Pre-Algebra Examples

$y = 2 \sqrt{2 x - 2}$

Step 1

Find the domain for $y = 2 \sqrt{2 x - 2}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the radical.

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Step 1.1

Set the radicand in $\sqrt{2 (x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (x - 1) \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Divide each term in $2 (x - 1) \geq 0$ by $2$ and simplify.

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Step 1.2.1.1

Divide each term in $2 (x - 1) \geq 0$ by $2$ .

$\frac{2 (x - 1)}{2} \geq \frac{0}{2}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 1)}{2} \geq \frac{0}{2}$

Step 1.2.1.2.1.2

Divide $x - 1$ by $1$ .

$x - 1 \geq \frac{0}{2}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $2$ .

$x - 1 \geq 0$

Step 1.2.2

Add $1$ to both sides of the inequality.

$x \geq 1$

Step 1.3

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

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Interval Notation:

$[1, \infty)$

Set-Builder Notation:

${x | x \geq 1}$

Step 2

To find the radical expression end point, substitute the $x$ value $1$ , which is the least value in the domain, into $f (x) = 2 \sqrt{2 (x - 1)}$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Subtract $1$ from $1$ .

$f (1) = 2 \sqrt{2 \cdot 0}$

Step 2.2.2

Multiply $2$ by $0$ .

$f (1) = 2 \sqrt{0}$

Step 2.2.3

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

$f (1) = 2 \sqrt{0^{2}}$

Step 2.2.4

Multiply by zero.

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Step 2.2.4.1

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

$f (1) = 2 \cdot 0$

Step 2.2.4.2

Multiply $2$ by $0$ .

$f (1) = 0$

Step 2.2.5

The final answer is $0$ .

$0$

Step 3

The radical expression end point is $(1, 0)$ .

$(1, 0)$

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.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Subtract $1$ from $2$ .

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

Step 4.1.2.2

Multiply $2$ by $1$ .

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

Step 4.1.2.3

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

$y = 2 \sqrt{2}$

Step 4.2

Substitute the $x$ value $3$ into $f (x) = 2 \sqrt{2 (x - 1)}$ . In this case, the point is $(3, 4)$ .

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Step 4.2.1

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

$f (3) = 2 \sqrt{2 ((3) - 1)}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Subtract $1$ from $3$ .

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

Step 4.2.2.2

Multiply $2$ by $2$ .

$f (3) = 2 \sqrt{4}$

Step 4.2.2.3

Rewrite $4$ as $2^{2}$ .

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

Step 4.2.2.4

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

$f (3) = 2 \cdot 2$

Step 4.2.2.5

Multiply $2$ by $2$ .

$f (3) = 4$

Step 4.2.2.6

The final answer is $4$ .

$y = 4$

Step 4.3

The square root can be graphed using the points around the vertex $(1, 0), (2, 2.83), (3, 4)$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.83 \\ 3 & 4 \end{array}$

Step 5