Precalculus Examples

$f (x) = x^{2} + x - 1$ ; $[- 2, - \frac{7}{4}]$

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.

$\frac{f (- \frac{7}{4}) - f (- 2)}{(- \frac{7}{4}) - (- 2)}$

Step 2.2

Substitute the equation $y = x^{2} + x - 1$ for $f (- \frac{7}{4})$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{{(- \frac{7}{4})}^{2} - \frac{7}{4} - 1 - ({(- 2)}^{2} - 2 - 1)}{(- \frac{7}{4}) - (- 2)}$

Step 3

Simplify the expression.

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Step 3.1

Remove parentheses.

$\frac{{(- \frac{7}{4})}^{2} - \frac{7}{4} - 1 - ({(- 2)}^{2} - 2 - 1)}{(- \frac{7}{4}) - (- 2)}$

Step 3.2

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

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Step 3.2.1

Multiply $\frac{{(- \frac{7}{4})}^{2} - \frac{7}{4} - 1 - ({(- 2)}^{2} - 2 - 1)}{- \frac{7}{4} - (- 2)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{{(- \frac{7}{4})}^{2} - \frac{7}{4} - 1 - ({(- 2)}^{2} - 2 - 1)}{- \frac{7}{4} - (- 2)}$

Step 3.2.2

Combine.

$\frac{4 ({(- \frac{7}{4})}^{2} - \frac{7}{4} - 1 - ({(- 2)}^{2} - 2 - 1))}{4 (- \frac{7}{4} - (- 2))}$

Step 3.3

Apply the distributive property.

$\frac{4 {(- \frac{7}{4})}^{2} + 4 (- \frac{7}{4}) + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (- \frac{7}{4}) + 4 (- (- 2))}$

Step 3.4

Simplify by cancelling.

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Step 3.4.1

Cancel the common factor of $4$ .

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Step 3.4.1.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$\frac{4 {(- \frac{7}{4})}^{2} + 4 (\frac{- 7}{4}) + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (- \frac{7}{4}) + 4 (- (- 2))}$

Step 3.4.1.2

Cancel the common factor.

$\frac{4 {(- \frac{7}{4})}^{2} + 4 (\frac{- 7}{4}) + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (- \frac{7}{4}) + 4 (- (- 2))}$

Step 3.4.1.3

Rewrite the expression.

$\frac{4 {(- \frac{7}{4})}^{2} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (- \frac{7}{4}) + 4 (- (- 2))}$

Step 3.4.2

Cancel the common factor of $4$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$\frac{4 {(- \frac{7}{4})}^{2} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (\frac{- 7}{4}) + 4 (- (- 2))}$

Step 3.4.2.2

Cancel the common factor.

$\frac{4 {(- \frac{7}{4})}^{2} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{4 (\frac{- 7}{4}) + 4 (- (- 2))}$

Step 3.4.2.3

Rewrite the expression.

$\frac{4 {(- \frac{7}{4})}^{2} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.5.1.1

Apply the product rule to $- \frac{7}{4}$ .

$\frac{4 ({(- 1)}^{2} {(\frac{7}{4})}^{2}) - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.1.2

Apply the product rule to $\frac{7}{4}$ .

$\frac{4 ({(- 1)}^{2} \frac{7^{2}}{4^{2}}) - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.2

Raise $- 1$ to the power of $2$ .

$\frac{4 (1 \frac{7^{2}}{4^{2}}) - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.3

Multiply $\frac{7^{2}}{4^{2}}$ by $1$ .

$\frac{4 \frac{7^{2}}{4^{2}} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.4

Cancel the common factor of $4$ .

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Step 3.5.4.1

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

$\frac{4 \frac{7^{2}}{4 \cdot 4} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.4.2

Cancel the common factor.

$\frac{4 \frac{7^{2}}{4 \cdot 4} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.4.3

Rewrite the expression.

$\frac{\frac{7^{2}}{4} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.5

Raise $7$ to the power of $2$ .

$\frac{\frac{49}{4} - 7 + 4 \cdot - 1 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.6

Multiply $4$ by $- 1$ .

$\frac{\frac{49}{4} - 7 - 4 + 4 (- ({(- 2)}^{2} - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.7

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

$\frac{\frac{49}{4} - 7 - 4 + 4 (- (4 - 2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.8

Subtract $2$ from $4$ .

$\frac{\frac{49}{4} - 7 - 4 + 4 (- (2 - 1))}{- 7 + 4 (- (- 2))}$

Step 3.5.9

Subtract $1$ from $2$ .

$\frac{\frac{49}{4} - 7 - 4 + 4 (- 1 \cdot 1)}{- 7 + 4 (- (- 2))}$

Step 3.5.10

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

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Step 3.5.10.1

Multiply $- 1$ by $1$ .

$\frac{\frac{49}{4} - 7 - 4 + 4 \cdot - 1}{- 7 + 4 (- (- 2))}$

Step 3.5.10.2

Multiply $4$ by $- 1$ .

$\frac{\frac{49}{4} - 7 - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.11

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{49}{4} - 7 \cdot \frac{4}{4} - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.12

Combine $- 7$ and $\frac{4}{4}$ .

$\frac{\frac{49}{4} + \frac{- 7 \cdot 4}{4} - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.13

Combine the numerators over the common denominator.

$\frac{\frac{49 - 7 \cdot 4}{4} - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.14

Simplify the numerator.

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Step 3.5.14.1

Multiply $- 7$ by $4$ .

$\frac{\frac{49 - 28}{4} - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.14.2

Subtract $28$ from $49$ .

$\frac{\frac{21}{4} - 4 - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.15

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

$\frac{\frac{21}{4} - 4 \cdot \frac{4}{4} - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.16

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

$\frac{\frac{21}{4} + \frac{- 4 \cdot 4}{4} - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.17

Combine the numerators over the common denominator.

$\frac{\frac{21 - 4 \cdot 4}{4} - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.18

Simplify the numerator.

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Step 3.5.18.1

Multiply $- 4$ by $4$ .

$\frac{\frac{21 - 16}{4} - 4}{- 7 + 4 (- (- 2))}$

Step 3.5.18.2

Subtract $16$ from $21$ .

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

Step 3.5.19

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

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

Step 3.5.20

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

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

Step 3.5.21

Combine the numerators over the common denominator.

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

Step 3.5.22

Simplify the numerator.

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Step 3.5.22.1

Multiply $- 4$ by $4$ .

$\frac{\frac{5 - 16}{4}}{- 7 + 4 (- (- 2))}$

Step 3.5.22.2

Subtract $16$ from $5$ .

$\frac{\frac{- 11}{4}}{- 7 + 4 (- (- 2))}$

Step 3.5.23

Move the negative in front of the fraction.

$\frac{- \frac{11}{4}}{- 7 + 4 (- (- 2))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $4 (- (- 2))$ .

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Step 3.6.1.1

Multiply $- 1$ by $- 2$ .

$\frac{- \frac{11}{4}}{- 7 + 4 \cdot 2}$

Step 3.6.1.2

Multiply $4$ by $2$ .

$\frac{- \frac{11}{4}}{- 7 + 8}$

Step 3.6.2

Add $- 7$ and $8$ .

$\frac{- \frac{11}{4}}{1}$

Step 3.7

Divide $- \frac{11}{4}$ by $1$ .

$- \frac{11}{4}$