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.
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Transforming the Graph of a Function
The graph of the parent function y=sqrt(x) and the graph of the radical function y=asqrt(x-h)+k are drawn on the same coordinate plane.
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Translating a Graph
The graph of the function y=sqrt(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.
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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=sqrt(x). cc Function & Vertical Translation & bykUnits y=sqrt(x) & y=sqrt(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.
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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! 
Function g(x)=0.5sqrt(x+2)-3 [1em]
Translation5Units Up y=g(x) + 5 ⇕ y=0.5sqrt(x+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.
Function h(x)=sqrt(x-2)+3 [1em]
Translation3Units Down and4Units to the Left y=h(x + 4) - 3 ⇕ y=sqrt(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.
Answer
a Equation: y=2sqrt(2x-6)-1
Graph:
b Equation: y=0.5sqrt(x+2)+2
Graph:
c Equation: y=sqrt(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
b To translate the graph of a function 5 units up, the number 5 must be added to the output of the function.
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.
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Stating the Translation
The graphs of the rational function y=sqrt(x) and a vertical or horizontal translation are shown in the coordinate plane.
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Stretching and Shrinking a Graph
The graph of the radical function y=asqrt(cx) 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.
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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=sqrt(x-1)+1.
Function y=sqrt(x-1)+1 [1em]
Vertical Stretch/Shrink by a Factor ofa y=a(sqrt(x-1)+1) ⇔ y=asqrt(x-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.
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=sqrt(x).
cc
Function & Horizontal Stretch/Shrink & by a Factor ofc y=sqrt(x) & y=sqrt(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. |
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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.
In order to help him, Jordan asks Ignacio to consider the following function. f(x)=sqrt(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=4sqrt(x-1)+12
Graph:
b Equation: y=sqrt(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.
b To horizontally shrink the graph of f by a factor of 2, the input of the function must be multiplied by 2.
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Stating the Factor of Stretch or Shrink
The graph of the parent function y=sqrt(x) is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
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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=sqrt(x)+1.
Function y=sqrt(x)+1 [1em]
Reflection in thex-axis y=- (sqrt(x)+1 ) ⇔ y=- sqrt(x)-1
This reflection can be shown on a coordinate plane.
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=sqrt(x).
cc
Function & Reflection in they-axis y=sqrt(x) & y=sqrt(- x)
This transformation can also be shown on a coordinate plane. 
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Reflecting Radical Functions
Ignacio is now feeling pretty confident about transformations of radical functions again. Now he turns his attention to reflections.
Ignacio considers the following radical function. f(x)=2sqrt(x-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=- 2sqrt(x-2)+1
Graph:
b Equation: y=2sqrt(- 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.
b To reflect the graph of the given function in the y-axis, the input needs to be multiplied by - 1.
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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=sqrt(- 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. Function f(x)=0.5sqrt(x+2)-3 [1em] Vertical Stretch by a Factor of2 y= 2f(x) ⇕ y= 2(0.5sqrt(x+2)-3) Simplify the right-hand side of the above equation by distributing the 2.y=2(0.5sqrt(x+2)-3)
Distr
Distribute 2
y=1sqrt(x+2)-6
IdPropMult
Identity Property of Multiplication
y=sqrt(x+2)-6
Reflection in the y-axis
To reflect the graph of a function in the y-axis, multiply the input by -1. Function g(x)=sqrt(x+2)-6 [1em] Reflection in they-axis y=g( -x) ⇕ y=sqrt(-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)=sqrt(- 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. Function h(x)=sqrt(- x+2)-6 [1em] Translation3Units to the Right y=h(x - 3) ⇕ y=sqrt(- (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.
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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=asqrt(x-h)+k and its corresponding parent function y=sqrt(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=sqrt(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=asqrt(x-h)+k, after moving 1 unit to the right of the starting point (- 3,0), the graph increased vertically 2 units.
y=2sqrt(x-(- 3)) + 0
IdPropAdd
Identity Property of Addition
y=2sqrt(x-(- 3))
SubNeg
a-(- b)=a+b
y=2sqrt(x+3)
Transformations of Radical Functions
Exercises
Consider the radical function. y=sqrt(x)-1 Which of the following graphs represents a vertical translation 2 units up from the given function?
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
| x | y=sqrt(x)-1 | y |
|---|---|---|
| 0 | sqrt(0)-1 | - 1 |
| 1 | sqrt(1)-1 | 0 |
| 2 | sqrt(2)-1 | ≈ 0.4 |
| 3 | sqrt(3)-1 | ≈ 0.7 |
| 4 | sqrt(4)-1 | 1 |
| 5 | sqrt(5)-1 | ≈ 1.2 |
| 10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this function 2 units up.
This graph corresponds to choice B.
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Consider the radical function.
y=sqrt(x)-1
What is the equation of a function whose graph is a vertical translation 3 units down of the graph of this radical function?
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We want to obtain a vertical translation 3 units down of the graph of the given radical function. To do this, we have to subtract 3 from the output of the function rule. Notice that sqrt(x)-1 represents the outputs of the function. cc Function & Translation3 Units Down y=sqrt(x)-1 & y=(sqrt(x)-1) -3 Finally, let's simplify the right-hand side of this equation.
