Precalculus Examples

$f (x) = 2 x^{2} - 4$ , $[5, 6]$

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.

$\frac{f (6) - f (5)}{(6) - (5)}$

Substitute the equation $y = 2 x^{2} - 4$ for $f (6)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(2 {(6)}^{2} - 4) - (2 {(5)}^{2} - 4)}{(6) - (5)}$

Simplify the expression.

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Simplify the numerator.

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Raise $6$ to the power of $2$ .

$\frac{2 \cdot 36 - 4 - (2 {(5)}^{2} - 4)}{(6) - (5)}$

Multiply $2$ by $36$ .

$\frac{72 - 4 - (2 {(5)}^{2} - 4)}{(6) - (5)}$

Simplify each term.

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Raise $5$ to the power of $2$ .

$\frac{72 - 4 - (2 \cdot 25 - 4)}{(6) - (5)}$

Multiply $2$ by $25$ .

$\frac{72 - 4 - (50 - 4)}{(6) - (5)}$

Subtract $4$ from $50$ .

$\frac{72 - 4 - 1 \cdot 46}{(6) - (5)}$

Multiply $- 1$ by $46$ .

$\frac{72 - 4 - 46}{(6) - (5)}$

Subtract $4$ from $72$ .

$\frac{68 - 46}{(6) - (5)}$

Subtract $46$ from $68$ .

$\frac{22}{(6) - (5)}$

Simplify the denominator.

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Multiply $- 1$ by $5$ .

$\frac{22}{6 - 5}$

Subtract $5$ from $6$ .

$\frac{22}{1}$

Divide $22$ by $1$ .

$22$

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