Precalculus Examples

$f (x) = \frac{x^{2}}{x + 2}$ ; $[- 1, 10]$

Step 1

Write $f (x) = \frac{x^{2}}{x + 2}$ as an equation.

$y = \frac{x^{2}}{x + 2}$

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.

$\frac{f (10) - f (- 1)}{(10) - (- 1)}$

Step 2.2

Substitute the equation $y = \frac{x^{2}}{x + 2}$ for $f (10)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{{(10)}^{2}}{(10) + 2}) - (\frac{{(- 1)}^{2}}{(- 1) + 2})}{(10) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(10 + 2) (- 1 + 2)$ .

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Step 3.1.1

Multiply $\frac{\frac{10^{2}}{10 + 2} - \frac{{(- 1)}^{2}}{- 1 + 2}}{10 - (- 1)}$ by $\frac{(10 + 2) (- 1 + 2)}{(10 + 2) (- 1 + 2)}$ .

$\frac{(10 + 2) (- 1 + 2)}{(10 + 2) (- 1 + 2)} \cdot \frac{\frac{10^{2}}{10 + 2} - \frac{{(- 1)}^{2}}{- 1 + 2}}{10 - (- 1)}$

Step 3.1.2

Combine.

$\frac{(10 + 2) (- 1 + 2) (\frac{10^{2}}{10 + 2} - \frac{{(- 1)}^{2}}{- 1 + 2})}{(10 + 2) (- 1 + 2) (10 - (- 1))}$

Step 3.2

Apply the distributive property.

$\frac{(10 + 2) (- 1 + 2) \frac{10^{2}}{10 + 2} + (10 + 2) (- 1 + 2) (- \frac{{(- 1)}^{2}}{- 1 + 2})}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $10 + 2$ .

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Step 3.3.1.1

Cancel the common factor.

$\frac{(10 + 2) (- 1 + 2) \frac{10^{2}}{10 + 2} + (10 + 2) (- 1 + 2) (- \frac{{(- 1)}^{2}}{- 1 + 2})}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) (- 1 + 2) (- \frac{{(- 1)}^{2}}{- 1 + 2})}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.2

Cancel the common factor of $- 1 + 2$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{{(- 1)}^{2}}{- 1 + 2}$ into the numerator.

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) (- 1 + 2) \frac{- {(- 1)}^{2}}{- 1 + 2}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.2.2

Factor $- 1 + 2$ out of $(10 + 2) (- 1 + 2)$ .

$\frac{(- 1 + 2) \cdot 10^{2} + (- 1 + 2) (10 + 2) \frac{- {(- 1)}^{2}}{- 1 + 2}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{(- 1 + 2) \cdot 10^{2} + (- 1 + 2) (10 + 2) \frac{- {(- 1)}^{2}}{- 1 + 2}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) (- {(- 1)}^{2})}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.3.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.3.3.1.1

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

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) ({(- 1)}^{1} {(- 1)}^{2})}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) {(- 1)}^{1 + 2}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.3.3.2

Add $1$ and $2$ .

$\frac{(- 1 + 2) \cdot 10^{2} + (10 + 2) {(- 1)}^{3}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Add $- 1$ and $2$ .

$\frac{1 \cdot 10^{2} + (10 + 2) {(- 1)}^{3}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.2

Multiply $10^{2}$ by $1$ .

$\frac{10^{2} + (10 + 2) {(- 1)}^{3}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.3

Raise $10$ to the power of $2$ .

$\frac{100 + (10 + 2) {(- 1)}^{3}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.4

Add $10$ and $2$ .

$\frac{100 + 12 {(- 1)}^{3}}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.5

Raise $- 1$ to the power of $3$ .

$\frac{100 + 12 \cdot - 1}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.6

Multiply $12$ by $- 1$ .

$\frac{100 - 12}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.4.7

Subtract $12$ from $100$ .

$\frac{88}{(10 + 2) (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Add $10$ and $2$ .

$\frac{88}{12 (- 1 + 2) \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.5.2

Add $- 1$ and $2$ .

$\frac{88}{12 \cdot 1 \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.5.3

Multiply $12 \cdot 1 \cdot 10$ .

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Step 3.5.3.1

Multiply $12$ by $1$ .

$\frac{88}{12 \cdot 10 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.5.3.2

Multiply $12$ by $10$ .

$\frac{88}{120 + (10 + 2) (- 1 + 2) (- (- 1))}$

Step 3.5.4

Add $10$ and $2$ .

$\frac{88}{120 + 12 (- 1 + 2) (- (- 1))}$

Step 3.5.5

Add $- 1$ and $2$ .

$\frac{88}{120 + 12 \cdot 1 (- (- 1))}$

Step 3.5.6

Multiply $12 \cdot 1 (- (- 1))$ .

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Step 3.5.6.1

Multiply $12$ by $1$ .

$\frac{88}{120 + 12 (- (- 1))}$

Step 3.5.6.2

Multiply $- 1$ by $- 1$ .

$\frac{88}{120 + 12 \cdot 1}$

Step 3.5.6.3

Multiply $12$ by $1$ .

$\frac{88}{120 + 12}$

Step 3.5.7

Add $120$ and $12$ .

$\frac{88}{132}$

Step 3.6

Cancel the common factor of $88$ and $132$ .

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Step 3.6.1

Factor $44$ out of $88$ .

$\frac{44 (2)}{132}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $44$ out of $132$ .

$\frac{44 \cdot 2}{44 \cdot 3}$

Step 3.6.2.2

Cancel the common factor.

$\frac{44 \cdot 2}{44 \cdot 3}$

Step 3.6.2.3

Rewrite the expression.

$\frac{2}{3}$