Precalculus Examples

$f (x) = \frac{x + 5}{x - 4}$ ; $[- 6, 2]$

Step 1

Write $f (x) = \frac{x + 5}{x - 4}$ as an equation.

$y = \frac{x + 5}{x - 4}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (2) - f (- 6)}{(2) - (- 6)}$

Step 2.2

Substitute the equation $y = \frac{x + 5}{x - 4}$ for $f (2)$ and $f (- 6)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{(2) + 5}{(2) - 4}) - (\frac{(- 6) + 5}{(- 6) - 4})}{(2) - (- 6)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(2 - 4) (- 6 - 4)$ .

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Step 3.1.1

Multiply $\frac{\frac{2 + 5}{2 - 4} - \frac{- 6 + 5}{- 6 - 4}}{2 - (- 6)}$ by $\frac{(2 - 4) (- 6 - 4)}{(2 - 4) (- 6 - 4)}$ .

$\frac{(2 - 4) (- 6 - 4)}{(2 - 4) (- 6 - 4)} \cdot \frac{\frac{2 + 5}{2 - 4} - \frac{- 6 + 5}{- 6 - 4}}{2 - (- 6)}$

Step 3.1.2

Combine.

$\frac{(2 - 4) (- 6 - 4) (\frac{2 + 5}{2 - 4} - \frac{- 6 + 5}{- 6 - 4})}{(2 - 4) (- 6 - 4) (2 - (- 6))}$

Step 3.2

Apply the distributive property.

$\frac{(2 - 4) (- 6 - 4) \frac{2 + 5}{2 - 4} + (2 - 4) (- 6 - 4) (- \frac{- 6 + 5}{- 6 - 4})}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $2 - 4$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{(2 - 4) (- 6 - 4) \frac{2 + 5}{2 - 4} + (2 - 4) (- 6 - 4) (- \frac{- 6 + 5}{- 6 - 4})}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{(- 6 - 4) (2 + 5) + (2 - 4) (- 6 - 4) (- \frac{- 6 + 5}{- 6 - 4})}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3.2

Cancel the common factor of $- 6 - 4$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{- 6 + 5}{- 6 - 4}$ into the numerator.

$\frac{(- 6 - 4) (2 + 5) + (2 - 4) (- 6 - 4) \frac{- (- 6 + 5)}{- 6 - 4}}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3.2.2

Factor $- 6 - 4$ out of $(2 - 4) (- 6 - 4)$ .

$\frac{(- 6 - 4) (2 + 5) + (- 6 - 4) (2 - 4) \frac{- (- 6 + 5)}{- 6 - 4}}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{(- 6 - 4) (2 + 5) + (- 6 - 4) (2 - 4) \frac{- (- 6 + 5)}{- 6 - 4}}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{(- 6 - 4) (2 + 5) + (2 - 4) (- (- 6 + 5))}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Subtract $4$ from $- 6$ .

$\frac{- 10 (2 + 5) + (2 - 4) (- (- 6 + 5))}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.2

Add $2$ and $5$ .

$\frac{- 10 \cdot 7 + (2 - 4) (- (- 6 + 5))}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.3

Multiply $- 10$ by $7$ .

$\frac{- 70 + (2 - 4) (- (- 6 + 5))}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.4

Subtract $4$ from $2$ .

$\frac{- 70 - 2 (- (- 6 + 5))}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.5

Add $- 6$ and $5$ .

$\frac{- 70 - 2 (- - 1)}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.6

Multiply $- 2 (- - 1)$ .

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Step 3.4.6.1

Multiply $- 1$ by $- 1$ .

$\frac{- 70 - 2 \cdot 1}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.6.2

Multiply $- 2$ by $1$ .

$\frac{- 70 - 2}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.4.7

Subtract $2$ from $- 70$ .

$\frac{- 72}{(2 - 4) (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Subtract $4$ from $2$ .

$\frac{- 72}{- 2 (- 6 - 4) \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.5.2

Subtract $4$ from $- 6$ .

$\frac{- 72}{- 2 \cdot - 10 \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.5.3

Multiply $- 2 \cdot - 10 \cdot 2$ .

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Step 3.5.3.1

Multiply $- 2$ by $- 10$ .

$\frac{- 72}{20 \cdot 2 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.5.3.2

Multiply $20$ by $2$ .

$\frac{- 72}{40 + (2 - 4) (- 6 - 4) (- (- 6))}$

Step 3.5.4

Subtract $4$ from $2$ .

$\frac{- 72}{40 - 2 (- 6 - 4) (- (- 6))}$

Step 3.5.5

Subtract $4$ from $- 6$ .

$\frac{- 72}{40 - 2 \cdot - 10 (- (- 6))}$

Step 3.5.6

Multiply $- 2 \cdot - 10 (- (- 6))$ .

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Step 3.5.6.1

Multiply $- 2$ by $- 10$ .

$\frac{- 72}{40 + 20 (- (- 6))}$

Step 3.5.6.2

Multiply $- 1$ by $- 6$ .

$\frac{- 72}{40 + 20 \cdot 6}$

Step 3.5.6.3

Multiply $20$ by $6$ .

$\frac{- 72}{40 + 120}$

Step 3.5.7

Add $40$ and $120$ .

$\frac{- 72}{160}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Cancel the common factor of $- 72$ and $160$ .

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Step 3.6.1.1

Factor $8$ out of $- 72$ .

$\frac{8 (- 9)}{160}$

Step 3.6.1.2

Cancel the common factors.

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Step 3.6.1.2.1

Factor $8$ out of $160$ .

$\frac{8 \cdot - 9}{8 \cdot 20}$

Step 3.6.1.2.2

Cancel the common factor.

$\frac{8 \cdot - 9}{8 \cdot 20}$

Step 3.6.1.2.3

Rewrite the expression.

$\frac{- 9}{20}$

Step 3.6.2

Move the negative in front of the fraction.

$- \frac{9}{20}$