Precalculus Examples

$f (x) = - \frac{1}{2} x^{2} + 3 x + 1$ ; $[8, 14]$

Step 1

Write $f (x) = - \frac{1}{2} x^{2} + 3 x + 1$ as an equation.

$y = - \frac{1}{2} x^{2} + 3 x + 1$

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.

$\frac{f (14) - f (8)}{(14) - (8)}$

Step 2.2

Substitute the equation $y = - \frac{1}{2} x^{2} + 3 x + 1$ for $f (14)$ and $f (8)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- \frac{1}{2} \cdot {(14)}^{2} + 3 (14) + 1) - (- \frac{1}{2} \cdot {(8)}^{2} + 3 (8) + 1)}{(14) - (8)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $14$ to the power of $2$ .

$\frac{- \frac{1}{2} \cdot 196 + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

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

$\frac{\frac{- 1}{2} \cdot 196 + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.2.2

Factor $2$ out of $196$ .

$\frac{\frac{- 1}{2} \cdot (2 (98)) + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.2.3

Cancel the common factor.

$\frac{\frac{- 1}{2} \cdot (2 \cdot 98) + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.2.4

Rewrite the expression.

$\frac{- 1 \cdot 98 + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.3

Multiply $- 1$ by $98$ .

$\frac{- 98 + 3 (14) + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.4

Multiply $3$ by $14$ .

$\frac{- 98 + 42 + 1 - (- \frac{1}{2} \cdot 8^{2} + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Raise $8$ to the power of $2$ .

$\frac{- 98 + 42 + 1 - (- \frac{1}{2} \cdot 64 + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.2

Cancel the common factor of $2$ .

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Step 3.1.5.2.1

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

$\frac{- 98 + 42 + 1 - (\frac{- 1}{2} \cdot 64 + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.2.2

Factor $2$ out of $64$ .

$\frac{- 98 + 42 + 1 - (\frac{- 1}{2} \cdot (2 (32)) + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.2.3

Cancel the common factor.

$\frac{- 98 + 42 + 1 - (\frac{- 1}{2} \cdot (2 \cdot 32) + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.2.4

Rewrite the expression.

$\frac{- 98 + 42 + 1 - (- 1 \cdot 32 + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.3

Multiply $- 1$ by $32$ .

$\frac{- 98 + 42 + 1 - (- 32 + 3 (8) + 1)}{14 - (8)}$

Step 3.1.5.4

Multiply $3$ by $8$ .

$\frac{- 98 + 42 + 1 - (- 32 + 24 + 1)}{14 - (8)}$

Step 3.1.6

Add $- 32$ and $24$ .

$\frac{- 98 + 42 + 1 - (- 8 + 1)}{14 - (8)}$

Step 3.1.7

Add $- 8$ and $1$ .

$\frac{- 98 + 42 + 1 - - 7}{14 - (8)}$

Step 3.1.8

Multiply $- 1$ by $- 7$ .

$\frac{- 98 + 42 + 1 + 7}{14 - (8)}$

Step 3.1.9

Add $- 98$ and $42$ .

$\frac{- 56 + 1 + 7}{14 - (8)}$

Step 3.1.10

Add $- 56$ and $1$ .

$\frac{- 55 + 7}{14 - (8)}$

Step 3.1.11

Add $- 55$ and $7$ .

$\frac{- 48}{14 - (8)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $8$ .

$\frac{- 48}{14 - 8}$

Step 3.2.2

Subtract $8$ from $14$ .

$\frac{- 48}{6}$

Step 3.3

Divide $- 48$ by $6$ .

$- 8$