4. Transformations of Radical Functions
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Chapter 3
4.
Transformations of Radical Functions
This lesson explores the transformations of radical functions, focusing on square root functions. It discusses how these functions can be translated, stretched, or shrunk horizontally and vertically. Understanding these transformations is crucial in fields like engineering, physics, and computer graphics. For instance, in engineering, you might need to adjust the curve of a radical function to fit specific structural requirements. The lesson provides practical exercises and examples, helping you master the art of transforming radical functions for various applications.
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Lesson Settings & Tools
| | 14 Theory slides |
| | 19 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Transformations of Radical Functions
Slide of 14
This lesson will discuss how to apply different transformations to radical functions. It will also show how to identify these transformations in the graphs of different radical functions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Transforming the Graph of a Function
The graph of the parent function y=x and the graph of the radical function y=ax−h+k are drawn on the same coordinate plane.
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Explore
Translating a Graph
The graph of the function y=x−h+k is shown in the coordinate plane. By changing the values of h and k, observe how the graph is horizontally and vertically translated.
Discussion
Translation of Radical Functions
A translation of a function is a transformation that shifts a graph vertically or horizontally. As with the graph of any other function, a vertical translation of the graph of a radical function is achieved by adding some number to every output value of the function rule. Consider the parent function y=x.
A horizontal translation is instead achieved by subtracting a number from every input value.
Functiony=xVertical Translationby k Unitsy=x+k
If k is a positive number, the translation is performed upwards. Conversely, if k is negative, the translation is performed downwards. If k=0, then there is no translation. This transformation can be shown on a coordinate plane.
Functiony=xHorizontal Translationby h Unitsy=x−h
In this case, if h is a positive number, the translation is performed to the right. Conversely, if h is negative, the translation is performed to the left. If h=0, then there is no translation. This transformation can also be shown on a coordinate plane. Example
Translating Radical Functions
Ignacio has just started to learn about transformations of radical functions. In order to get some extra practice, he goes online to to find a worksheet for the topic. He found a pretty good exercise, which is divided into parts A, B, and C.
Help Ignacio solve the exercise! 
Answer
a Equation: y=22x−6−1
Graph:
b Equation: y=0.5x+2+2
Graph:
c Equation: y=x+2
Graph:
Hint
a To translate the graph of f to the right 3 units, find an equation for f(x−3).
b To translate the graph of g up 5 units, find an equation for g(x)+5.
c To translate the graph of h down 3 units and to the left 4 units, find an equation for h(x+4)−3.
Solution
Functionf(x)=22x−1Translation 3 Units to the Righty=f(x−3)⇕y=22(x−3)−1
The right-hand side of the equation can be simplified by distributing the 2 in the radicand.
Now both functions can be drawn on the same coordinate plane.
b To translate the graph of a function 5 units up, the number 5 must be added to the output of the function.
Functiong(x)=0.5x+2−3Translation 5 Units Upy=g(x)+5⇕y=0.5x+2−3+5
The right-hand side of the equation can be simplified by adding the terms.
Now both functions can be drawn on the same coordinate plane.
c To translate the graph of a function 3 units down and 4 units to the left, 3 must be subtracted from the output and 4 must be added to the input.
Functionh(x)=x−2+3Translation 3 Units Downand 4 Units to the Lefty=h(x+4)−3⇕y=x+4−2+3−3
The right-hand side of the equation can be simplified by subtracting the terms.
Now both functions can be drawn on the same coordinate plane.
Pop Quiz
Stating the Translation
The graphs of the rational function y=x and a vertical or horizontal translation are shown in the coordinate plane.
Explore
Stretching and Shrinking a Graph
The graph of the radical function y=acx is shown in the coordinate plane. By changing the values of a and c, observe how the graph is vertically and horizontally stretched and shrunk.
Discussion
Stretch and Shrink of Radical Functions
A function graph is vertically stretched or shrunk by multiplying the output of the function rule by some positive constant a. Consider the radical function y=x−1+1.
Similarly, the graph of a radical function is horizontally stretched or shrunk by multiplying the input of the function rule by some positive constant c. Consider this time the parent function y=x.
Functiony=x−1+1Vertical Stretch/Shrink by a Factor of ay=a(x−1+1) ⇔ y=ax−1+a
If a is greater than 1, the graph is vertically stretched by a factor of a. Conversely, if a is less than 1, the graph is vertically shrunk by a factor of a. If a=1, then there is neither stretch nor shrink. All vertical distances from the graph to the x-axis are changed by the factor a.
