Finite Math Examples

$f (x) = x^{2} + \frac{1}{2}$ , $(- 3, 4)$

Substitute using the average rate of change formula.

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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.

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

Substitute the equation $y = x^{2} + \frac{1}{2}$ for $f (4)$ and $f (- 3)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Simplify the expression.

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

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

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Cancel the common factor of $2$ .

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Write $2$ as a fraction with denominator $1$ .

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

Factor out the greatest common factor $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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Multiply $\frac{1}{1}$ and $\frac{1}{1}$ .

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

Divide $1$ by $1$ .

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

Simplify the numerator.

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Remove parentheses around $4$ .

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

Raise $4$ to the power of $2$ .

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

Multiply $2$ by $16$ .

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

Raise $- 3$ to the power of $2$ .

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

To write $\frac{9}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Cancel the common factor of $2$ .

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

Simplify.

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

Add $32$ and $1$ .

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

Subtract $19$ from $33$ .

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

Simplify the denominator.

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Rewrite.

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

Apply the distributive property.

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

Multiply $2$ by $4$ .

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

Multiply $- 1$ by $- 3$ .

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

Multiply $2$ by $3$ .

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

Add $8$ and $6$ .

$\frac{14}{14}$

Divide $14$ by $14$ .

$1$

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