radical functions 10 - hhs algebra ii · 2018. 9. 7. · 772 chapter 10 radical functions 10 if the...

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763 © Carnegie Learning 10 Radical Functions 10.1 With Great Power . . . Inverses of Power Functions ..................... 765 10.2 The Root of the Matter Radical Functions .............................. 777 10.3 Making Waves Transformations of Radical Functions .............. 791 10.4 Keepin’ It Real Extracting Roots and Rewriting Radicals ............ 801 10.5 Time to Operate! Multiplying, Dividing, Adding, and Subtracting Radicals ........................... 817 10.6 Look to the Horizon Solving Radical Equations ....................... 829 This picture shows a surfer in a “barrel ride”—one of surfing's most sought-after experiences. Given the right conditions, a surfer can ride inside a wave as it breaks.

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  • 763

    © Carnegie Learning

    10Radical Functions

    10.1 With Great Power . . .

    Inverses of Power Functions . . . . . . . . . . . . . . . . . . . . . 765

    10.2 The Root of the Matter

    Radical Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

    10.3 Making Waves

    Transformations of Radical Functions . . . . . . . . . . . . . . 791

    10.4 Keepin’ It Real

    Extracting Roots and Rewriting Radicals . . . . . . . . . . . . 801

    10.5 Time to Operate!

    Multiplying, Dividing, Adding, and

    Subtracting Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . 817

    10.6 Look to the Horizon

    Solving Radical Equations . . . . . . . . . . . . . . . . . . . . . . . 829

    This picture shows a surfer in a “barrel ride”—one of

    surfing's most sought-after experiences. Given the

    right conditions, a surfercan ride inside a wave

    as it breaks.

  • © Carnegie Learning

    764

    10

  • © Carnegie Learning

    765

    10.1

    LEARNING GOALS

    The word transpose means to switch two or more items. The word combines the

    Latin prefix trans-, meaning “across” or “over” and ponere, meaning “to put” or

    “place.” The word interchange means the same thing as transpose.

    Like many words, transpose is used in different ways in different fields:

    • In music, the word transpose is most often used to mean rewriting a song in a

    different key—either higher or lower.

    • In biology, a transposable element is a sequence of DNA that can move from one

    location to another in a gene.

    • Magicians use transposition when they make two objects appear to switch places.

    Keep an eye out for the word transpose in these lessons! What different ways can you

    use the word transpose?

    KEY TERMS

    • inverse of a function

    • invertible function

    • Horizontal Line Test

    In this lesson, you will:

    • Graph the inverses of power functions.

    • Use the Vertical Line Test to determine whether an inverse relation is a function.

    • Use graphs to determine whether a function is invertible.

    • Use the Horizontal Line Test to determine whether a function is invertible.

    • Graph inverses of higher-degreepower functions.

    • Generalize about inverses of even- and odd-degree power functions.

    With Great Power . . .Inverses of Power Functions

  • © Carnegie Learning

    766 Chapter 10 Radical Functions

    10

    PROBLEM 1 Strike That, Invert It

    Recall that a power function is a polynomial function of the form P(x) 5 a x n, where n is a

    non-negative integer.

    The graphs at the end of this lesson show these 6 power functions.

    L(x) 5 x, Q(x) 5 x2, C(x) 5 x3, F(x) 5 x4, V(x) 5 x5, S(x) 5 x6

    Cut out the graphs.

    1. The graph of the linear function L(x) 5 x models the width

    of a square as the independent quantity and the height of

    the square as the dependent quantity.

    x

    width

    (1)

    width

    (2)

    height

    (1)

    height

    (2)

    y

    L(x) 5 x

    a. Transform the cutout so that it shows the height as

    the independent quantity on the horizontal axis

    and the width as the dependent quantity on the vertical

    axis. Then sketch the resulting graph and label the axes.

    x

    y

    Resulting Graph

    How do I know when I’ve got the right graph?

    What partor parts of this graphdon’t make sense in

    terms of the quantitiesin this situation?

  • © Carnegie Learning

    10.1 Inverses of Power Functions 767

    10

    b. Describe the transformations you used to transpose the independent and

    dependent quantities.

    c. Is the resulting graph a function? Explain your reasoning.

    d. Compare the graph of L(x) 5 x to the resulting graph. Interpret both graphs in terms

    of the width and height of a square.

