Precalculus Examples

$\begin{array}{c} x & y \\ - 2 & - 2 \\ - 1 & 2 \\ 0 & 6 \\ 1 & 10 \\ 2 & 14 \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 2, - 2), (2, 14)$

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.

$m = \frac{14 - (- 2)}{2 - (- 2)}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.1.1

Cancel the common factor of $14 - (- 2)$ and $2 - (- 2)$ .

Tap for more steps...

Step 6.1.1.1

Rewrite $14$ as $- 1 (- 14)$ .

$m = \frac{- 1 \cdot - 14 - (- 2)}{2 - (- 2)}$

Step 6.1.1.2

Factor $- 1$ out of $- 1 (- 14) - (- 2)$ .

$m = \frac{- 1 (- 14 - 2)}{2 - (- 2)}$

Step 6.1.1.3

Reorder terms.

$m = \frac{- 1 (- 14 - 2)}{2 - 2 \cdot - 1}$

Step 6.1.1.4

Factor $2$ out of $- 1 (- 14 - 2)$ .

$m = \frac{2 (- 1 (- 7 - 1))}{2 - 2 \cdot - 1}$

Step 6.1.1.5

Cancel the common factors.

Tap for more steps...

Step 6.1.1.5.1

Factor $2$ out of $2$ .

$m = \frac{2 (- 1 (- 7 - 1))}{2 (1) - 2 \cdot - 1}$

Step 6.1.1.5.2

Factor $2$ out of $- 2 \cdot - 1$ .

$m = \frac{2 (- 1 (- 7 - 1))}{2 (1) + 2 (1)}$

Step 6.1.1.5.3

Factor $2$ out of $2 (1) + 2 (- - 1)$ .

$m = \frac{2 (- 1 (- 7 - 1))}{2 (1 + 1)}$

Step 6.1.1.5.4

Cancel the common factor.

$m = \frac{2 (- 1 (- 7 - 1))}{2 (1 + 1)}$

Step 6.1.1.5.5

Rewrite the expression.

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

Step 6.1.2

Subtract $1$ from $- 7$ .

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

Step 6.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.2

Add $1$ and $1$ .

$m = \frac{- 1 \cdot - 8}{2}$

Step 6.3

Simplify the expression.

Tap for more steps...

Step 6.3.1

Multiply $- 1$ by $- 8$ .

$m = \frac{8}{2}$

Step 6.3.2

Divide $8$ by $2$ .

$m = 4$

Step 7