Precalculus Examples

$\begin{array}{c} x & y \\ 0 & 4 \\ 1 & 2 \\ 2 & 1 \\ 3 & \frac{1}{2} \\ 4 & \frac{1}{4} \end{array}$

Step 1

Choose the starting and ending values from the table.

$(0, 4), (4, \frac{1}{4})$

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.

$m = \frac{\frac{1}{4} - (4)}{4 - (0)}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply the numerator and denominator of the fraction by $4$ .

Tap for more steps...

Step 6.1.1

Multiply $\frac{\frac{1}{4} - (4)}{4 - (0)}$ by $\frac{4}{4}$ .

$m = \frac{4}{4} \cdot \frac{\frac{1}{4} - (4)}{4 - (0)}$

Step 6.1.2

Combine.

$m = \frac{4 (\frac{1}{4} - (4))}{4 (4 - (0))}$

Step 6.2

Apply the distributive property.

$m = \frac{4 (\frac{1}{4}) + 4 (- (4))}{4 \cdot 4 + 4 (- (0))}$

Step 6.3

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.3.1

Cancel the common factor.

$m = \frac{4 (\frac{1}{4}) + 4 (- (4))}{4 \cdot 4 + 4 (- (0))}$

Step 6.3.2

Rewrite the expression.

$m = \frac{1 + 4 (- (4))}{4 \cdot 4 + 4 (- (0))}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Multiply $4 (- (4))$ .

Tap for more steps...

Step 6.4.1.1

Multiply $- 1$ by $4$ .

$m = \frac{1 + 4 \cdot - 4}{4 \cdot 4 + 4 (- (0))}$

Step 6.4.1.2

Multiply $4$ by $- 4$ .

$m = \frac{1 - 16}{4 \cdot 4 + 4 (- (0))}$

Step 6.4.2

Subtract $16$ from $1$ .

$m = \frac{- 15}{4 \cdot 4 + 4 (- (0))}$

Step 6.5

Simplify the denominator.

Tap for more steps...

Step 6.5.1

Multiply $4$ by $4$ .

$m = \frac{- 15}{16 + 4 (- (0))}$

Step 6.5.2

Multiply $4 (- (0))$ .

Tap for more steps...

Step 6.5.2.1

Multiply $- 1$ by $0$ .

$m = \frac{- 15}{16 + 4 \cdot 0}$

Step 6.5.2.2

Multiply $4$ by $0$ .

$m = \frac{- 15}{16 + 0}$

Step 6.5.3

Add $16$ and $0$ .

$m = \frac{- 15}{16}$

Step 6.6

Move the negative in front of the fraction.

$m = - \frac{15}{16}$

Step 7