Calculus Examples

$x = 2 \sqrt{y}$

Step 1

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

$[0, \infty)$

${y | y \geq 0}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

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

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

Step 2.2.3

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

$f (0) = 2 \cdot 0$

Step 2.2.4

Multiply $2$ by $0$ .

$f (0) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

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

$(0, 0)$

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 $1$ into $2 \sqrt{y}$ . In this case, the point is $(2, 1)$ .

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Remove parentheses.

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

Step 4.1.2.2

Any root of $1$ is $1$ .

$f (1) = 2 \cdot 1$

Step 4.1.2.3

Multiply $2$ by $1$ .

$f (1) = 2$

Step 4.1.2.4

The final answer is $2$ .

$y = 2$

Step 4.2

Substitute the $y$ value $2$ into $2 \sqrt{y}$ . In this case, the point is $(2.83, 2)$ .

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Step 4.2.1

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

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

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Remove parentheses.

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

Step 4.2.2.2

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

$y = 2 \sqrt{2}$

Step 4.3

The square root can be graphed using the points around the vertex $(0, 0), (2, 1), (2.83, 2)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 2 & 1 \\ 2.83 & 2 \end{array}$

Step 5