Graph y=-1/3 square root of -x+2

$y = - \frac{1}{3} \sqrt{- x} + 2$

Step 1

Find the domain for $y = - \frac{1}{3} \cdot \sqrt{- x} + 2$ 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 $\sqrt{- x}$ greater than or equal to $0$ to find where the expression is defined.

$- x \geq 0$

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x \leq 0$

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

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Step 2

To find the radical expression end point, substitute the $x$ value $0$ , which is the least value in the domain, into $f (x) = - \frac{\sqrt{- x}}{3} + 2$ .

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

$f (0) = - \frac{\sqrt{- (0)}}{3} + 2$

Simplify the result.

Simplify each term.

Simplify the numerator.

Multiply $- 1$ by $0$ .

$f (0) = - \frac{\sqrt{0}}{3} + 2$

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

$f (0) = - \frac{\sqrt{0^{2}}}{3} + 2$

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

$f (0) = - \frac{0}{3} + 2$

Divide $0$ by $3$ .

$f (0) = - 0 + 2$

Multiply $- 1$ by $0$ .

$f (0) = 0 + 2$

Add $0$ and $2$ .

$f (0) = 2$

The final answer is $2$ .

$2$

Step 3

The radical expression end point is $(0, 2)$ .

$(0, 2)$

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

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

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

Simplify the result.

Multiply $- 1$ by $- 2$ .

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

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

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

Substitute the $x$ value $- 1$ into $f (x) = - \frac{\sqrt{- x}}{3} + 2$ . In this case, the point is $(- 1, \frac{5}{3})$ .

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

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

Simplify the result.

Simplify the numerator.

Multiply $- 1$ by $- 1$ .

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

Any root of $1$ is $1$ .

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $2$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$f (- 1) = \frac{- 1 + 6}{3}$

Add $- 1$ and $6$ .

$f (- 1) = \frac{5}{3}$

The final answer is $\frac{5}{3}$ .

$y = \frac{5}{3}$

The square root can be graphed using the points around the vertex $(0, 2), (- 2, 1.53), (- 1, 1.67)$

$\begin{array}{c} x & y \\ - 2 & 1.53 \\ - 1 & 1.67 \\ 0 & 2 \end{array}$

Step 5

Graph y=-1/3 square root of -x+2

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