Algebra Examples

$y = \frac{1}{2} \sqrt{4 - x} + 4$

Step 1

Find the domain for $y = \frac{1}{2} \cdot \sqrt{4 - x} + 4$ 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 $\sqrt{4 - x}$ greater than or equal to $0$ to find where the expression is defined.

$4 - x \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $4$ from both sides of the inequality.

$- x \geq - 4$

Step 1.2.2

Divide each term in $- x \geq - 4$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- x \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 4}{- 1}$

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.

$\frac{x}{1} \leq \frac{- 4}{- 1}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x \leq \frac{- 4}{- 1}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $- 4$ by $- 1$ .

$x \leq 4$

Step 1.3

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

Interval Notation:

$(- \infty, 4]$

Set-Builder Notation:

${x | x \leq 4}$

Interval Notation:

$(- \infty, 4]$

Set-Builder Notation:

${x | x \leq 4}$

Step 2

To find the radical expression end point, substitute the $x$ value $4$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{4 - x} + 8}{2}$ .

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Step 2.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{\sqrt{4 - (4)} + 8}{2}$

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 $4$ .

$f (4) = \frac{\sqrt{4 - 4} + 8}{2}$

Step 2.2.1.2

Subtract $4$ from $4$ .

$f (4) = \frac{\sqrt{0} + 8}{2}$

Step 2.2.1.3

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

$f (4) = \frac{\sqrt{0^{2}} + 8}{2}$

Step 2.2.1.4

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

$f (4) = \frac{0 + 8}{2}$

Step 2.2.1.5

Add $0$ and $8$ .

$f (4) = \frac{8}{2}$

Step 2.2.2

Divide $8$ by $2$ .

$f (4) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 3

The radical expression end point is $(4, 4)$ .

$(4, 4)$

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 $2$ into $f (x) = \frac{\sqrt{4 - x} + 8}{2}$ . In this case, the point is $(2, \frac{\sqrt{2} + 8}{2})$ .

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Step 4.1.1

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

$f (2) = \frac{\sqrt{4 - (2)} + 8}{2}$

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 $2$ .

$f (2) = \frac{\sqrt{4 - 2} + 8}{2}$

Step 4.1.2.1.2

Subtract $2$ from $4$ .

$f (2) = \frac{\sqrt{2} + 8}{2}$

Step 4.1.2.2

The final answer is $\frac{\sqrt{2} + 8}{2}$ .

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

Step 4.2

Substitute the $x$ value $3$ into $f (x) = \frac{\sqrt{4 - x} + 8}{2}$ . In this case, the point is $(3, \frac{9}{2})$ .

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Step 4.2.1

Replace the variable $x$ with $3$ in the expression.

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

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 $3$ .

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

Step 4.2.2.1.2

Subtract $3$ from $4$ .

$f (3) = \frac{\sqrt{1} + 8}{2}$

Step 4.2.2.1.3

Any root of $1$ is $1$ .

$f (3) = \frac{1 + 8}{2}$

Step 4.2.2.1.4

Add $1$ and $8$ .

$f (3) = \frac{9}{2}$

Step 4.2.2.2

The final answer is $\frac{9}{2}$ .

$y = \frac{9}{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(4, 4), (2, 4.71), (3, 4.5)$

$\begin{array}{c} x & y \\ 2 & 4.71 \\ 3 & 4.5 \\ 4 & 4 \end{array}$

Step 5