  • © Carnegie Learning

    768 Chapter 10 Radical Functions

    10

    What partor parts of this

    graph don’t make sensein terms of the quantities

    in this situation?

    2. The graph of the quadratic function Q(x) 5 x 2 models the

    side length of a square as the independent quantity and the

    area of the square as the dependent quantity.

    a. Transform the cutout so that it shows the area as the

    independent quantity on the horizontal axis and the side

    length as the dependent quantity on the vertical axis.

    Then sketch the resulting graph and label the axes.

    x

    side length

    (1)

    side length

    (2)

    are

    a

    (1)

    are

    a

    (2)

    y

    x

    y

    Q(x) 5 x 2 Resulting Graph

    b. Describe the transformations you used to transpose the independent and

    dependent quantities.

    c. Is the resulting graph a function? Explain your reasoning.

  • © Carnegie Learning

    10.1 Inverses of Power Functions 769

    10

    d. Cole used an incorrect strategy to transpose the independent and

    dependent quantities.

    Cole

    I can rotate the graph 90° clockwiseto transpose the independent anddependent quantities.

    x

    side length

    (1)

    side length

    (2)

    are

    a

    (1)

    are

    a

    (2)

    y

    xsid

    e le

    ng

    th

    (1)

    sid

    e le

    ng

    th

    (2)

    area

    (1)

    area

    (2)

    y

    Describe why Cole’s strategy is incorrect.

    e. Compare the graph of Q(x) 5 x 2 to the resulting graph you

    sketched. Interpret both graphs in terms of the side length

    and area of a square.

    What units are used to

    describe area?

  • © Carnegie Learning

    770 Chapter 10 Radical Functions

    10

    3. The graph of the cubic function C(x) 5 x 3 models the side length of a cube as the

    independent quantity and the volume of the cube as the dependent quantity.

    a. Transform the cutout so that it shows the volume as the independent quantity on the

    horizontal axis and the side length as the dependent quantity on the vertical axis.

    Then sketch the resulting graph and label the axes.

    x

    side length

    (1)

    side length

    (2)

    vo

    lum

    e

    (1)

    vo

    lum

    e

    (2)

    y

    x

    y

    C(x) 5 x 3 Resulting Graph

    b. Describe the transformations you used to transpose the independent and

    dependent quantities.

    c. Is the resulting graph a function? Explain your reasoning.

    d. Compare the graph of C(x) 5 x 3 to the resulting graph. Interpret both graphs in terms

    of the side length and volume of a cube.

  • © Carnegie Learning

    10.1 Inverses of Power Functions 771

    10

    PROBLEM 2 Across the Line

    Recall that a function f is the set of all ordered pairs (x, y), or (x, f(x)), where for every value of

    x there is one and only one value of y, or f(x). The inverse of a function is the set of all

    ordered pairs (y, x), or (f(x), x).

    By transforming the cutouts in Problem 1, you were able to see and sketch the inverses of

    the functions L(x) 5 x, Q(x) 5 x2, and C(x) 5 x3.

    1. Deanna discovered a way to use just one re#ection to transpose the independent and

    dependent quantities.

    Use your cutouts and Deanna’s strategy to sketch the graphs of the inverses of F(x) 5 x 4 ,

    V(x) 5 x 5 , and S(x) 5 x 6 .

    x

    y

    x

    y

    x

    y

    inverse of F(x) 5 x 4 inverse of V(x) 5 x 5 inverse of S(x) 5 x 6

    Deanna

    I can re"ect the graph across the line y = x by folding it diagonally to switch the independent and dependent

    quantities.

    x

    width

    (1)

    width

    (2)

    height

    (1)

    height

    (2)

    y

    x

    height

    (1)

    width

    (2)

    height

    (1)

    width

    (2)

    y

    x

    width

    (2)

    width

    (1)

    height

    (1)

    height

    (2)

    y

    xx xxxx

  • © Carnegie Learning

    772 Chapter 10 Radical Functions

    10

    If the inverse of a function f is also a function, then f is an invertible function, and its inverse

    is written as f 21 (x).

    2 Which of the 6 power functions that you explored are invertible

    functions? Explain your reasoning.

    3. You used the Vertical Line Test to determine whether or not the inverse of a power

    function was also a function. What test could you use on the original power function to

    determine if its inverse is also a function? Explain your reasoning.

    Talk the Talk

    1. How does the graph of a power function and the graph of its inverse demonstrate

    symmetry? Explain your reasoning.

