Graph y=1/2 square root of x-4-2

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

Step 1

Find the domain for $y = \frac{1}{2} \cdot \sqrt{x - 4} - 2$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Set the radicand in $\sqrt{x - 4}$ greater than or equal to $0$ to find where the expression is defined.

$x - 4 \geq 0$

Add $4$ to both sides of the inequality.

$x \geq 4$

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

Interval Notation:

$[4, \infty)$

Set-Builder Notation:

${x | x \geq 4}$

Interval Notation:

$[4, \infty)$

Set-Builder Notation:

${x | x \geq 4}$

Step 2

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

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

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $4$ .

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

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

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

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

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

Subtract $4$ from $0$ .

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

Divide $- 4$ by $2$ .

$f (4) = - 2$

The final answer is $- 2$ .

$- 2$

Step 3

The radical expression end point is $(4, - 2)$ .

$(4, - 2)$

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.

Substitute the $x$ value $5$ into $f (x) = \frac{\sqrt{x - 4} - 4}{2}$ . In this case, the point is $(5, - \frac{3}{2})$ .

Replace the variable $x$ with $5$ in the expression.

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $5$ .

$f (5) = \frac{\sqrt{1} - 4}{2}$

Any root of $1$ is $1$ .

$f (5) = \frac{1 - 4}{2}$

Subtract $4$ from $1$ .

$f (5) = \frac{- 3}{2}$

Move the negative in front of the fraction.

$f (5) = - \frac{3}{2}$

The final answer is $- \frac{3}{2}$ .

$y = - \frac{3}{2}$

Substitute the $x$ value $6$ into $f (x) = \frac{\sqrt{x - 4} - 4}{2}$ . In this case, the point is $(6, \frac{\sqrt{2} - 4}{2})$ .

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{\sqrt{(6) - 4} - 4}{2}$

Simplify the result.

Subtract $4$ from $6$ .

$f (6) = \frac{\sqrt{2} - 4}{2}$

The final answer is $\frac{\sqrt{2} - 4}{2}$ .

$y = \frac{\sqrt{2} - 4}{2}$

The square root can be graphed using the points around the vertex $(4, - 2), (5, - 1.5), (6, - 1.29)$

$\begin{array}{c} x & y \\ 4 & - 2 \\ 5 & - 1.5 \\ 6 & - 1.29 \end{array}$

Step 5

Graph y=1/2 square root of x-4-2

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