Pre-Algebra Examples

$\sqrt{y + 13}$

Step 1

The domain in terms of $y$ are all the $y$ values that make the radicand non-negative.

$[- 13, \infty)$

${y | y \geq - 13}$

Step 2

To find the radical expression end point, substitute the $y$ value $- 13$ , which is the least value in the domain, into $f (y) = \sqrt{y + 13}$ .

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Step 2.1

Replace the variable $y$ with $- 13$ in the expression.

$f (- 13) = \sqrt{(- 13) + 13}$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 13$ and $13$ .

$f (- 13) = \sqrt{0}$

Step 2.2.2

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

$f (- 13) = \sqrt{0^{2}}$

Step 2.2.3

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

$f (- 13) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(0, - 13)$ .

$(0, - 13)$

Step 4

Select a few $y$ values from the domain. It would be more useful to select the values so that they are next to the $y$ value of the radical expression end point.

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Step 4.1

Substitute the $y$ value $- 13$ into $\sqrt{y + 13}$ . In this case, the point is $(0, - 13)$ .

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Step 4.1.1

Replace the variable $y$ with $- 13$ in the expression.

$f (- 13) = \sqrt{(- 13) + 13}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Add $- 13$ and $13$ .

$f (- 13) = \sqrt{0}$

Step 4.1.2.2

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

$f (- 13) = \sqrt{0^{2}}$

Step 4.1.2.3

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

$f (- 13) = 0$

Step 4.1.2.4

The final answer is $0$ .

$y = 0$

Step 4.2

Substitute the $y$ value $- 12$ into $\sqrt{y + 13}$ . In this case, the point is $(1, - 12)$ .

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Step 4.2.1

Replace the variable $y$ with $- 12$ in the expression.

$f (- 12) = \sqrt{(- 12) + 13}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Add $- 12$ and $13$ .

$f (- 12) = \sqrt{1}$

Step 4.2.2.2

Any root of $1$ is $1$ .

$f (- 12) = 1$

Step 4.2.2.3

The final answer is $1$ .

$y = 1$

Step 4.3

Substitute the $y$ value $- 11$ into $\sqrt{y + 13}$ . In this case, the point is $(1.41, - 11)$ .

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Step 4.3.1

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

$f (- 11) = \sqrt{(- 11) + 13}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Add $- 11$ and $13$ .

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

Step 4.3.2.2

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

$y = \sqrt{2}$

Step 4.4

The square root can be graphed using the points around the vertex $(0, - 13), (0, - 13), (1, - 12), (1.41, - 11)$

$\begin{array}{c} x & y \\ 0 & - 13 \\ 0 & - 13 \\ 1 & - 12 \\ 1.41 & - 11 \end{array}$

Step 5