Functiony=xHorizontal Stretch/Shrinkby a Factor of cy=cx
In this case, if c is greater than 1, the graph is horizontally shrunk by a factor of c. Conversely, if c is less than 1, the graph is horizontally stretched by a factor of c. If c=1, then there is neither a stretch nor shrink of the graph.
Extra
Finding a and cSuppose that a function is horizontally or vertically stretched/shrunk, and that the graphs of the transformed and the original function are both drawn on the same coordinate plane. Then, the values of a or c can be found by following these procedures.
| Finding a | Select two points with the same x-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of a is the quotient of the y-coordinate of Q and the y-coordinate of P. |
|---|---|
| Finding c | Select two points with the same y-coordinate, one point P on the parent function and the other point Q on the transformed function. The value of c is the quotient of the x-coordinate of P and the x-coordinate of Q. |
Example
Stretching and Shrinking Radical Functions
After mastering vertical and horizontal translations of radical functions, Ignacio is having a hard time understanding vertical and horizontal stretches and shrinks of this type of function. He asked for some help from his very good friend Jordan.
f(x)=x−1+3
She then tells him to complete the next two exercises. a Write the equation and draw the graph of a function whose graph is a vertical stretch of the graph of f by a factor of 4.
b Write the equation and draw the graph of a function whose graph is a horizontal shrink of the graph of f by a factor of 2.
Answer
a Equation: y=4x−1+12
Graph:
b Equation: y=2x−1+3
Graph:
Solution
a To vertically stretch the graph of f by a factor of 4, the output of the function must be multiplied by 4.
Functionf(x)=x−1+3Vertical Stretch by a Factor of 4y=4f(x)⇕y=4(x−1+3)
The right-hand side of the function can be simplified by distributing the 4.
Now both functions can be drawn on the same coordinate plane.
b To horizontally shrink the graph of f by a factor of 2, the input of the function must be multiplied by 2.
Functionf(x)=x−1+3Horizontal Shrink by a Factor of 2y=f(2x)⇕y=2x−1+3
Now both functions can be drawn on the same coordinate plane.
Pop Quiz
Stating the Factor of Stretch or Shrink
The graph of the parent function y=x is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
Discussion
Reflection of Radical Functions
A reflection of a function is a transformation that flips a graph over a line called the line of reflection. A reflection in the x-axis is achieved by changing the sign of every output value. This means changing the sign of the y-coordinate of every point on the graph of a function. Consider the radical function y=x+1.
A reflection in the y-axis is instead achieved by changing the sign of every input value. In this case, note that the domain of the radical function needs to be modified from all non-negative numbers to all non-positive numbers. Consider this time the parent function y=x.
Functiony=x+1Reflection in the x-axisy=-(x+1) ⇔ y=-x−1
This reflection can be shown on a coordinate plane.
Functiony=xReflection in the y-axisy=-x
This transformation can also be shown on a coordinate plane. Example
Reflecting Radical Functions
Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.
f(x)=2x−2−1
He wants to answer two practice exercises about reflection of radical functions. a Write the equation and draw the graph of a function whose graph is a reflection in the x-axis of the graph of f.
b Write the equation and draw the graph of a function whose graph is a reflection in the y-axis of the graph of f.
Answer
a Equation: y=-2x−2+1
Graph:
b Equation: y=2-x−2−1
Graph:
Solution
a To reflect the graph of the given function in the x-axis, the output needs to be multiplied by -1.
Functionf(x)=2x−2−1Reflection in the x-axisy=-f(x)⇕y=-(2x−2−1)
The right-hand side of the function can be simplified by distributing the -1.
Now both functions can be graphed on the same coordinate plane.
b To reflect the graph of the given function in the y-axis, the input needs to be multiplied by -1.
Functionf(x)=2x−2−1Reflection in the y-axisy=f(-x)⇕y=2-x−2−1
Now, both functions can be graphed on the same coordinate plane.
Example
Combining Transformations
Ignacio confidently states that he can now solve any exercise about transformations of radical functions. Jordan skeptically challenges her friend to solve an exercise that combines transformations.
Answer
Equation: y=-x+5−6
Graph:
Hint
Start by multiplying the output by 2 to find the function that represents the vertical stretch. Then, multiply the input by -1 to reflect the graph in the y-axis. Finally, subtract 3 units from the input.
