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Wize High School Algebra II Textbook (Common Core) > Radical Expressions and Functions

Transformations of Radical Functions

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Applications of Radical FunctionsNext Section
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Radical Functions

A radical function is defined as:
y=  nf(x)\boxed{y=~~^n\sqrt {f(x)}}
where f(x)f(x) is a function and n∈Wn\in\mathbb{W}.

Square-Root Functions

y=xy=\sqrt{x}

xy00114293164\begin{array}{|c|c|}\hline x&y\\\hline 0&0\\\hline 1&1\\\hline 4&2\\\hline 9&3\\\hline 16&4\\\hline \end{array}

xx-Intercept: (0,0)(0,0)
yy-Intercept: (0,0)(0,0)
Domain: [0,∞)[0,\infin)
Range: [0,∞)[0,\infin)
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Transformations of Radical Functions

If y=ab(x−h)+ky=a\sqrt{b(x-h)}+k , then the following transformations are applied to y=xy=\sqrt{x}:

HorizontalVerticalb<0‾:a<0‾:horizontal reflection about thevertical reflection about they-axisx-axis∣b∣>1‾:∣a∣>1‾:horizontal compression of  1b  unitsvertical expansion of ’a’ units∣b∣<1‾:∣a∣<1‾:horizontal expansion of  1b  unitsvertical compression of ’a’ unitsh>0‾:k>0‾:horizontal translation ’h’ units rightvertical translation ’k units uph<0‾:k<0‾:horizontal translation ’h’ units leftvertical translation ’k’ units down\begin{array}{|l c c |l|} \hline \text{Horizontal}&&&\text{Vertical}\\ \hline\\ \underline{b<0}:&&&\underline{a<0}:\\ \text{horizontal reflection about the}&&&\text{vertical reflection about the}\\ \text{y-axis}&&&\text{x-axis}\\\\ \hline\\ \underline{|b|>1}:&&&\underline{|a|>1}:\\ \text{horizontal compression of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical expansion of 'a' units}\\\\ \hline\\ \underline{|b|<1}:&&&\underline{|a|<1}:\\ \text{horizontal expansion of}~~\displaystyle\frac{1}{b}~~\text{units}&&&\text{vertical compression of 'a' units}\\\\ \hline\\ \underline{h>0}:&&&\underline{k>0}:\\ \text{horizontal translation 'h' units right}&&&\text{vertical translation 'k units up}\\\\ \hline\\ \underline{h<0}:&&&\underline{k<0}:\\ \text{horizontal translation 'h' units left}&&&\text{vertical translation 'k' units down}\\ \\\hline \end{array}


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Example

Graph y=−2−(x+1)+1y=-2\sqrt{-(x+1)}+1, stating the domain and range.


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

Table of Values of Parent Function:
xy00114293164\begin{array}{|c|c|}\hline x&y\\\hline 0&0\\\hline 1&1\\\hline 4&2\\\hline 9&3\\\hline 16&4\\\hline \end{array}

Transformations applied to parent function:
Horizontal‾Vertical‾Reflection about the y-axisReflection about the x-axisExpansion by a factor of 2Translation 1 unit leftTranslation 1 unit up\begin{array}{lccl} \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\\\ \text{Reflection about the y-axis}&&&\text{Reflection about the x-axis}\\\\ &&&\text{Expansion by a factor of 2}\\\\ \text{Translation 1 unit left}&&&\text{Translation 1 unit up} \end{array}

Sketch & Table of Values of Transformed Function:

xy−11−2−1−5−3−10−5−177\begin{array}{|c|c|}\hline x&y\\\hline -1&1\\\hline -2&-1\\\hline -5&-3\\\hline -10&-5\\\hline -17&7\\\hline \end{array}

Domain: (−∞,−1](-\infin,-1]

Range: (−∞,1](-\infin,1]
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Example: Transformations of Radical Functions

Use transformations to sketch the graph of y=−2(x−1)+4y=-\sqrt{2(x-1)}+4, stating the domain and range.


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

Table of Values of Parent Function:
xy00114293164\begin{array}{|c|c|}\hline x&y\\\hline 0&0\\\hline 1&1\\\hline 4&2\\\hline 9&3\\\hline 16&4\\\hline \end{array}

Transformations applied to parent function:
Horizontal‾Vertical‾Reflection about the x-axisCompression by a factor of  12Translation 1 unit rightTranslation 4 units up\begin{array}{|lcc|l|} \hline \underline{\text{Horizontal}}&&&\underline{\text{Vertical}}\\ \hline\\ &&&\text{Reflection about the x-axis}\\ \\ \hline\\ \text{Compression by a factor of}~~\frac{1}{2}&&&\\ \\ \hline\\ \text{Translation 1 unit right}&&&\text{Translation 4 units up}\\ \\ \hline \end{array}

Sketch & Table of Values of Transformed Function:
xy141.53325.5190\begin{array}{|c|c|}\hline x&y\\\hline 1&4\\\hline 1.5&3\\\hline 3&2\\\hline 5.5&1\\\hline 9&0\\\hline \end{array}

Domain: [1,∞)[1,\infin)

Range: (−∞,4](-\infin,4]

Practice: Transformations of Radical Functions

If the point (x,16x4) (x,16x^4)~ lies on the graph y=f(x),y=f(x), then what point lies on the graph y=f(x)y=\sqrt{f(x)}?

Practice: Transformations of Radical Functions

Which of the following is the graph of the function y=−2−2x+8+1y=-2\sqrt{-2x+8}+1?

Practice: Transformations of Radical Functions

If the point (1, 4) is on the function y=14f(2x−4)+3y=\frac{1}{4}f(2x-4)+3, then what point must be on the function y=f(x)y=\sqrt{f(x)}?
Extra Practice
Radical Functions
Graph the function y=−34(x−2)+3y=-3\sqrt{4(x-2)}+3.
Solving Radical Equations
Practice: Transformations
Let y=f(x)y=\sqrt{f(x)}.
Match the appropriate transformation with its transformed function.
Solving Radical Equations
Practice: Transformations
Let y=f(x)y=\sqrt{f(x)}.
Match the appropriate transformation with its transformed function.
Transformations of Radical Functions
Practice: Transformations of Radical Functions
If the point (-4,5) is on the function y=23f(−x−8)+13y=\frac{2}{3}f(-x-8)+\dfrac{1}{3}, then what point must be on the function y=f(x)y=\sqrt{f(x)}?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
If the point (2, 3) is on the function y=2f(x−1)−3y=2f(x-1)-3, then what point must be on the function y=f(x)y=\sqrt{f(x)}?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
Which of the following is the graph of the function y=−−3x+6−2y=-\sqrt{-3x+6}-2?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
Which of the following is the graph of the function y=4x−1y=\sqrt{4x}-1?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
If the point (x,64x52) (x,64x^{\frac{5}{2}})~ lies on the graph y=f(x),y=f(x), then what point lies on the graph y=f(x)y=\sqrt{f(x)}?
Transformations of Radical Functions
Practice: Transformations of Radical Functions
If the point (x,81x16) (x,81x^{16})~ lies on the graph y=f(x),y=f(x), then what point lies on the graph y=f(x)y=\sqrt{f(x)}?