Precalculus Examples

$(2, - 5)$ and $(- 10, 3)$

Step 1

To find the average rate of change, find the slope.

Step 2

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 3

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 4

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{3 - (- 5)}{- 10 - (2)}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by $- 5$ .

$m = \frac{3 + 5}{- 10 - (2)}$

Step 5.1.2

Add $3$ and $5$ .

$m = \frac{8}{- 10 - (2)}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $2$ .

$m = \frac{8}{- 10 - 2}$

Step 5.2.2

Subtract $2$ from $- 10$ .

$m = \frac{8}{- 12}$

Step 5.3

Reduce the expression by cancelling the common factors.

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Step 5.3.1

Cancel the common factor of $8$ and $- 12$ .

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Step 5.3.1.1

Factor $4$ out of $8$ .

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

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $4$ out of $- 12$ .

$m = \frac{4 \cdot 2}{4 \cdot - 3}$

Step 5.3.1.2.2

Cancel the common factor.

$m = \frac{4 \cdot 2}{4 \cdot - 3}$

Step 5.3.1.2.3

Rewrite the expression.

$m = \frac{2}{- 3}$

Step 5.3.2

Move the negative in front of the fraction.

$m = - \frac{2}{3}$

Step 6