Examples

f (x) = x^{2} + \frac{1}{2}

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

(- 3, 4)

$(- 3, 4)$

Step 1

Write

f (x) = x^{2} + \frac{1}{2}

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

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

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

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

$y$ values of the two points divided by the change in

x

$x$ values of the two points.

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

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

Step 2.2

Substitute the equation

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

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

f (4)

$f (4)$ and

f (- 3)

$f (- 3)$ , replacing

x

$x$ in the function with the corresponding

x

$x$ value.

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

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

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

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by

2

$2$ .

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Step 3.1.1

Multiply

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

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

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.1.2

Combine.

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

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

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

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

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))}

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

Step 3.3

Cancel the common factor of

2

$2$ .

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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))}

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

Step 3.3.2

Rewrite the expression.

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

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

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

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

Raise

4

$4$ to the power of

2

$2$ .

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

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

Step 3.4.2

Multiply

2

$2$ by

16

$16$ .

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

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

Step 3.4.3

Raise

- 3

$- 3$ to the power of

2

$2$ .

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

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

Step 3.4.4

To write

9

$9$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

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

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

Step 3.4.5

Combine

9

$9$ and

\frac{2}{2}

$\frac{2}{2}$ .

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

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

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))}

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

Step 3.4.7

Simplify the numerator.

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Step 3.4.7.1

Multiply

9

$9$ by

2

$2$ .

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

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

Step 3.4.7.2

Add

18

$18$ and

1

$1$ .

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

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

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

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

Step 3.4.8

Cancel the common factor of

2

$2$ .

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Step 3.4.8.1

Move the leading negative in

- \frac{19}{2}

$- \frac{19}{2}$ into the numerator.

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

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

Step 3.4.8.2

Cancel the common factor.

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

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

Step 3.4.8.3

Rewrite the expression.

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

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

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

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

Step 3.4.9

Add

32

$32$ and

1

$1$ .

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

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

Step 3.4.10

Subtract

19

$19$ from

33

$33$ .

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

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

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

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply

2

$2$ by

4

$4$ .

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

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

Step 3.5.2

Multiply

2 (- (- 3))

$2 (- (- 3))$ .

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Step 3.5.2.1

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

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

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

Step 3.5.2.2

Multiply

2

$2$ by

3

$3$ .

\frac{14}{8 + 6}

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

\frac{14}{8 + 6}

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

Step 3.5.3

Add

8

$8$ and

6

$6$ .

\frac{14}{14}

$\frac{14}{14}$

\frac{14}{14}

$\frac{14}{14}$

Step 3.6

Divide

14

$14$ by

14

$14$ .

1

$1$

1

$1$

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