Precalculus Examples

$\begin{array}{c} x & y \\ - 2 & \frac{1}{5} \\ - 1 & 1 \\ 0 & 5 \\ 1 & 25 \\ 2 & 125 \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 2, \frac{1}{5}), (2, 125)$

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.

$m = \frac{125 - (\frac{1}{5})}{2 - (- 2)}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply the numerator and denominator of the fraction by $5$ .

Tap for more steps...

Step 6.1.1

Multiply $\frac{125 - \frac{1}{5}}{2 - (- 2)}$ by $\frac{5}{5}$ .

$m = \frac{5}{5} \cdot \frac{125 - \frac{1}{5}}{2 - (- 2)}$

Step 6.1.2

Combine.

$m = \frac{5 (125 - \frac{1}{5})}{5 (2 - (- 2))}$

Step 6.2

Apply the distributive property.

$m = \frac{5 \cdot 125 + 5 (- \frac{1}{5})}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 6.3.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$m = \frac{5 \cdot 125 + 5 (\frac{- 1}{5})}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.3.2

Cancel the common factor.

$m = \frac{5 \cdot 125 + 5 (\frac{- 1}{5})}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.3.3

Rewrite the expression.

$m = \frac{5 \cdot 125 - 1}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Multiply $5$ by $125$ .

$m = \frac{625 - 1}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.4.2

Subtract $1$ from $625$ .

$m = \frac{624}{5 \cdot 2 + 5 (- (- 2))}$

Step 6.5

Simplify the denominator.

Tap for more steps...

Step 6.5.1

Multiply $5$ by $2$ .

$m = \frac{624}{10 + 5 (- (- 2))}$

Step 6.5.2

Multiply $5 (- (- 2))$ .

Tap for more steps...

Step 6.5.2.1

Multiply $- 1$ by $- 2$ .

$m = \frac{624}{10 + 5 \cdot 2}$

Step 6.5.2.2

Multiply $5$ by $2$ .

$m = \frac{624}{10 + 10}$

Step 6.5.3

Add $10$ and $10$ .

$m = \frac{624}{20}$

Step 6.6

Cancel the common factor of $624$ and $20$ .

Tap for more steps...

Step 6.6.1

Factor $4$ out of $624$ .

$m = \frac{4 (156)}{20}$

Step 6.6.2

Cancel the common factors.

Tap for more steps...

Step 6.6.2.1

Factor $4$ out of $20$ .

$m = \frac{4 \cdot 156}{4 \cdot 5}$

Step 6.6.2.2

Cancel the common factor.

$m = \frac{4 \cdot 156}{4 \cdot 5}$

Step 6.6.2.3

Rewrite the expression.

$m = \frac{156}{5}$

Step 7