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Consider the radical function. y=sqrt(x)-1 Which of the following graphs is the graph of a horizontal translation 3 units to the right of this function?
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
| x | y=sqrt(x)-1 | y |
|---|---|---|
| 0 | sqrt(0)-1 | - 1 |
| 1 | sqrt(1)-1 | 0 |
| 2 | sqrt(2)-1 | ≈ 0.4 |
| 3 | sqrt(3)-1 | ≈ 0.7 |
| 4 | sqrt(4)-1 | 1 |
| 5 | sqrt(5)-1 | ≈ 1.2 |
| 10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this graph 3 units to the right.
This graph corresponds to choice A.
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Consider the radical function.
y=sqrt(x)-1
What is the equation of a function whose graph is a horizontal translation 5 units to the left of the graph of this radical function?
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We want to obtain a horizontal translation 5 units to the left of the graph of the given radical function. To do this, let's first recall how to perform vertical and horizontal translations.
| Translations of f(x) | |
|---|---|
| Vertical Translations | Translation up k units, k>0 y=f(x)+ k |
| Translation down k units, k>0 y=f(x)- k | |
| Horizontal Translations | Translation right h units, h>0 y=f(x- h) |
| Translation left h units, h>0 y=f(x+ h) | |
This review lets us know that we need to add 5 to the input of the function rule. Notice that the input is represented with x. cc Function & Translation5 Units to the Left y=sqrt(x)-1 & y=sqrt(x + 5)-1
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Consider the radical function. y=sqrt(x)-1 Which of the following graphs is the graph of a translation 1 unit down and 2 units to the left of this function?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use non-negative values for the variable x.
| x | y=sqrt(x)-1 | y |
|---|---|---|
| 0 | sqrt(0)-1 | - 1 |
| 1 | sqrt(1)-1 | 0 |
| 2 | sqrt(2)-1 | ≈ 0.4 |
| 3 | sqrt(3)-1 | ≈ 0.7 |
| 4 | sqrt(4)-1 | 1 |
| 5 | sqrt(5)-1 | ≈ 1.2 |
| 10 | sqrt(10)-1 | ≈ 2.2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's translate this function 1 unit down and 2 units to the left.
This graph corresponds to choice D.
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Consider the radical function.
y=sqrt(x)-1
What is the equation of a function whose graph is a horizontal translation 3 units to the right and 2 units down of the graph of this radical function?
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We want to obtain a translation 3 units to the right and 2 units down of the graph of the given radical function. To do this, we have to subtract 3 from the input and subtract 2 from the output of the function rule. Notice that x represents the input and sqrt(x)-1 represents the outputs of the given function. cc Function & Translation3 Units to the & Right and2 Units Down y=sqrt(x)-1 & y=sqrt(x- 3)-1- 2 Let's finally simplify the right-hand side of the obtained equation.
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Consider the radical function. y=sqrt(x-1)+2 Which of the following graphs is the graph of a vertical stretch by a factor of 2 of this function?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":0}
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a square root must be non-negative, we will only use values greater than or equal to 1 for the variable x.
| x | sqrt(x-1)+2 | y |
|---|---|---|
| 1 | sqrt(1-1)+2 | 2 |
| 2 | sqrt(2-1)+2 | 3 |
| 5 | sqrt(5-1)+2 | 4 |
| 10 | sqrt(10-1)+2 | 5 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's vertically stretch this graph by a factor of 2.
This graph matches choice A.
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Consider the radical function.
y=sqrt(2x+1)-3
What is the equation of a function whose graph is a vertical stretch by a factor of 5 of the graph of this radical function?
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We want to obtain a vertical stretch by a factor of 5 of the graph of the given radical function. To do this, we need to multiply by 5 the output of the function rule. cc Function & Vertical Stretch & by a Factor of5 y=sqrt(2x+1)-3 & y= 5(sqrt(2x+1)-3) Let's now simplify the right-hand side of the obtained equation. To do so, we will distribute the 5.
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Consider the radical function.
y=sqrt(2x+1)-3
What is the equation of a function whose graph is a horizontal shrink by a factor of 3 of the graph of this radical function?
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We want to obtain a horizontal shrink by a factor of 3 of the graph of the given radical function. To do this, we need to multiply by 3 the input of the function. Notice that the input of this function is represented by x. cc Function & Horizontal Shrink & by a Factor of3 y=sqrt(2x+1)-3 & y=sqrt(2( 3x)+1)-3 Let's now simplify the right-hand side of the obtained equation.
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Consider the radical function. y=sqrt(1/2x-1)+2 Which of the following graphs is the graph of a horizontal shrink by a factor of 4 of this function?
