Precalculus Examples

$f (x) = \sqrt{2 x + 8}$ on the interval $[- 2, 5]$

Step 1

Write $f (x) = \sqrt{2 x + 8}$ as an equation.

$y = \sqrt{2 x + 8}$

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.

$\frac{f (5) - f (- 2)}{(5) - (- 2)}$

Step 2.2

Substitute the equation $y = \sqrt{2 x + 8}$ for $f (5)$ and $f (- 2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{2 (5) + 8}) - (\sqrt{2 (- 2) + 8})}{(5) - (- 2)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $2$ by $5$ .

$\frac{\sqrt{10 + 8} - \sqrt{2 (- 2) + 8}}{5 - (- 2)}$

Step 3.1.2

Add $10$ and $8$ .

$\frac{\sqrt{18} - \sqrt{2 (- 2) + 8}}{5 - (- 2)}$

Step 3.1.3

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.1.3.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)} - \sqrt{2 (- 2) + 8}}{5 - (- 2)}$

Step 3.1.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 2} - \sqrt{2 (- 2) + 8}}{5 - (- 2)}$

Step 3.1.4

Pull terms out from under the radical.

$\frac{3 \sqrt{2} - \sqrt{2 (- 2) + 8}}{5 - (- 2)}$

Step 3.1.5

Multiply $2$ by $- 2$ .

$\frac{3 \sqrt{2} - \sqrt{- 4 + 8}}{5 - (- 2)}$

Step 3.1.6

Add $- 4$ and $8$ .

$\frac{3 \sqrt{2} - \sqrt{4}}{5 - (- 2)}$

Step 3.1.7

Rewrite $4$ as $2^{2}$ .

$\frac{3 \sqrt{2} - \sqrt{2^{2}}}{5 - (- 2)}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\frac{3 \sqrt{2} - 1 \cdot 2}{5 - (- 2)}$

Step 3.1.9

Multiply $- 1$ by $2$ .

$\frac{3 \sqrt{2} - 2}{5 - (- 2)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 2$ .

$\frac{3 \sqrt{2} - 2}{5 + 2}$

Step 3.2.2

Add $5$ and $2$ .

$\frac{3 \sqrt{2} - 2}{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{2} - 2}{7}$

Decimal Form:

$0.32037724 \dots$