    The Horizontal Line Test is a visual method to determine whether a function has an inverse

    that is also a function. To apply the horizontal line test, consider all the horizontal lines that

    could be drawn on the graph of the function. If any of the horizontal lines intersect the graph

    of the function at more than one point, then the inverse of the function is not a function.

    2. If a graph passes both the Horizontal Line Test and the Vertical Line Test, what can you

    conclude about the graph?

    Is there a pattern

    here?

  • © Carnegie Learning

    10.1 Inverses of Power Functions 773

    10

    3. If a graph passes the Vertical Line Test but not the Horizontal Line Test, what can you

    conclude about the graph?

    4. Given any point (x, y) on a graph, use a single transformation to transform the point to

    its inverse location. What do you notice?

    Be prepared to share your solutions and methods.

  • © Carnegie Learning

    774 Chapter 10 Radical Functions

    10

  • © Carnegie Learning

    10.1 Inverses of Power Functions 775

    10

    L(x) 5 x

    height

    (2)

    height

    (1)

    width

    (1)

    width

    (2) x

    y

    Q(x) 5 x 2

    are

    a

    (2)

    are

    a

    (1)

    side length

    (1)

    side length

    (2) x

    y

    C(x) 5 x 3

    vo

    lum

    e

    (2)

    vo

    lum

    e

    (1)

    side length

    (1)

    side length

    (2) x

    y

    F(x) 5 x 4

    dependent

    (2)

    dependent

    (1)

    independent

    (1)

    independent

    (2) x

    y

    V(x) 5 x 5

    dependent

    (2)

    dependent

    (1)

    independent

    (1)

    independent

    (2) x

    y

    S(x) 5 x 6

    dependent

    (2)

    dependent

    (1)

    independent

    (1)

    independent

    (2) x

    y

  • © Carnegie Learning

    776 Chapter 10 Radical Functions

    10

    are

    a

    (2)

    are

    a

    (1)

    side length

    (2)

    side length

    (1) x

    y

    height

    (2)

    height

    (1)

    width

    (2)

    width

    (1) x

    y

    dependent

    (2)

    dependent

    (1)

    independent

    (2)

    independent

    (1) x

    y

    vo

    lum

    e

    (2)

    vo

    lum

    e

    (1)

    side length

    (2)

    side length

    (1) x

    y

    dependent

    (2)

    dependent

    (1)

    independent

    (2)

    independent

    (1) x

    y

    dependent

    (2)

    dependent

    (1)

    independent

    (2)

    independent

    (1) x

    y

  • © Carnegie Learning

    777

    LEARNING GOALS

    10.2

    Many science museums display what is known

    as a Foucault pendulum. French physicist

    Léon Foucault used a device like this to

    demonstrate in 1851 that the Earth was rotating in

    space—although it was known long before that the

    Earth rotated on its axis.

    As a Foucault pendulum swings back and forth

    throughout the day, the Earth’s rotation causes it to

    appear to move in a circular direction. At the North

    Pole, a Foucault pendulum would appear to move

    clockwise during the day. At the South Pole, it

    would appear to move counterclockwise.

    The time it takes for one swing of a pendulum can

    be modeled by the inverse of a power function.

    KEY TERMS

    • square root function

    • cube root function

    • radical function

    • composition of functions

    In this lesson, you will:

    • Restrict the domain of f(x) 5 x 2 to graph the square root function.

    • Determine equations for the inverses of power functions.

    • Identify characteristics of square rootand cube root functions, such as domain and range.

    • Use composition of functions to determine whether two functions are inverses ofeach other.

    • Solve real-world problems using the square root and cube root functions.

    The Root of the MatterRadical Functions

    1 2

    Wire 2

    00 f

    eet

    long

    3

    4

    5 hours

    Foucault pendulum

    Swing

    Steel

    ball

    Apparent m

    otion of the pendulum

  • © Carnegie Learning

    778 Chapter 10 Radical Functions

    10

    PROBLEM 1 The Square Root Function

    In the previous lesson, you learned that the inverse of a power function de$ned by the set of

    all points (x, y), or (x, f(x)) is the set of all points (y, x), or (f(x), x).

    Thus, to determine the equation of the inverse of a power function, you can transpose x and

    y in the equation and solve for y.

    Determine the inverse of the power function f(x) 5 x 2 , or y 5 x 2 .

