Algebra Examples

$y = \sqrt{\frac{1}{2} x}$

Step 1

The parent function is the simplest form of the type of function given.

$y = \sqrt{x}$

Step 2

Simplify $\sqrt{\frac{1}{2} x}$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.2

Rewrite $\sqrt{\frac{x}{2}}$ as $\frac{\sqrt{x}}{\sqrt{2}}$ .

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

Step 2.3

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \frac{\sqrt{x}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{\sqrt{x}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.5

Add $1$ and $1$ .

$y = \frac{\sqrt{x} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = \frac{\sqrt{x} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \frac{\sqrt{x} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$y = \frac{\sqrt{x} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{x} \sqrt{2}}{2^{1}}$

Step 2.4.6.5

Evaluate the exponent.

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

Step 2.5

Combine using the product rule for radicals.

$y = \frac{\sqrt{x \cdot 2}}{2}$

Step 2.6

Reorder factors in $\frac{\sqrt{x \cdot 2}}{2}$ .

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

Step 3

Assume that $y = \sqrt{x}$ is $f (x) = \sqrt{x}$ and $y = \sqrt{\frac{1}{2} x}$ is $g (x) = \frac{\sqrt{2 x}}{2}$ .

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

$g (x) = \frac{\sqrt{2 x}}{2}$

Step 4

The transformation from the first equation to the second one can be found by finding $a$ , $h$ , and $k$ for each equation.

$y = a \sqrt{x - h} + k$

Step 5

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

$y = \sqrt{x}$

Step 6

Factor a $2$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

$y = (\frac{\sqrt{2}}{2}) \sqrt{x}$

Step 7

Find $a$ , $h$ , and $k$ for $y = (\frac{\sqrt{2}}{2}) \sqrt{x}$ .

$a = 0.70710678$

$h = 0$

$k = 0$

Step 8

The horizontal shift depends on the value of $h$ . When $h > 0$ , the horizontal shift is described as:

$g (x) = f (x + h)$ - The graph is shifted to the left $h$ units.

$g (x) = f (x - h)$ - The graph is shifted to the right $h$ units.

Horizontal Shift: None

Step 9

The vertical shift depends on the value of $k$ . When $k > 0$ , the vertical shift is described as:

$g (x) = f (x) + k$ - The graph is shifted up $k$ units.

$g (x) = f (x) - k$ - The graph is shifted down $k$ units.

Vertical Shift: None

Step 10

The sign of $a$ describes the reflection across the x-axis. $- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 11

The value of $a$ describes the vertical stretch or compression of the graph.

$a > 1$ is a vertical stretch (makes it narrower)

$0 < a < 1$ is a vertical compression (makes it wider)

Vertical Compression: Compressed

Step 12

To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.

Parent Function: $y = \sqrt{x}$

Horizontal Shift: None

Vertical Shift: None

Reflection about the x-axis: None

Vertical Compression: Compressed

Step 13