Precalculus Examples

$\begin{array}{c} x & f (x) \\ - 4 & - 8 \\ - 2 & 5 \\ - 1 & 3 \\ 0 & 0 \\ 1 & - 3 \\ 2 & - 5 \\ 4 & 8 \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 4, - 8), (4, 8)$

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.

$m = \frac{8 - (- 8)}{4 - (- 4)}$

Step 6

Simplify.

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Step 6.1

Reduce the expression by cancelling the common factors.

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Step 6.1.1

Cancel the common factor of $8 - (- 8)$ and $4 - (- 4)$ .

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Step 6.1.1.1

Rewrite $8$ as $- 1 (- 8)$ .

$m = \frac{- 1 \cdot - 8 - (- 8)}{4 - (- 4)}$

Step 6.1.1.2

Factor $- 1$ out of $- 1 (- 8) - (- 8)$ .

$m = \frac{- 1 (- 8 - 8)}{4 - (- 4)}$

Step 6.1.1.3

Reorder terms.

$m = \frac{- 1 (- 8 - 8)}{4 - 4 \cdot - 1}$

Step 6.1.1.4

Factor $4$ out of $- 1 (- 8 - 8)$ .

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

Step 6.1.1.5

Cancel the common factors.

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Step 6.1.1.5.1

Factor $4$ out of $4$ .

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

Step 6.1.1.5.2

Factor $4$ out of $- 4 \cdot - 1$ .

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

Step 6.1.1.5.3

Factor $4$ out of $4 (1) + 4 (- - 1)$ .

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

Step 6.1.1.5.4

Cancel the common factor.

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

Step 6.1.1.5.5

Rewrite the expression.

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

Step 6.1.2

Subtract $2$ from $- 2$ .

$m = \frac{- 1 \cdot - 4}{1 + 1}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 \cdot - 4}{1 + 1}$

Step 6.2.2

Add $1$ and $1$ .

$m = \frac{- 1 \cdot - 4}{2}$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $- 1$ by $- 4$ .

$m = \frac{4}{2}$

Step 6.3.2

Divide $4$ by $2$ .

$m = 2$

Step 7