Precalculus Examples

$f (x) = x^{2} - x - 1$ ; $[3, \frac{13}{4}]$

Step 1

Write $f (x) = x^{2} - x - 1$ as an equation.

$y = x^{2} - x - 1$

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.

$\frac{f (\frac{13}{4}) - f (3)}{(\frac{13}{4}) - (3)}$

Step 2.2

Substitute the equation $y = x^{2} - x - 1$ for $f (\frac{13}{4})$ and $f (3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(\frac{13}{4})}^{2} - (\frac{13}{4}) - 1) - ({(3)}^{2} - (3) - 1)}{(\frac{13}{4}) - (3)}$

Step 3

Simplify the expression.

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Step 3.1

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

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Step 3.1.1

Multiply $\frac{{(\frac{13}{4})}^{2} - \frac{13}{4} - 1 - (3^{2} - (3) - 1)}{\frac{13}{4} - (3)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{{(\frac{13}{4})}^{2} - \frac{13}{4} - 1 - (3^{2} - (3) - 1)}{\frac{13}{4} - (3)}$

Step 3.1.2

Combine.

$\frac{4 ({(\frac{13}{4})}^{2} - \frac{13}{4} - 1 - (3^{2} - (3) - 1))}{4 (\frac{13}{4} - (3))}$

Step 3.2

Apply the distributive property.

$\frac{4 {(\frac{13}{4})}^{2} + 4 (- \frac{13}{4}) + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{4 (\frac{13}{4}) + 4 (- (3))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $4$ .

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Step 3.3.1.1

Move the leading negative in $- \frac{13}{4}$ into the numerator.

$\frac{4 {(\frac{13}{4})}^{2} + 4 (\frac{- 13}{4}) + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{4 (\frac{13}{4}) + 4 (- (3))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{4 {(\frac{13}{4})}^{2} + 4 (\frac{- 13}{4}) + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{4 (\frac{13}{4}) + 4 (- (3))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{4 {(\frac{13}{4})}^{2} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{4 (\frac{13}{4}) + 4 (- (3))}$

Step 3.3.2

Cancel the common factor of $4$ .

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Step 3.3.2.1

Cancel the common factor.

$\frac{4 {(\frac{13}{4})}^{2} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{4 (\frac{13}{4}) + 4 (- (3))}$

Step 3.3.2.2

Rewrite the expression.

$\frac{4 {(\frac{13}{4})}^{2} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Apply the product rule to $\frac{13}{4}$ .

$\frac{4 \frac{13^{2}}{4^{2}} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.2

Cancel the common factor of $4$ .

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Step 3.4.2.1

Factor $4$ out of $4^{2}$ .

$\frac{4 \frac{13^{2}}{4 \cdot 4} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{4 \frac{13^{2}}{4 \cdot 4} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{\frac{13^{2}}{4} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.3

Raise $13$ to the power of $2$ .

$\frac{\frac{169}{4} - 13 + 4 \cdot - 1 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.4

Multiply $4$ by $- 1$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 (- (3^{2} - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.5

Simplify each term.

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Step 3.4.5.1

Raise $3$ to the power of $2$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 (- (9 - (3) - 1))}{13 + 4 (- (3))}$

Step 3.4.5.2

Multiply $- 1$ by $3$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 (- (9 - 3 - 1))}{13 + 4 (- (3))}$

Step 3.4.6

Subtract $3$ from $9$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 (- (6 - 1))}{13 + 4 (- (3))}$

Step 3.4.7

Subtract $1$ from $6$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 (- 1 \cdot 5)}{13 + 4 (- (3))}$

Step 3.4.8

Multiply $4 (- 1 \cdot 5)$ .

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Step 3.4.8.1

Multiply $- 1$ by $5$ .

$\frac{\frac{169}{4} - 13 - 4 + 4 \cdot - 5}{13 + 4 (- (3))}$

Step 3.4.8.2

Multiply $4$ by $- 5$ .

$\frac{\frac{169}{4} - 13 - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.9

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{169}{4} - 13 \cdot \frac{4}{4} - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.10

Combine $- 13$ and $\frac{4}{4}$ .

$\frac{\frac{169}{4} + \frac{- 13 \cdot 4}{4} - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.11

Combine the numerators over the common denominator.

$\frac{\frac{169 - 13 \cdot 4}{4} - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.12

Simplify the numerator.

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Step 3.4.12.1

Multiply $- 13$ by $4$ .

$\frac{\frac{169 - 52}{4} - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.12.2

Subtract $52$ from $169$ .

$\frac{\frac{117}{4} - 4 - 20}{13 + 4 (- (3))}$

Step 3.4.13

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{117}{4} - 4 \cdot \frac{4}{4} - 20}{13 + 4 (- (3))}$

Step 3.4.14

Combine $- 4$ and $\frac{4}{4}$ .

$\frac{\frac{117}{4} + \frac{- 4 \cdot 4}{4} - 20}{13 + 4 (- (3))}$

Step 3.4.15

Combine the numerators over the common denominator.

$\frac{\frac{117 - 4 \cdot 4}{4} - 20}{13 + 4 (- (3))}$

Step 3.4.16

Simplify the numerator.

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Step 3.4.16.1

Multiply $- 4$ by $4$ .

$\frac{\frac{117 - 16}{4} - 20}{13 + 4 (- (3))}$

Step 3.4.16.2

Subtract $16$ from $117$ .

$\frac{\frac{101}{4} - 20}{13 + 4 (- (3))}$

Step 3.4.17

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{101}{4} - 20 \cdot \frac{4}{4}}{13 + 4 (- (3))}$

Step 3.4.18

Combine $- 20$ and $\frac{4}{4}$ .

$\frac{\frac{101}{4} + \frac{- 20 \cdot 4}{4}}{13 + 4 (- (3))}$

Step 3.4.19

Combine the numerators over the common denominator.

$\frac{\frac{101 - 20 \cdot 4}{4}}{13 + 4 (- (3))}$

Step 3.4.20

Simplify the numerator.

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Step 3.4.20.1

Multiply $- 20$ by $4$ .

$\frac{\frac{101 - 80}{4}}{13 + 4 (- (3))}$

Step 3.4.20.2

Subtract $80$ from $101$ .

$\frac{\frac{21}{4}}{13 + 4 (- (3))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $4 (- (3))$ .

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Step 3.5.1.1

Multiply $- 1$ by $3$ .

$\frac{\frac{21}{4}}{13 + 4 \cdot - 3}$

Step 3.5.1.2

Multiply $4$ by $- 3$ .

$\frac{\frac{21}{4}}{13 - 12}$

Step 3.5.2

Subtract $12$ from $13$ .

$\frac{\frac{21}{4}}{1}$

Step 3.6

Divide $\frac{21}{4}$ by $1$ .

$\frac{21}{4}$