Algebra Examples

$y = \sqrt{x}$ , $(- 4, 1)$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

$\frac{f (1) - f (- 4)}{(1) - (- 4)}$

Step 1.2

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

$\frac{(\sqrt{1}) - (\sqrt{- 4})}{(1) - (- 4)}$

Step 2

Simplify the expression.

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Step 2.1

Pull out imaginary unit $i$ .

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Step 2.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{\sqrt{1} - \sqrt{- 1 (4)}}{(1) - (- 4)}$

Step 2.1.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$\frac{\sqrt{1} - (\sqrt{- 1} \cdot \sqrt{4})}{(1) - (- 4)}$

Step 2.1.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{\sqrt{1} - (i \cdot \sqrt{4})}{(1) - (- 4)}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Any root of $1$ is $1$ .

$\frac{1 - i \cdot \sqrt{4}}{1 - (- 4)}$

Step 2.2.2

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

$\frac{1 - i \cdot \sqrt{2^{2}}}{1 - (- 4)}$

Step 2.2.3

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

$\frac{1 - i \cdot 2}{1 - (- 4)}$

Step 2.2.4

Multiply $2$ by $- 1$ .

$\frac{1 - 2 i}{1 - (- 4)}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Multiply $- 1$ by $- 4$ .

$\frac{1 - 2 i}{1 + 4}$

Step 2.3.2

Add $1$ and $4$ .

$\frac{1 - 2 i}{5}$

Step 2.4

Split the fraction $\frac{1 - 2 i}{5}$ into two fractions.

$\frac{1}{5} + \frac{- 2 i}{5}$

Step 2.5

Move the negative in front of the fraction.

$\frac{1}{5} - \frac{2 i}{5}$