    First, transpose x and y.

    y 5 x 2 xxxx22 x 5 y 2

    Then, solve for y.

    √__ x 5 √

    __ y 2

    y 5 6 √__ x

    The inverse of f(x) 5 x 2 is y 5 6 √__ x .

    1. Why must the symbol 6 be written in front of the radical to write the inverse of the

    function f(x) 5 x2?

    2. Why is the inverse of the function f(x) 5 x2 not written with the notation f21(x)?

    Explain your reasoning.

    Is the function f (x) 5 x 2

    invertible?

    yyy 5555 xxx22

  • © Carnegie Learning

    10

    10.2 Radical Functions 779

    3. The table shows several coordinates of the function f(x) 5 x 2 .

    a. Use the ordered pairs in the table and what you know about inverses to graph the

    function and the inverse of the function, y 5 6 √__ x . Explain your reasoning.

    x f(x) 5 x 2

    23 9

    22 4

    21 1

    0 0

    1 1

    2 4

    3 9

    x

    2

    4

    22

    420

    24 22 8628 26

    y

    24

    26

    28

    6

    8

    b. What point or points do the two graphs have in common? Why?

    4. Describe the key characteristics of each function:

    Function: f(x) 5 x 2 Inverse function: y 5 √__ x

    Domain: Domain:

    Range: Range:

    x-intercept(s): x-intercept(s):

    y-intercept(s): y-intercept(s):

    The graph in Question 3 shows that every positive real number has 2 square roots—a

    positive square root and a negative square root. For example, 9 has 2 square roots, because

    (23 ) 2 5 9 and 3 2 5 9. The two square roots of 9 are 3 and 23.

    When you restrict the domain of the power function f(x) 5 x 2 to values greater than or equal

    to 0, the inverse of the function is called the square root function and is written as:

    f 21 (x) 5 √__ x , for x $ 0.

    Now thefunction and its

    inverse will be on one coordinate plane. How dœs each point (x, y) of the

    function map tothe inverse?

  • © Carnegie Learning

    780 Chapter 10 Radical Functions

    10

    5. Draw dashed line segments between the plotted points on the function for the

    restricted domain x $ 0 and the corresponding inverse points.

    a. List the ordered pairs of the points you connected.

    b. List the ordered pairs of the points that you did not connect.

    Explain why these points are not connected.

    6. Graph the square root function f 21 (x) 5 √__ x by restricting the

    domain of f(x) 5 x 2 .

    2

    4

    22

    420

    24 22 8628 26

    24

    26

    28

    6

    8

    x

    y

    Dœs restrictingthe domain of the function restrict

    the range ofthe inverse?

  • © Carnegie Learning

    10

    10.2 Radical Functions 781

    7.

    Brent

    f –1 (x) = 1 ___ f (x)

    Explain why Brent’s equation is incorrect.

    8. Describe the key characteristics of each function:

    Function: f(x) 5 x 2 , for x $ 0 Inverse function: f 21 (x) 5 √__ x

    Domain: Domain:

    Range: Range:

    x-intercept(s): x-intercept(s):

    y-intercept(s): y-intercept(s):

    9. Does the inverse function f 21 (x) 5 √__ x have an asymptote?

    Explain your reasoning.

    You’ve explored the relationship between the function f(x) 5 x2 and its inverse, both with a

    domain restriction and without a domain restriction.

    10. Make a conjecture about the relationship between the domain and range of a quadratic

    function and its inverse.

    Let’s look at more quadratic functions to explore domain restrictions and the relationship

    between the domain and range of a quadratic function and its inverse.

    Keep in mindthe restrictions placed

    on f (x) toproduce f 21 (x).

  • © Carnegie Learning

    782 Chapter 10 Radical Functions

    10

    11. Consider the function g(x) 5 x2 2 4 shown on the coordinate plane.

    a. How is g(x) transformed from the basic quadratic function f(x) 5 x2?

    b. Write the equation for the inverse of g(x) and sketch its graph.

    c. Is the inverse of g(x) a function? Explain.

    d. How is the inverse of g(x) transformed

    from the basic square root relation y 5 6 √__ x ?

    e. List the domain and range of g(x) and the inverse of g(x).

    Function: g(x) 5 x2 2 4 Inverse of g(x): y 5

    Domain: Domain:

    Range: Range:

    f. What conclusion can you make about the relationship between the domain and

    range of a quadratic function and its inverse when the domain is not restricted?

    g. How can you restrict the domain of g(x) so that its inverse is also a function?

    h. List the domain and range for both the quadratic function with the domain restriction

    and the inverse function.

