Precalculus Examples

$\begin{array}{c} x & Y \\ - 4 & 6 \\ 0 & 4 \\ 4 & 2 \\ 8 & 0 \\ 12 & - 2 \end{array}$

Step 1

Choose the starting and ending values from the table.

$(- 4, 6), (12, - 2)$

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.

$m = \frac{- 2 - (6)}{12 - (- 4)}$

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 $- 2 - (6)$ and $12 - (- 4)$ .

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Step 6.1.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$m = \frac{- 1 \cdot 2 - (6)}{12 - (- 4)}$

Step 6.1.1.2

Factor $- 1$ out of $- 1 (2) - (6)$ .

$m = \frac{- 1 (2 + 6)}{12 - (- 4)}$

Step 6.1.1.3

Reorder terms.

$m = \frac{- 1 (2 + 6)}{12 - 4 \cdot - 1}$

Step 6.1.1.4

Factor $2$ out of $- 1 (2 + 6)$ .

$m = \frac{2 (- 1 (1 + 3))}{12 - 4 \cdot - 1}$

Step 6.1.1.5

Cancel the common factors.

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Step 6.1.1.5.1

Factor $2$ out of $12$ .

$m = \frac{2 (- 1 (1 + 3))}{2 (6) - 4 \cdot - 1}$

Step 6.1.1.5.2

Factor $2$ out of $- 4 \cdot - 1$ .

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

Step 6.1.1.5.3

Factor $2$ out of $2 (6) + 2 (- 2 \cdot - 1)$ .

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

Step 6.1.1.5.4

Cancel the common factor.

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

Step 6.1.1.5.5

Rewrite the expression.

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

Step 6.1.2

Add $1$ and $3$ .

$m = \frac{- 1 \cdot 4}{6 - 2 \cdot - 1}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Multiply $- 2$ by $- 1$ .

$m = \frac{- 1 \cdot 4}{6 + 2}$

Step 6.2.2

Add $6$ and $2$ .

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

Step 6.3

Reduce the expression by cancelling the common factors.

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Step 6.3.1

Multiply $- 1$ by $4$ .

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

Step 6.3.2

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

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Step 6.3.2.1

Factor $4$ out of $- 4$ .

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

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor $4$ out of $8$ .

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

Step 6.3.2.2.2

Cancel the common factor.

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

Step 6.3.2.2.3

Rewrite the expression.

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

Step 6.3.3

Move the negative in front of the fraction.

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

Step 7