Graph y=4+ square root of 6-x

$y = 4 + \sqrt{6 - x}$

Step 1

Reorder $6$ and $- x$ .

$y = 4 + \sqrt{- x + 6}$

Step 2

Find the domain for $y = 4 + \sqrt{6 - x}$ 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 $\sqrt{- x + 6}$ greater than or equal to $0$ to find where the expression is defined.

$- x + 6 \geq 0$

Solve for $x$ .

Subtract $6$ from both sides of the inequality.

$- x \geq - 6$

Divide each term in $- x \geq - 6$ by $- 1$ and simplify.

Divide each term in $- x \geq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x \leq 6$

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

Interval Notation:

$(- \infty, 6]$

Set-Builder Notation:

${x | x \leq 6}$

Interval Notation:

$(- \infty, 6]$

Set-Builder Notation:

${x | x \leq 6}$

Step 3

To find the radical expression end point, substitute the $x$ value $6$ , which is the least value in the domain, into $f (x) = 4 + \sqrt{- x + 6}$ .

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $6$ .

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

Add $- 6$ and $6$ .

$f (6) = 4 + \sqrt{0}$

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

$f (6) = 4 + \sqrt{0^{2}}$

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

$f (6) = 4 + 0$

Add $4$ and $0$ .

$f (6) = 4$

The final answer is $4$ .

$4$

Step 4

The radical expression end point is $(6, 4)$ .

$(6, 4)$

Step 5

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

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

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

Simplify the result.

Simplify each term.

Multiply $- 1$ by $4$ .

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

Add $- 4$ and $6$ .

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

The final answer is $4 + \sqrt{2}$ .

$y = 4 + \sqrt{2}$

Substitute the $x$ value $5$ into $f (x) = 4 + \sqrt{- x + 6}$ . In this case, the point is $(5, 5)$ .

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

$f (5) = 4 + \sqrt{- (5) + 6}$

Simplify the result.

Simplify each term.

Multiply $- 1$ by $5$ .

$f (5) = 4 + \sqrt{- 5 + 6}$

Add $- 5$ and $6$ .

$f (5) = 4 + \sqrt{1}$

Any root of $1$ is $1$ .

$f (5) = 4 + 1$

Add $4$ and $1$ .

$f (5) = 5$

The final answer is $5$ .

$y = 5$

The square root can be graphed using the points around the vertex $(6, 4), (4, 5.41), (5, 5)$

$\begin{array}{c} x & y \\ 4 & 5.41 \\ 5 & 5 \\ 6 & 4 \end{array}$

Step 6

Graph y=4+ square root of 6-x

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