Solution
To help Ignacio, the transformations will be applied one at a time.
Vertical Stretch by a Factor of 2
First, the equation of the vertical stretch by a factor of 2 will be found. To do so, multiply the output of the function by 2.Functionf(x)=0.5x+2−3Vertical Stretch by a Factor of 2y=2f(x)⇕y=2(0.5x+2−3)
Simplify the right-hand side of the above equation by distributing the 2.
The graph of the function is a vertical stretch by a factor of 2 of the graph of f. Let g(x) be this function.
g(x)=x+2−6
Reflection in the y-axis
To reflect the graph of a function in the y-axis, multiply the input by -1.Functiong(x)=x+2−6Reflection in the y-axisy=g(-x)⇕y=-x+2−6
The graph of this function is a reflection in the y-axis of the graph of g. Let h(x) be this function.
h(x)=-x+2−6
Translation 3 Units to the Right
Finally, to horizontally translate the graph of a function 3 units to the right, subtract 3 from the input.Functionh(x)=-x+2−6Translation 3 Units to the Righty=h(x−3)⇕y=-(x−3)+2−6
Simplify the right-hand side of the equation by distributing the -1 and adding the terms in the radicand.
The graph of this final function is a vertical stretch by a factor of 2, followed by a reflection in the y-axis, and a translation 3 units to the right of the graph of f. Graph
The transformations can be shown on a coordinate plane.
Closure
Transforming the Graph of a Function
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The graphs of the radical function y=ax−h+k and its corresponding parent function y=x are given.
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Hint
Start by considering a vertical stretch. Then, consider vertical and horizontal translations.
Solution
Disregard translations for a moment. In the graph of y=x it can be understood as, after moving 1 unit to the right of the starting point (0,0), the graph increases vertically 1 unit. Conversely, in the graph of y=ax−h+k, after moving 1 unit to the right of the starting point (-3,0), the graph increased vertically 2 units.
Functiony=xVertical Stretch by a Factor of 2y=2x
Next, pay close attention to the starting point of each curve. It can be concluded that the graph of the parent function y=x needs to be translated 3 units to the left to match the other graph. Therefore, the value of h is -3.
Functiony=2xHorizontal Translation3 Units to the Lefty=2x−(-3)
Finally, note that the y-coordinate of the initial point of both graphs is 0. Therefore, there is no vertical translation. This means that k=0.
Functiony=2x−(-3)Vertical Translation0 Units Upy=2x−(-3)+0
It has been found that a=2, h=-3, and k=0. The obtained function can be simplified.
The transformations can be shown on a graph.
Transformations of Radical Functions
Exercises
In the coordinate plane, two graphs are shown. One curve is the graph of y=x+1 and the other shows the same curve after a single transformation.
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Note that the graphs have different orientations. This means that there is no translation that maps the graph of y=sqrt(x+1) onto the other graph. Furthermore, the transformed graph is neither a stretch nor a shrink of the graph of y=sqrt(x+1). However, we can see that the y-axis acts as a line of symmetry of the two graphs.
Therefore, the transformed graph is a reflection in the y-axis of the graph of y=sqrt(x+1). To obtain its equation, we need to multiply the variable x — the input of the radical function — by - 1. cc Function & Reflection in they-axis y=sqrt(x+1) & y=sqrt(- x+1)
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In the coordinate plane, two graphs are shown. One curve is the graph of y=3x−1 and the other shows the same curve after a single transformation.
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Note that there is no translation that maps the graph of y=sqrt(x-1) onto the other graph. Furthermore, there is no line of symmetry between the graphs. This means that the transformed graph is not a reflection of the graph of y=sqrt(x-1). cc Translation & Reflection * & * However, we can see that the transformed graph is a vertical stretch of the given radical function's graph. To find the factor, we need to pay close attention to the y-coordinates of the points on both curves that have the same x-coordinate.
We see that if two points on the curves have the same x-coordinate, the y-coordinate of the point on the transformed curve is three times the y-coordinate of the point on the original curve. Therefore, the transformed graph is a vertical stretch by a factor of 3 of the graph of y=sqrt(x-1). cc Point on the & Point on the Original Curve & Transformed Curve (x,y) & (x, 3y) To obtain the equation of the new curve, we need to multiply the function's output by 3. cc Function & Vertical Stretch & by a Factor of3 y=sqrt(x-1) & y= 3sqrt(x-1)
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Transformations of Radical Functions
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