Finite Math Examples

$f (x) = 3 x^{2} - \frac{x}{2}$ , $[8, 9]$

Step 1

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

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

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.

$\frac{f (9) - f (8)}{(9) - (8)}$

Step 2.2

Substitute the equation $y = 3 x^{2} - \frac{x}{2}$ for $f (9)$ and $f (8)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(3 {(9)}^{2} - \frac{9}{2}) - (3 {(8)}^{2} - \frac{8}{2})}{(9) - (8)}$

Step 3

Simplify the expression.

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Step 3.1

Cancel the common factor of $8$ and $2$ .

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Step 3.1.1

Factor $2$ out of $8$ .

$\frac{3 {(9)}^{2} - \frac{9}{2} - (3 {(8)}^{2} - \frac{2 \cdot 4}{2})}{(9) - (8)}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $2$ out of $2$ .

$\frac{3 {(9)}^{2} - \frac{9}{2} - (3 {(8)}^{2} - \frac{2 \cdot 4}{2 (1)})}{(9) - (8)}$

Step 3.1.2.2

Cancel the common factor.

$\frac{3 {(9)}^{2} - \frac{9}{2} - (3 {(8)}^{2} - \frac{2 \cdot 4}{2 \cdot 1})}{(9) - (8)}$

Step 3.1.2.3

Rewrite the expression.

$\frac{3 {(9)}^{2} - \frac{9}{2} - (3 {(8)}^{2} - \frac{4}{1})}{(9) - (8)}$

Step 3.1.2.4

Divide $4$ by $1$ .

$\frac{3 {(9)}^{2} - \frac{9}{2} - (3 {(8)}^{2} - 1 \cdot 4)}{(9) - (8)}$

Step 3.2

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

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Step 3.2.1

Multiply $\frac{3 \cdot 9^{2} - \frac{9}{2} - (3 \cdot 8^{2} - 1 \cdot 4)}{9 - (8)}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{3 \cdot 9^{2} - \frac{9}{2} - (3 \cdot 8^{2} - 1 \cdot 4)}{9 - (8)}$

Step 3.2.2

Combine.

$\frac{2 (3 \cdot 9^{2} - \frac{9}{2} - (3 \cdot 8^{2} - 1 \cdot 4))}{2 (9 - (8))}$

Step 3.3

Apply the distributive property.

$\frac{2 (3 \cdot 9^{2}) + 2 (- \frac{9}{2}) + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Move the leading negative in $- \frac{9}{2}$ into the numerator.

$\frac{2 (3 \cdot 9^{2}) + 2 (\frac{- 9}{2}) + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.4.2

Cancel the common factor.

$\frac{2 (3 \cdot 9^{2}) + 2 (\frac{- 9}{2}) + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.4.3

Rewrite the expression.

$\frac{2 (3 \cdot 9^{2}) - 9 + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $3$ .

$\frac{6 \cdot 9^{2} - 9 + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.2

Raise $9$ to the power of $2$ .

$\frac{6 \cdot 81 - 9 + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.3

Multiply $6$ by $81$ .

$\frac{486 - 9 + 2 (- (3 \cdot 8^{2} - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

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

$\frac{486 - 9 + 2 (- (3 \cdot 64 - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.4.2

Multiply $3$ by $64$ .

$\frac{486 - 9 + 2 (- (192 - 1 \cdot 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.4.3

Multiply $- 1$ by $4$ .

$\frac{486 - 9 + 2 (- (192 - 4))}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.5

Subtract $4$ from $192$ .

$\frac{486 - 9 + 2 (- 1 \cdot 188)}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.6

Multiply $2 (- 1 \cdot 188)$ .

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Step 3.5.6.1

Multiply $- 1$ by $188$ .

$\frac{486 - 9 + 2 \cdot - 188}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.6.2

Multiply $2$ by $- 188$ .

$\frac{486 - 9 - 376}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.7

Subtract $9$ from $486$ .

$\frac{477 - 376}{2 \cdot 9 + 2 (- (8))}$

Step 3.5.8

Subtract $376$ from $477$ .

$\frac{101}{2 \cdot 9 + 2 (- (8))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $2$ by $9$ .

$\frac{101}{18 + 2 (- (8))}$

Step 3.6.2

Multiply $2 (- (8))$ .

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Step 3.6.2.1

Multiply $- 1$ by $8$ .

$\frac{101}{18 + 2 \cdot - 8}$

Step 3.6.2.2

Multiply $2$ by $- 8$ .

$\frac{101}{18 - 16}$

Step 3.6.3

Subtract $16$ from $18$ .

$\frac{101}{2}$