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

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

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

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${x | x \geq 2}$

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

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

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

Simplify the result.

Simplify the numerator.

Subtract $2$ from $2$ .

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

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

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

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

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

Subtract $4$ from $0$ .

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

Simplify the expression.

Divide $- 4$ by $2$ .

$f (2) = 2$

Multiply $- 1$ by $- 2$ .

$f (2) = 2$

The final answer is $2$ .

$2$

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

$(2, 2)$

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

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

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

Simplify the result.

Simplify the numerator.

Subtract $2$ from $3$ .

$f (3) = - \frac{\sqrt{1} - 4}{2}$

Any root of $1$ is $1$ .

$f (3) = - \frac{1 - 4}{2}$

Subtract $4$ from $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

Simplify the result.

Subtract $2$ from $4$ .

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

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 $(2, 2), (3, 1.5), (4, 1.29)$

$\begin{array}{c} x & y \\ 2 & 2 \\ 3 & 1.5 \\ 4 & 1.29 \end{array}$

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

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