Graph y = square root of x/2

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

Step 1

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

$2 x \geq 0$

Divide each term in $2 x \geq 0$ by $2$ and simplify.

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x \geq 0$

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = \frac{\sqrt{2 x}}{2}$ .

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

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

Simplify the result.

Simplify the numerator.

Multiply $2$ by $0$ .

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

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

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

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

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

Divide $0$ by $2$ .

$f (0) = 0$

The final answer is $0$ .

$0$

Step 3

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

$(0, 0)$

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

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

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

Simplify the result.

Multiply $2$ by $1$ .

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

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

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

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

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

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

Simplify the result.

Simplify the numerator.

Multiply $2$ by $2$ .

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

Rewrite $4$ as $2^{2}$ .

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

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

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

Divide $2$ by $2$ .

$f (2) = 1$

The final answer is $1$ .

$y = 1$

The square root can be graphed using the points around the vertex $(0, 0), (1, 0.71), (2, 1)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0.71 \\ 2 & 1 \end{array}$

Step 5

Graph y = square root of x/2

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