    Function: g(x) 5 x2 2 4 Inverse of g(x): g21 (x) 5

    Domain restriction:

    Domain: Domain:

    Range: Range:

    28 26 24 22

    22

    24

    20 4 6 8

    28

    26

    8

    6

    4

    2

    x

    y

  • © Carnegie Learning

    10

    10.2 Radical Functions 783

    i. What conclusion can you make about the relationship between the domain and

    range of a quadratic function and its inverse when the domain is restricted?

    12. Consider the function h(x) 5 (x 2 4)2 shown on the coordinate plane.

    a. How is h(x) transformed from the basic quadratic function f(x) 5 x2?

    b. Write the equation for the inverse of h(x) and sketch its graph.

    c. Is the inverse of h(x) a function? Explain.

    d. How is the inverse of h(x) transformed

    from the basic square root relation y 5 6 √__

    x ?

    e. List the domain and range of h(x) and the inverse of h(x).

    Function: h(x) 5 (x 2 4)2 Inverse of h(x): y 5

    Domain: Domain:

    Range: Range:

    f. What conclusion can you make about the relationship between the domain and

    range of a quadratic function and its inverse when the domain is not restricted?

    g. How can you restrict the domain of h(x) so that its inverse is also a function?

    28 26 24 22

    22

    24

    20 4 6 8

    28

    26

    8

    6

    4

    2

    x

    y

  • © Carnegie Learning

    784 Chapter 10 Radical Functions

    10

    h. List the domain and range for both the quadratic function with the domain restriction

    and the inverse function.

    Function: h(x) 5 (x 2 4)2 Inverse of h(x): h21 (x) 5

    Domain restriction:

    Domain: Domain:

    Range: Range:

    i. What conclusion can you make about the relationship between the domain and

    range of a quadratic function and its inverse when the domain is restricted?

    13. When the domain of a quadratic function is restricted to create an inverse function,

    what is the lower bound of the domain? Explain your reasoning.

    14. Complete the table to describe the effect of each transformation on the inverse of the

    quadratic function.

    Transformation of

    Quadratic Function, f(x)

    Transformation of

    Inverse Function, f21(x)

    translation up D units

    translation down D units

    translation right C units

    translation left C units

  • © Carnegie Learning

    10

    10.2 Radical Functions 785

    15. Write the equation for the inverse of each quadratic function and identify the appropriate

    domain restrictions. Then, describe the domain and range of each function and its inverse

    without graphing the functions.

    a. f(x) 5 x2 2 2 b. f(x) 5 (x 1 2)2

    PROBLEM 2 The Cube Root Function

    The cube root function is the inverse of the power function f(x) 5 x 3 and can be written as

    f 21 (x) 5 3 Ï·· x .

    1. The table shows several coordinates of the function c(x) 5 x 3 .

    a. Use these points to graph the function and the inverse of the function, c 21 (x).

    x c(x) 5 x3

    22 28

    21 21

    0 0

    1 1

    2 8

    x

    2

    4

    22

    420

    24 22 8628 26

    y

    24

    26

    28

    6

    8

    b. Explain how you determined the coordinates for the points on the inverse of

    the function.

  • © Carnegie Learning

    786 Chapter 10 Radical Functions

    10

    c. What point or points do the two graphs have in common? Why?

    2. Why is the symbol 6 not written in front of the radical to write the inverse of the

    function c(x) 5 x 3 ?

    3. Why do you not need to restrict the domain of the function c(x) 5 x 3 to write the inverse

    with the notation c 21 (x)?

    4. Describe the key characteristics of each function:

    Function: c(x) 5 x 3 Inverse function: c 21 (x) 5 3 Ï·· x

    Domain: Domain:

    Range: Range:

    x-intercept(s): x-intercept(s):

    y-intercept(s): y-intercept(s):

    5. Does the inverse function c 21 (x) 5 3 Ï·· x have an asymptote? Explain your reasoning.

    The inverses of power functions with exponents greater than or equal to 2, such as the

    square root function and the cube root function, are called radical functions. Radical

    functions are used in many areas of science, including physics and computer science.

