Precalculus Examples

$g (x) = - x^{2} + 24 x - 109$ , $s i \cdot 8 \leq x \leq 14$

Step 1

Write $g (x) = - x^{2} + 24 x - 109$ as an equation.

$y = - x^{2} + 24 x - 109$

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.

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

Step 2.2

Substitute the equation $y = - x^{2} + 24 x - 109$ for $f (14)$ and $f (8 s i)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- {(14)}^{2} + 24 (14) - 109) - (- {(8 s i)}^{2} + 24 (8 s i) - 109)}{(14) - (8 s i)}$

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 $2$ .

$\frac{- 1 \cdot 196 + 24 (14) - 109 - (- {(8 s i)}^{2} + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.2

Multiply $- 1$ by $196$ .

$\frac{- 196 + 24 (14) - 109 - (- {(8 s i)}^{2} + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.3

Multiply $24$ by $14$ .

$\frac{- 196 + 336 - 109 - (- {(8 s i)}^{2} + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1.4.1.1

Apply the product rule to $8 s i$ .

$\frac{- 196 + 336 - 109 - (- ({(8 s)}^{2} i^{2}) + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.1.2

Apply the product rule to $8 s$ .

$\frac{- 196 + 336 - 109 - (- (8^{2} s^{2} i^{2}) + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.2

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

$\frac{- 196 + 336 - 109 - (- (64 s^{2} i^{2}) + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.3

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 196 + 336 - 109 - (- (64 s^{2} \cdot - 1) + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.4

Multiply $- 1$ by $64$ .

$\frac{- 196 + 336 - 109 - (- (- 64 s^{2}) + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.5

Multiply $- 64$ by $- 1$ .

$\frac{- 196 + 336 - 109 - (64 s^{2} + 24 \cdot 8 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.4.6

Multiply $24$ by $8$ .

$\frac{- 196 + 336 - 109 - (64 s^{2} + 192 s i - 109)}{14 - 1 \cdot 8 s i}$

Step 3.1.5

Apply the distributive property.

$\frac{- 196 + 336 - 109 - (64 s^{2}) - (192 s i) - - 109}{14 - 1 \cdot 8 s i}$

Step 3.1.6

Simplify.

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Step 3.1.6.1

Multiply $64$ by $- 1$ .

$\frac{- 196 + 336 - 109 - 64 s^{2} - (192 s i) - - 109}{14 - 1 \cdot 8 s i}$

Step 3.1.6.2

Multiply $192$ by $- 1$ .

$\frac{- 196 + 336 - 109 - 64 s^{2} - 192 (s i) - - 109}{14 - 1 \cdot 8 s i}$

Step 3.1.6.3

Multiply $- 1$ by $- 109$ .

$\frac{- 196 + 336 - 109 - 64 s^{2} - 192 (s i) + 109}{14 - 1 \cdot 8 s i}$

Step 3.1.7

Add $- 196$ and $336$ .

$\frac{140 - 109 - 64 s^{2} - 192 s i + 109}{14 - 1 \cdot 8 s i}$

Step 3.1.8

Subtract $109$ from $140$ .

$\frac{31 - 64 s^{2} - 192 s i + 109}{14 - 1 \cdot 8 s i}$

Step 3.1.9

Add $31$ and $109$ .

$\frac{- 64 s^{2} - 192 s i + 140}{14 - 1 \cdot 8 s i}$

Step 3.1.10

Factor $4$ out of $- 64 s^{2} - 192 s i + 140$ .

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Step 3.1.10.1

Factor $4$ out of $- 64 s^{2}$ .

$\frac{4 (- 16 s^{2}) - 192 s i + 140}{14 - 1 \cdot 8 s i}$

Step 3.1.10.2

Factor $4$ out of $- 192 s i$ .

$\frac{4 (- 16 s^{2}) + 4 (- 48 s i) + 140}{14 - 1 \cdot 8 s i}$

Step 3.1.10.3

Factor $4$ out of $140$ .

$\frac{4 (- 16 s^{2}) + 4 (- 48 s i) + 4 (35)}{14 - 1 \cdot 8 s i}$

Step 3.1.10.4

Factor $4$ out of $4 (- 16 s^{2}) + 4 (- 48 s i)$ .

$\frac{4 (- 16 s^{2} - 48 s i) + 4 (35)}{14 - 1 \cdot 8 s i}$

Step 3.1.10.5

Factor $4$ out of $4 (- 16 s^{2} - 48 s i) + 4 (35)$ .

$\frac{4 (- 16 s^{2} - 48 s i + 35)}{14 - 1 \cdot 8 s i}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $8$ .

$\frac{4 (- 16 s^{2} - 48 s i + 35)}{14 - 8 s i}$

Step 3.2.2

Factor $2$ out of $14 - 8 s i$ .

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Step 3.2.2.1

Factor $2$ out of $14$ .

$\frac{4 (- 16 s^{2} - 48 s i + 35)}{2 (7) - 8 s i}$

Step 3.2.2.2

Factor $2$ out of $- 8 s i$ .

$\frac{4 (- 16 s^{2} - 48 s i + 35)}{2 (7) + 2 (- 4 s i)}$

Step 3.2.2.3

Factor $2$ out of $2 (7) + 2 (- 4 s i)$ .

$\frac{4 (- 16 s^{2} - 48 s i + 35)}{2 (7 - 4 s i)}$

Step 3.3

Simplify terms.

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Step 3.3.1

Cancel the common factor of $4$ and $2$ .

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Step 3.3.1.1

Factor $2$ out of $4 (- 16 s^{2} - 48 s i + 35)$ .

$\frac{2 (2 (- 16 s^{2} - 48 s i + 35))}{2 (7 - 4 s i)}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Cancel the common factor.

$\frac{2 (2 (- 16 s^{2} - 48 s i + 35))}{2 (7 - 4 s i)}$

Step 3.3.1.2.2

Rewrite the expression.

$\frac{2 (- 16 s^{2} - 48 s i + 35)}{7 - 4 s i}$

Step 3.3.2

Factor $- 1$ out of $- 16 s^{2}$ .

$\frac{2 (- (16 s^{2}) - 48 s i + 35)}{7 - 4 s i}$

Step 3.3.3

Factor $- 1$ out of $- 48 s i$ .

$\frac{2 (- (16 s^{2}) - (48 s i) + 35)}{7 - 4 s i}$

Step 3.3.4

Factor $- 1$ out of $- (16 s^{2}) - (48 s i)$ .

$\frac{2 (- (16 s^{2} + 48 s i) + 35)}{7 - 4 s i}$

Step 3.3.5

Rewrite $35$ as $- 1 (- 35)$ .

$\frac{2 (- (16 s^{2} + 48 s i) - 1 (- 35))}{7 - 4 s i}$

Step 3.3.6

Factor $- 1$ out of $- (16 s^{2} + 48 s i) - 1 (- 35)$ .

$\frac{2 (- (16 s^{2} + 48 s i - 35))}{7 - 4 s i}$

Step 3.3.7

Simplify the expression.

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Step 3.3.7.1

Rewrite $- (16 s^{2} + 48 s i - 35)$ as $- 1 (16 s^{2} + 48 s i - 35)$ .

$\frac{2 (- 1 (16 s^{2} + 48 s i - 35))}{7 - 4 s i}$

Step 3.3.7.2

Move the negative in front of the fraction.

$- \frac{2 (16 s^{2} + 48 s i - 35)}{7 - 4 s i}$