5}{8}) + 32 (- (1))}$

Step 6.3.2

Cancel the common factor of $8$ .

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Step 6.3.2.1

Factor $8$ out of $32$ .

$m = \frac{5 + 32 (- (2))}{8 (4) (\frac{5}{8}) + 32 (- (1))}$

Step 6.3.2.2

Cancel the common factor.

$m = \frac{5 + 32 (- (2))}{8 \cdot (4 (\frac{5}{8})) + 32 (- (1))}$

Step 6.3.2.3

Rewrite the expression.

$m = \frac{5 + 32 (- (2))}{4 \cdot 5 + 32 (- (1))}$

Step 6.3.3

Multiply $4$ by $5$ .

$m = \frac{5 + 32 (- (2))}{20 + 32 (- (1))}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $32 (- (2))$ .

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Step 6.4.1.1

Multiply $- 1$ by $2$ .

$m = \frac{5 + 32 \cdot - 2}{20 + 32 (- (1))}$

Step 6.4.1.2

Multiply $32$ by $- 2$ .

$m = \frac{5 - 64}{20 + 32 (- (1))}$

Step 6.4.2

Subtract $64$ from $5$ .

$m = \frac{- 59}{20 + 32 (- (1))}$

Step 6.5

Simplify the denominator.

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Step 6.5.1

Multiply $32 (- (1))$ .

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Step 6.5.1.1

Multiply $- 1$ by $1$ .

$m = \frac{- 59}{20 + 32 \cdot - 1}$

Step 6.5.1.2

Multiply $32$ by $- 1$ .

$m = \frac{- 59}{20 - 32}$

Step 6.5.2

Subtract $32$ from $20$ .

$m = \frac{- 59}{- 12}$

Step 6.6

Dividing two negative values results in a positive value.

$m = \frac{59}{12}$

Step 7