  • © Carnegie Learning

    10

    PROBLEM 3 Inverse by Composition

    You know that when the domain is restricted to x $ 0, the function f(x) 5 √__x is the inverse of

    the power function g(x) 5 x2. You also know that the function h(x) 5 3Ï··x is the inverse of the

    power function q(x) 5 x3.

    The process of evaluating one function inside of another function is called the composition

    of functions. For two functions f and g, the composition of functions uses the output of g(x)

    as the input of f(x). It is notated as (f + g)(x) or f(g(x)).

    To write a composition of the functions g(x) 5 x 2 and f(x) 5 √__ x when the domain of g(x) is

    restricted to x $ 0, substitute the value of one of the functions for the argument, x, of the

    other function.

    f(x) 5 √__ x g(x) 5 x 2 5 √√√√xx√√√√

    f(g(x)) 5 √__ x 2 5 x, for x $ 0

    You can write the composition of these two functions as f(g(x)) 5 x for x $ 0.

    5 √__√x√ g(x( ) x 55 xx225 √√x√

    1. Determine g(f(x)) for the functions g(x) 5 x2 and f(x) 5 √__x for x $ 0.

    If f(g(x)) 5 g(f(x)) 5 x, then f(x) and g(x) are inverse functions.

    2. Are f(x) and g(x) inverse functions? Explain your reasoning.

    10.2 Radical Functions 787

  • © Carnegie Learning

    788 Chapter 10 Radical Functions

    10

    3. Algebraically determine whether each pair of functions are inverses. Show your work.

    a. Verify that h(x) 5 3 Ï·· x is the inverse of q(x) 5 x 3 .

    b. Determine if k(x) 5 2 x 2 1 5 and j(x) 5 22 x 2 2 5 are inverse functions.

    ? 4. Mike said that all linear functions are inverses of themselves because f(x) 5 x is the

    inverse of g(x) 5 x.

    Is Mike correct? Explain your reasoning.

  • © Carnegie Learning

    10

    PROBLEM 4 Pendula

    The time it takes for one complete swing of a pendulum depends on the length of the

    pendulum and the acceleration due to gravity.

    The formula for the time it takes a pendulum to complete one swing is T 5 2p √__

    L __ g ,

    where T is time in seconds, L is the length of the pendulum in meters, and g is the

    acceleration due to gravity in meters per second squared.

    1. If the acceleration due to gravity on Earth is 9.8 m/ s 2 , write a function T(L) that

    represents the time of one pendulum swing.

    2. Graph the function T(L).

    x

    2

    4

    22

    42 6

    Length of Pendulum (meters)

    Tim

    e (seco

    nd

    s)

    120 8

    y

    24

    26

    28

    6

    8

    22 10 14

    3. Describe the characteristics of the function, such as its domain, range, and intercepts.

    Explain your reasoning.

    10.2 Radical Functions 789

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    4. How long does it take for one complete swing when the length of the pendulum is

    0.5 meter?

    5. A typical grandfather clock pendulum completes a full swing in 2 seconds. Use your

    graph to determine the approximate length of a grandfather clock pendulum.

    Talk the Talk

    1. How can knowing the domain, range, intercepts, and other key characteristics of a

    power function help you determine those characteristics for the function’s inverse?

    Explain your reasoning.

    2. When a function has an asymptote, will its inverse have an asymptote? If so, describe

    the location of the asymptote for the function’s inverse.

    Be prepared to share your solutions and methods.

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    LEARNING GOALS

    Some people think that they won’t need math if they choose to work in an artistic

    career. Not so! Much of the graphic and animation work you see on television, in

    movies, and even in print and art galleries is done on the computer, using

    sophisticated graphic design software.

    To use many graphic design programs, a knowledge of transformations, like

    reflections and rotations, coordinate systems, ratios, and on and on, is essential to

    working efficiently and accurately—and to get just the right effect.

    How do you think knowledge about power functions and radical functions can be

    used in graphic design?

    In this lesson, you will:

    • Graph transformations of radical functions.

    • Analyze transformations of radical functions using transformational function form.

    • Describe transformations of radical functions using algebraic, graphical, andverbal representations.

    • Generalize about the effects of transformations on power functions and their inverses.

    Making WavesTransformations of Radical Functions

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    PROBLEM 1 Shifting Sands

    You have already explored transformations with functions in many function families.

