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

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

Step 1

Find the domain for $y = - \frac{1}{2} \cdot \sqrt{x - 4}$ 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 $f (x) = - \frac{\sqrt{x - 4}}{2}$ .

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

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $4$ .

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

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

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

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

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

Simplify the expression.

Divide $0$ by $2$ .

$f (4) = - 0$

Multiply $- 1$ by $0$ .

$f (4) = 0$

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(4, 0)$ .

$(4, 0)$

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 $f (x) = - \frac{\sqrt{x - 4}}{2}$ . In this case, the point is $(5, - \frac{1}{2})$ .

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

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

Simplify the result.

Simplify the numerator.

Subtract $4$ from $5$ .

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

Any root of $1$ is $1$ .

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

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

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

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

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

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

Simplify the result.

Subtract $4$ from $6$ .

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

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

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

The square root can be graphed using the points around the vertex $(4, 0), (5, - 0.5), (6, - 0.71)$

$\begin{array}{c} x & y \\ 4 & 0 \\ 5 & - 0.5 \\ 6 & - 0.71 \end{array}$

Step 5

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

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