Precalculus Examples

$f (x) = - x^{2} - x + 1$ ;, $[2, \frac{7}{3}]$

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.

$\frac{f (\frac{7}{3}) - f (2)}{(\frac{7}{3}) - (2)}$

Step 2.2

Substitute the equation $y = - x^{2} - x + 1$ for $f (\frac{7}{3})$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- {(\frac{7}{3})}^{2} - (\frac{7}{3}) + 1) - (- {(2)}^{2} - (2) + 1)}{(\frac{7}{3}) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 3.1.1

Multiply $\frac{- {(\frac{7}{3})}^{2} - \frac{7}{3} + 1 - (- 2^{2} - (2) + 1)}{\frac{7}{3} - (2)}$ by $\frac{3}{3}$ .

$\frac{3}{3} \cdot \frac{- {(\frac{7}{3})}^{2} - \frac{7}{3} + 1 - (- 2^{2} - (2) + 1)}{\frac{7}{3} - (2)}$

Step 3.1.2

Combine.

$\frac{3 (- {(\frac{7}{3})}^{2} - \frac{7}{3} + 1 - (- 2^{2} - (2) + 1))}{3 (\frac{7}{3} - (2))}$

Step 3.2

Apply the distributive property.

$\frac{3 (- {(\frac{7}{3})}^{2}) + 3 (- \frac{7}{3}) + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{3 (\frac{7}{3}) + 3 (- (2))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $3$ .

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Step 3.3.1.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$\frac{3 (- {(\frac{7}{3})}^{2}) + 3 (\frac{- 7}{3}) + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{3 (\frac{7}{3}) + 3 (- (2))}$

Step 3.3.1.2

Cancel the common factor.

$\frac{3 (- {(\frac{7}{3})}^{2}) + 3 (\frac{- 7}{3}) + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{3 (\frac{7}{3}) + 3 (- (2))}$

Step 3.3.1.3

Rewrite the expression.

$\frac{3 (- {(\frac{7}{3})}^{2}) - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{3 (\frac{7}{3}) + 3 (- (2))}$

Step 3.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.1

Cancel the common factor.

$\frac{3 (- {(\frac{7}{3})}^{2}) - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{3 (\frac{7}{3}) + 3 (- (2))}$

Step 3.3.2.2

Rewrite the expression.

$\frac{3 (- {(\frac{7}{3})}^{2}) - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Apply the product rule to $\frac{7}{3}$ .

$\frac{3 (- \frac{7^{2}}{3^{2}}) - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.2

Cancel the common factor of $3$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{7^{2}}{3^{2}}$ into the numerator.

$\frac{3 \frac{- 7^{2}}{3^{2}} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.2.2

Factor $3$ out of $3^{2}$ .

$\frac{3 \frac{- 7^{2}}{3 \cdot 3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.2.3

Cancel the common factor.

$\frac{3 \frac{- 7^{2}}{3 \cdot 3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.2.4

Rewrite the expression.

$\frac{\frac{- 7^{2}}{3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.3

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

$\frac{\frac{- 1 \cdot 49}{3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.4

Multiply $- 1$ by $49$ .

$\frac{\frac{- 49}{3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.5

Move the negative in front of the fraction.

$\frac{- \frac{49}{3} - 7 + 3 \cdot 1 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.6

Multiply $3$ by $1$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- (- 2^{2} - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.7

Simplify each term.

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Step 3.4.7.1

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

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- (- 1 \cdot 4 - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.7.2

Multiply $- 1$ by $4$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- (- 4 - (2) + 1))}{7 + 3 (- (2))}$

Step 3.4.7.3

Multiply $- 1$ by $2$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- (- 4 - 2 + 1))}{7 + 3 (- (2))}$

Step 3.4.8

Subtract $2$ from $- 4$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- (- 6 + 1))}{7 + 3 (- (2))}$

Step 3.4.9

Add $- 6$ and $1$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 (- - 5)}{7 + 3 (- (2))}$

Step 3.4.10

Multiply $3 (- - 5)$ .

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Step 3.4.10.1

Multiply $- 1$ by $- 5$ .

$\frac{- \frac{49}{3} - 7 + 3 + 3 \cdot 5}{7 + 3 (- (2))}$

Step 3.4.10.2

Multiply $3$ by $5$ .

$\frac{- \frac{49}{3} - 7 + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.11

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- \frac{49}{3} - 7 \cdot \frac{3}{3} + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.12

Combine $- 7$ and $\frac{3}{3}$ .

$\frac{- \frac{49}{3} + \frac{- 7 \cdot 3}{3} + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.13

Combine the numerators over the common denominator.

$\frac{\frac{- 49 - 7 \cdot 3}{3} + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.14

Simplify the numerator.

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Step 3.4.14.1

Multiply $- 7$ by $3$ .

$\frac{\frac{- 49 - 21}{3} + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.14.2

Subtract $21$ from $- 49$ .

$\frac{\frac{- 70}{3} + 3 + 15}{7 + 3 (- (2))}$

Step 3.4.15

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{- 70}{3} + 3 \cdot \frac{3}{3} + 15}{7 + 3 (- (2))}$

Step 3.4.16

Combine $3$ and $\frac{3}{3}$ .

$\frac{\frac{- 70}{3} + \frac{3 \cdot 3}{3} + 15}{7 + 3 (- (2))}$

Step 3.4.17

Combine the numerators over the common denominator.

$\frac{\frac{- 70 + 3 \cdot 3}{3} + 15}{7 + 3 (- (2))}$

Step 3.4.18

Simplify the numerator.

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Step 3.4.18.1

Multiply $3$ by $3$ .

$\frac{\frac{- 70 + 9}{3} + 15}{7 + 3 (- (2))}$

Step 3.4.18.2

Add $- 70$ and $9$ .

$\frac{\frac{- 61}{3} + 15}{7 + 3 (- (2))}$

Step 3.4.19

To write $15$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{- 61}{3} + 15 \cdot \frac{3}{3}}{7 + 3 (- (2))}$

Step 3.4.20

Combine $15$ and $\frac{3}{3}$ .

$\frac{\frac{- 61}{3} + \frac{15 \cdot 3}{3}}{7 + 3 (- (2))}$

Step 3.4.21

Combine the numerators over the common denominator.

$\frac{\frac{- 61 + 15 \cdot 3}{3}}{7 + 3 (- (2))}$

Step 3.4.22

Simplify the numerator.

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Step 3.4.22.1

Multiply $15$ by $3$ .

$\frac{\frac{- 61 + 45}{3}}{7 + 3 (- (2))}$

Step 3.4.22.2

Add $- 61$ and $45$ .

$\frac{\frac{- 16}{3}}{7 + 3 (- (2))}$

Step 3.4.23

Move the negative in front of the fraction.

$\frac{- \frac{16}{3}}{7 + 3 (- (2))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $3 (- (2))$ .

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Step 3.5.1.1

Multiply $- 1$ by $2$ .

$\frac{- \frac{16}{3}}{7 + 3 \cdot - 2}$

Step 3.5.1.2

Multiply $3$ by $- 2$ .

$\frac{- \frac{16}{3}}{7 - 6}$

Step 3.5.2

Subtract $6$ from $7$ .

$\frac{- \frac{16}{3}}{1}$

Step 3.6

Divide $- \frac{16}{3}$ by $1$ .

$- \frac{16}{3}$