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
| x | sqrt(1/2x-1)+2 | y |
|---|---|---|
| - 5 | sqrt(1/2( - 5)-1)+2 | ≈ 0.5 |
| - 3 | sqrt(1/2( - 3)-1)+2 | ≈ 0.6 |
| - 1 | sqrt(1/2( - 1)-1)+2 | ≈ 0.9 |
| 0 | sqrt(1/2( 0)-1)+2 | 1 |
| 1 | sqrt(1/2( 1)-1)+2 | ≈ 1.2 |
| 3 | sqrt(1/2( 3)-1)+2 | ≈ 2.8 |
| 5 | sqrt(1/2( 5)-1)+2 | ≈ 3.1 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Finally, let's horizontally shrink this graph by a factor of 4. Graphically, this means that the x-coordinate of each point on the transformed curve will be one quarter the x-coordinate of its corresponding point on the original curve. Let's consider the points (2,2) and (4,3).
| Point | Transformed Point |
|---|---|
| ( 2, 2) | (1/4( 2), 2) ⇕ (1/2,2) |
| ( 4, 3) | (1/4( 4), 3) ⇕ (1,3) |
Note that, for each point on the curve, the y-coordinate will remain the same but the x-coordinate will be one quarter its original value.
This graph matches choice B.
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Consider the following radical function. y=2sqrt(x+1)-2 Which of the given graphs is a reflection of this function's graph in the x-axis?
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
| x | 2sqrt(x+1)-2 | y |
|---|---|---|
| - 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
| - 2 | 2sqrt(- 2+1)-2 | - 4 |
| - 1 | 2sqrt(- 1+1)-2 | - 2 |
| 0 | 2sqrt(0+1)-2 | 0 |
| 3 | 2sqrt(3+1)-2 | ≈ 1.2 |
| 5 | 2sqrt(5+1)-2 | ≈ 1.6 |
| 7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's finally reflect this graph in the x-axis.
This graph matches choice C.
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Consider the radical function.
y=2sqrt(x+1)-2
What is the equation of a function whose graph is a reflection in the x-axis of the graph of this radical function?
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We want to obtain the equation of a function whose graph is a reflection in the x-axis of the graph of the given radical function. To do so, we need to multiply the function rule by - 1. cc Function & Reflection in thex -axis y=2sqrt(x+1)-2 & y= - (2sqrt(x+1)-2) Now we can distribute - 1 to simplify the right-hand side of the equation.
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Consider the radical function.
y=2sqrt(x+1)-2
What is the equation of a function whose graph is a reflection in the y-axis of the graph of this radical function?
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We want to obtain the equation of a function whose graph is a reflection in the y-axis of the graph of the given radical function. To do so, we need to multiply the input of the function — the variable x — by - 1. cc Function & Reflection in they -axis y=2sqrt(x+1)-2 & y=2sqrt(-x+1)-2
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Consider the radical function. y=2sqrt(x+1)-2 Which of the following graphs is a reflection of the given function's graph in the y-axis?
{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
| x | 2sqrt(x+1)-2 | y |
|---|---|---|
| - 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
| - 2 | 2sqrt(- 2+1)-2 | - 4 |
| - 1 | 2sqrt(- 1+1)-2 | - 2 |
| 0 | 2sqrt(0+1)-2 | 0 |
| 3 | 2sqrt(3+1)-2 | ≈ 1.2 |
| 5 | 2sqrt(5+1)-2 | ≈ 1.6 |
| 7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's finally reflect this graph in the y-axis.
This graph matches choice D.
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Consider the radical function. y=2sqrt(x+1)-2 Which of the following graphs is a translation 2 units to the right, followed by a reflection in the x-axis of this function's graph?
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Let's start by drawing the graph of the given radical function. To do this, we will make a table of values. Since the radicand of a cube root can be any real number, we can use positive values, negative values, and zero for the variable x.
| x | 2sqrt(x+1)-2 | y |
|---|---|---|
| - 5 | 2sqrt(- 5+1)-2 | ≈ - 5.2 |
| - 2 | 2sqrt(- 2+1)-2 | - 4 |
| - 1 | 2sqrt(- 1+1)-2 | - 2 |
| 0 | 2sqrt(0+1)-2 | 0 |
| 3 | 2sqrt(3+1)-2 | ≈ 1.2 |
| 5 | 2sqrt(5+1)-2 | ≈ 1.6 |
| 7 | 2sqrt(7+1)-2 | 2 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Let's now translate this graph 2 units to the right.
Finally, let's reflect the obtained graph in the x-axis.
This graph matches choice A.
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Consider the radical function.
y=2sqrt(x+1)-2
What is the equation of a function whose graph is a translation 2 units to the right followed by a reflection in the x-axis of the graph of this radical function? Write the answer in its simplest form.
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We want to obtain the equation of a function whose graph is a translation 2 units to the right followed by a reflection in the x-axis of graph of the given radical function. Let's first write the equation of the translation. To do so, we need to subtract 2 from the function's input. cc Function & Translation2 Units & to the Right y=2sqrt(x+1)-2 & y=2sqrt(x- 2+1)-2 We can simplify the right-hand side of the equation.
Next, let's find the equation that corresponds to a reflection in the x-axis of the translated graph. To do this, we need to multiply the function rule — the function's output — by - 1. cc Function & Reflection in thex -axis y=2sqrt(x-1)-2 & y= - (2sqrt(x-1)-2) We can finally simplify the right-hand side of the last equation by distributing the - 1.
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Transformations of Radical Functions
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