Precalculus Examples

$f (x) = 0.05 x^{2} - 1.3 x + 22.8$ on $13$ , $23$

Step 1

Write $f (x) = 0.05 x^{2} - 1.3 x + 22.8$ as an equation.

$y = 0.05 x^{2} - 1.3 x + 22.8$

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.

$\frac{f (23) - f (13)}{(23) - (13)}$

Step 2.2

Substitute the equation $y = 0.05 x^{2} - 1.3 x + 22.8$ for $f (23)$ and $f (13)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(0.05 {(23)}^{2} - 1.3 \cdot 23 + 22.8) - (0.05 {(13)}^{2} - 1.3 \cdot 13 + 22.8)}{(23) - (13)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $23$ to the power of $2$ .

$\frac{0.05 \cdot 529 - 1.3 \cdot 23 + 22.8 - (0.05 \cdot 13^{2} - 1.3 \cdot 13 + 22.8)}{23 - (13)}$

Step 3.1.2

Multiply $0.05$ by $529$ .

$\frac{26.45 - 1.3 \cdot 23 + 22.8 - (0.05 \cdot 13^{2} - 1.3 \cdot 13 + 22.8)}{23 - (13)}$

Step 3.1.3

Multiply $- 1.3$ by $23$ .

$\frac{26.45 - 29.9 + 22.8 - (0.05 \cdot 13^{2} - 1.3 \cdot 13 + 22.8)}{23 - (13)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Raise $13$ to the power of $2$ .

$\frac{26.45 - 29.9 + 22.8 - (0.05 \cdot 169 - 1.3 \cdot 13 + 22.8)}{23 - (13)}$

Step 3.1.4.2

Multiply $0.05$ by $169$ .

$\frac{26.45 - 29.9 + 22.8 - (8.45 - 1.3 \cdot 13 + 22.8)}{23 - (13)}$

Step 3.1.4.3

Multiply $- 1.3$ by $13$ .

$\frac{26.45 - 29.9 + 22.8 - (8.45 - 16.9 + 22.8)}{23 - (13)}$

Step 3.1.5

Subtract $16.9$ from $8.45$ .

$\frac{26.45 - 29.9 + 22.8 - (- 8.45 + 22.8)}{23 - (13)}$

Step 3.1.6

Add $- 8.45$ and $22.8$ .

$\frac{26.45 - 29.9 + 22.8 - 1 \cdot 14.35}{23 - (13)}$

Step 3.1.7

Multiply $- 1$ by $14.35$ .

$\frac{26.45 - 29.9 + 22.8 - 14.35}{23 - (13)}$

Step 3.1.8

Subtract $29.9$ from $26.45$ .

$\frac{- 3.45 + 22.8 - 14.35}{23 - (13)}$

Step 3.1.9

Add $- 3.45$ and $22.8$ .

$\frac{19.35 - 14.35}{23 - (13)}$

Step 3.1.10

Subtract $14.35$ from $19.35$ .

$\frac{5}{23 - (13)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $13$ .

$\frac{5}{23 - 13}$

Step 3.2.2

Subtract $13$ from $23$ .

$\frac{5}{10}$

Step 3.3

Cancel the common factor of $5$ and $10$ .

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Step 3.3.1

Factor $5$ out of $5$ .

$\frac{5 (1)}{10}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 3.3.2.2

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 2}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{2}$