Graph y = square root of 3-x

$y = \sqrt{3 - x}$

Step 1

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

$3 - x \geq 0$

Solve for $x$ .

Subtract $3$ from both sides of the inequality.

$- x \geq - 3$

Divide each term in $- x \geq - 3$ by $- 1$ and simplify.

Divide each term in $- x \geq - 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x \leq 3$

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

Interval Notation:

$(- \infty, 3]$

Set-Builder Notation:

${x | x \leq 3}$

Interval Notation:

$(- \infty, 3]$

Set-Builder Notation:

${x | x \leq 3}$

Step 2

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

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

$f (3) = \sqrt{3 - (3)}$

Simplify the result.

Multiply $- 1$ by $3$ .

$f (3) = \sqrt{3 - 3}$

Subtract $3$ from $3$ .

$f (3) = \sqrt{0}$

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

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

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

$f (3) = 0$

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(3, 0)$ .

$(3, 0)$

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 $f (x) = \sqrt{3 - x}$ . In this case, the point is $(1, \sqrt{2})$ .

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

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

Simplify the result.

Multiply $- 1$ by $1$ .

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

Subtract $1$ from $3$ .

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

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

$y = \sqrt{2}$

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

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

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

Simplify the result.

Multiply $- 1$ by $2$ .

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

Subtract $2$ from $3$ .

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

Any root of $1$ is $1$ .

$f (2) = 1$

The final answer is $1$ .

$y = 1$

The square root can be graphed using the points around the vertex $(3, 0), (1, 1.41), (2, 1)$

$\begin{array}{c} x & y \\ 1 & 1.41 \\ 2 & 1 \\ 3 & 0 \end{array}$

Step 5

Graph y = square root of 3-x

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