Algebra Examples

$f (x) = - \frac{1}{4} \sqrt{8 - 4 x} + 1$

Step 1

Find the domain for $y = - \frac{1}{4} \cdot \sqrt{8 - 4 x} + 1$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{2 - x}$ greater than or equal to $0$ to find where the expression is defined.

$2 - x \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $2$ from both sides of the inequality.

$- x \geq - 2$

Step 1.2.2

Divide each term in $- x \geq - 2$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 2}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 2}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 2}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 2$ by $- 1$ .

$x \leq 2$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 2]$

Set-Builder Notation:

${x | x \leq 2}$

Interval Notation:

$(- \infty, 2]$

Set-Builder Notation:

${x | x \leq 2}$

Step 2

To find the radical expression end point, substitute the $x$ value $2$ , which is the least value in the domain, into $f (x) = - \frac{\sqrt{2 - x} - 2}{2}$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{\sqrt{2 - (2)} - 2}{2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $- 1$ by $2$ .

$f (2) = - \frac{\sqrt{2 - 2} - 2}{2}$

Step 2.2.1.2

Subtract $2$ from $2$ .

$f (2) = - \frac{\sqrt{0} - 2}{2}$

Step 2.2.1.3

Rewrite $0$ as $0^{2}$ .

$f (2) = - \frac{\sqrt{0^{2}} - 2}{2}$

Step 2.2.1.4

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

$f (2) = - \frac{0 - 2}{2}$

Step 2.2.1.5

Subtract $2$ from $0$ .

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

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Divide $- 2$ by $2$ .

$f (2) = 1$

Step 2.2.2.2

Multiply $- 1$ by $- 1$ .

$f (2) = 1$

Step 2.2.3

The final answer is $1$ .

$1$

Step 3

The radical expression end point is $(2, 1)$ .

$(2, 1)$

Step 4

Select a few $x$ values from the domain. It would be more useful to select the values so that they are next to the $x$ value of the radical expression end point.

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Step 4.1

Substitute the $x$ value $0$ into $f (x) = - \frac{\sqrt{2 - x} - 2}{2}$ . In this case, the point is $(0, - \frac{\sqrt{2} - 2}{2})$ .

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Step 4.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{\sqrt{2 - (0)} - 2}{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Multiply $- 1$ by $0$ .

$f (0) = - \frac{\sqrt{2 + 0} - 2}{2}$

Step 4.1.2.1.2

Add $2$ and $0$ .

$f (0) = - \frac{\sqrt{2} - 2}{2}$

Step 4.1.2.2

The final answer is $- \frac{\sqrt{2} - 2}{2}$ .

$y = - \frac{\sqrt{2} - 2}{2}$

Step 4.2

Substitute the $x$ value $1$ into $f (x) = - \frac{\sqrt{2 - x} - 2}{2}$ . In this case, the point is $(1, \frac{1}{2})$ .

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Step 4.2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{\sqrt{2 - (1)} - 2}{2}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Multiply $- 1$ by $1$ .

$f (1) = - \frac{\sqrt{2 - 1} - 2}{2}$

Step 4.2.2.1.2

Subtract $1$ from $2$ .

$f (1) = - \frac{\sqrt{1} - 2}{2}$

Step 4.2.2.1.3

Any root of $1$ is $1$ .

$f (1) = - \frac{1 - 2}{2}$

Step 4.2.2.1.4

Subtract $2$ from $1$ .

$f (1) = - \frac{- 1}{2}$

Step 4.2.2.2

Move the negative in front of the fraction.

$f (1) = \frac{1}{2}$

Step 4.2.2.3

The final answer is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(2, 1), (0, 0.29), (1, 0.5)$

$\begin{array}{c} x & y \\ 0 & 0.29 \\ 1 & 0.5 \\ 2 & 1 \end{array}$

Step 5