Examples

$f (x) = x^{2} + \frac{1}{2}$ , $(- 3, 4)$

Step 1

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

$y = x^{2} + \frac{1}{2}$

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

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

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

Step 3.1.2

Combine.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Raise $4$ to the power of $2$ .

$\frac{2 \cdot 16 + 1 + 2 (- ({(- 3)}^{2} + \frac{1}{2}))}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.2

Multiply $2$ by $16$ .

$\frac{32 + 1 + 2 (- ({(- 3)}^{2} + \frac{1}{2}))}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.3

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

$\frac{32 + 1 + 2 (- (9 + \frac{1}{2}))}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{32 + 1 + 2 (- (9 \cdot \frac{2}{2} + \frac{1}{2}))}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.5

Combine $9$ and $\frac{2}{2}$ .

$\frac{32 + 1 + 2 (- (\frac{9 \cdot 2}{2} + \frac{1}{2}))}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.6

Combine the numerators over the common denominator.

$\frac{32 + 1 + 2 (- \frac{9 \cdot 2 + 1}{2})}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.7

Simplify the numerator.

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Step 3.4.7.1

Multiply $9$ by $2$ .

$\frac{32 + 1 + 2 (- \frac{18 + 1}{2})}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.7.2

Add $18$ and $1$ .

$\frac{32 + 1 + 2 (- \frac{19}{2})}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.8

Cancel the common factor of $2$ .

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Step 3.4.8.1

Move the leading negative in $- \frac{19}{2}$ into the numerator.

$\frac{32 + 1 + 2 (\frac{- 19}{2})}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.8.2

Cancel the common factor.

$\frac{32 + 1 + 2 (\frac{- 19}{2})}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.8.3

Rewrite the expression.

$\frac{32 + 1 - 19}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.9

Add $32$ and $1$ .

$\frac{33 - 19}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.4.10

Subtract $19$ from $33$ .

$\frac{14}{2 \cdot 4 + 2 (- (- 3))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $2$ by $4$ .

$\frac{14}{8 + 2 (- (- 3))}$

Step 3.5.2

Multiply $2 (- (- 3))$ .

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Step 3.5.2.1

Multiply $- 1$ by $- 3$ .

$\frac{14}{8 + 2 \cdot 3}$

Step 3.5.2.2

Multiply $2$ by $3$ .

$\frac{14}{8 + 6}$

Step 3.5.3

Add $8$ and $6$ .

$\frac{14}{14}$

Step 3.6

Divide $14$ by $14$ .

$1$

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