Precalculus Examples

$\begin{array}{c} x & y \\ 1 & \frac{3}{2} \\ 2 & \frac{9}{8} \\ 3 & \frac{27}{32} \\ 4 & \frac{81}{128} \end{array}$

Step 1

Choose the starting and ending values from the table.

$(1, \frac{3}{2}), (4, \frac{81}{128})$

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.

$m = \frac{\frac{81}{128} - (\frac{3}{2})}{4 - (1)}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply the numerator and denominator of the fraction by $128$ .

Tap for more steps...

Step 6.1.1

Multiply $\frac{\frac{81}{128} - \frac{3}{2}}{4 - (1)}$ by $\frac{128}{128}$ .

$m = \frac{128}{128} \cdot \frac{\frac{81}{128} - \frac{3}{2}}{4 - (1)}$

Step 6.1.2

Combine.

$m = \frac{128 (\frac{81}{128} - \frac{3}{2})}{128 (4 - (1))}$

Step 6.2

Apply the distributive property.

$m = \frac{128 (\frac{81}{128}) + 128 (- \frac{3}{2})}{128 \cdot 4 + 128 (- (1))}$

Step 6.3

Simplify by cancelling.

Tap for more steps...

Step 6.3.1

Cancel the common factor of $128$ .

Tap for more steps...

Step 6.3.1.1

Cancel the common factor.

$m = \frac{128 (\frac{81}{128}) + 128 (- \frac{3}{2})}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.1.2

Rewrite the expression.

$m = \frac{81 + 128 (- \frac{3}{2})}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$m = \frac{81 + 128 (\frac{- 3}{2})}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.2.2

Factor $2$ out of $128$ .

$m = \frac{81 + 2 (64) (\frac{- 3}{2})}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.2.3

Cancel the common factor.

$m = \frac{81 + 2 \cdot (64 (\frac{- 3}{2}))}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.2.4

Rewrite the expression.

$m = \frac{81 + 64 \cdot - 3}{128 \cdot 4 + 128 (- (1))}$

Step 6.3.3

Multiply $64$ by $- 3$ .

$m = \frac{81 - 192}{128 \cdot 4 + 128 (- (1))}$

Step 6.4

Subtract $192$ from $81$ .

$m = \frac{- 111}{128 \cdot 4 + 128 (- (1))}$

Step 6.5

Simplify the denominator.

Tap for more steps...

Step 6.5.1

Multiply $128$ by $4$ .

$m = \frac{- 111}{512 + 128 (- (1))}$

Step 6.5.2

Multiply $128 (- (1))$ .

Tap for more steps...

Step 6.5.2.1

Multiply $- 1$ by $1$ .

$m = \frac{- 111}{512 + 128 \cdot - 1}$

Step 6.5.2.2

Multiply $128$ by $- 1$ .

$m = \frac{- 111}{512 - 128}$

Step 6.5.3

Subtract $128$ from $512$ .

$m = \frac{- 111}{384}$

Step 6.6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.6.1

Cancel the common factor of $- 111$ and $384$ .

Tap for more steps...

Step 6.6.1.1

Factor $3$ out of $- 111$ .

$m = \frac{3 (- 37)}{384}$

Step 6.6.1.2

Cancel the common factors.

Tap for more steps...

Step 6.6.1.2.1

Factor $3$ out of $384$ .

$m = \frac{3 \cdot - 37}{3 \cdot 128}$

Step 6.6.1.2.2

Cancel the common factor.

$m = \frac{3 \cdot - 37}{3 \cdot 128}$

Step 6.6.1.2.3

Rewrite the expression.

$m = \frac{- 37}{128}$

Step 6.6.2

Move the negative in front of the fraction.

$m = - \frac{37}{128}$

Step 7