Trigonometry Examples

$h (x) = - \sqrt{x + 11}$

Step 1

Find the domain for $y = - \sqrt{x + 11}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

Tap for more steps...

Step 1.1

Set the radicand in $\sqrt{x + 11}$ greater than or equal to $0$ to find where the expression is defined.

$x + 11 \geq 0$

Step 1.2

Subtract $11$ from both sides of the inequality.

$x \geq - 11$

Step 1.3

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

Interval Notation:

$[- 11, \infty)$

Set-Builder Notation:

${x | x \geq - 11}$

Interval Notation:

$[- 11, \infty)$

Set-Builder Notation:

${x | x \geq - 11}$

Step 2

To find the radical expression end point, substitute the $x$ value $- 11$ , which is the least value in the domain, into $f (x) = - \sqrt{x + 11}$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $- 11$ in the expression.

$f (- 11) = - \sqrt{(- 11) + 11}$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

Add $- 11$ and $11$ .

$f (- 11) = - \sqrt{0}$

Step 2.2.2

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

$f (- 11) = - \sqrt{0^{2}}$

Step 2.2.3

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

$f (- 11) = - 0$

Step 2.2.4

Multiply $- 1$ by $0$ .

$f (- 11) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(- 11, 0)$ .

$(- 11, 0)$

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.

Tap for more steps...

Step 4.1

Substitute the $x$ value $- 10$ into $f (x) = - \sqrt{x + 11}$ . In this case, the point is $(- 10, - 1)$ .

Tap for more steps...

Step 4.1.1

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = - \sqrt{(- 10) + 11}$

Step 4.1.2

Simplify the result.

Tap for more steps...

Step 4.1.2.1

Add $- 10$ and $11$ .

$f (- 10) = - \sqrt{1}$

Step 4.1.2.2

Any root of $1$ is $1$ .

$f (- 10) = - 1 \cdot 1$

Step 4.1.2.3

Multiply $- 1$ by $1$ .

$f (- 10) = - 1$

Step 4.1.2.4

The final answer is $- 1$ .

$y = - 1$

Step 4.2

Substitute the $x$ value $- 9$ into $f (x) = - \sqrt{x + 11}$ . In this case, the point is $(- 9, - \sqrt{2})$ .

Tap for more steps...

Step 4.2.1

Replace the variable $x$ with $- 9$ in the expression.

$f (- 9) = - \sqrt{(- 9) + 11}$

Step 4.2.2

Simplify the result.

Tap for more steps...

Step 4.2.2.1

Add $- 9$ and $11$ .

$f (- 9) = - \sqrt{2}$

Step 4.2.2.2

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

$y = - \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(- 11, 0), (- 10, - 1), (- 9, - 1.41)$

$\begin{array}{c} x & y \\ - 11 & 0 \\ - 10 & - 1 \\ - 9 & - 1.41 \end{array}$

Step 5