    You learned that transformations performed on any function f(x) to form a new function g(x)

    can be described by the transformational function:

    g(x) 5 Af(B(x 2 C)) 1 D

    Recall that this transformational function generalizes to any function. Changes to the A- or

    D-values dilate, translate, or re#ect a function vertically. Changes to the B- or C-values

    dilate, translate, or re#ect a function horizontally.

    For the square root function and cube root function respectively, the transformational

    function can be written as:

    s(x) 5 A √________

    B(x 2 C) 1 D c(x) 5 A 3 Ï········ B(x 2 C) 1 D

    1. Determine how the values of A, B, C, or D transform the graph of f(x) 5 √__ x or q(x) 5 3 Ï·· x .

    a. g(x) 5 A √__

    x or r(x) 5 A 3 Ï·· x , for positive and negative values of A.

    b. h(x) 5 √__ x 1 D or u(x) 5 3 Ï·· x 1 D, for positive and negative values of D.

    c. j(x) 5 √______ x 2 C or v(x) 5 3 Ï······ x 2 C , for positive and negative values of C.

    d. k(x) 5 √___ Bx or w(x) 5 3 Ï··· Bx , for positive and negative values of B.

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    A group of art students had the idea to use transformations of

    radical functions to create a logo for the Radical Sur$ng School.

    To start, they graphed the function f(x) 5 √__ x , for 0 # x # 14, and

    shifted copies of the curve to create the waves g(x), h(x), and k(x).

    f(x) g(x) h(x) k(x)

    x0 5

    y

    5

    10

    10

    2. Do the transformations of f(x) shown on the graph take

    place inside the function or outside the function?

    Explain your reasoning.

    3. What value or values in the transformation function were changed to create

    these curves? Explain your reasoning.

    4. Write the domain of each transformed function as an inequality statement using the

    dimensions of the logo.

    10.3 Transformations of Radical Functions 793

    The square root function has

    a restricted domain. Now the dimensions ofthe logo will restrict it

    even more!

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    ? 5. Devin, Stuart, and Kristen each wrote an equation for a function that was added to the

    graph $rst using the transformational function form of f(x), and then in terms of x.

    • Devin’s equation: g(x) 5 f(x) 2 45 √

    __ x 2 4

    • Stuart’s equation: h(x) 5 f(x 2 8)

    5 √______ x 2 8

    • Kristen’s equation: k(x) 5 f(x 1 12)

    5 √_______ x 1 12

    a. Describe whether each student’s equation is correct or incorrect.

    Explain your reasoning.

    b. Write the correct equations to describe the 3 new functions shown in the graph $rst

    using transformational function form of f(x), and then in terms of x. Finally, write their

    domains as inequality statements.

    f(x) 5 √__ x Domain:

    g(x) 5 5 Domain:

    h(x) 5 5 Domain:

    k(x) 5 5 Domain:

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    10.3 Transformations of Radical Functions 795

    6. The students decide that re#ecting each curve, g(x), h(x), and k(x), across the respective

    lines where x 5 C will make them look more like waves crashing on the beach.

    a. Graph the resulting functions f9(x), g9(x), h9(x), and k9(x). Write each function $rst in

    terms of their transformations of f(x), g(x), h(x), and k(x), and then in terms of x.

    Finally, state the domain of each.

    x0 5

    y

    5

    10

    10

    b. Describe how you used the transformation function to determine the equations of

    the new functions.

    c. How did the domain of each transformed function change as a result of the

    re#ection across x 5 C?

    d. Why does your graph show only 3 curves when the original graph had 4?

    Explain your reasoning.

    You canuse the prime symbol (’) to

    indicate that a functionis a transformation of

    another function.

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    7. Suppose the students wanted to re#ect the 3 new waves g9(x), h9(x), and k9(x) across the

    line y 5 0.

    a. Describe how you can use the transformational function to determine the equations

    of the re#ected functions.

    b. Write three new functions using transformational form to represent each re#ection

    of g9(x), h9(x), and k9(x), and then each in terms of x. Use the double prime symbol (0)

    to indicate each transformed function. Finally, write the domain of each

    transformed function.

    8. Jamal wants to add waves below the 3 waves as

    shown. These waves should be copies of g9(x), h9(x),

    and k9(x), except half as high and shifted to the left

    2 units.

    a. Write 3 new functions q(x), r(x), and s(x) in terms of

    g9(x), h9(x), and k9(x) to create the waves that Jamal

    wants. Make sure to write the domains of each

    transformed function.

    b. Describe how you used what you know about transformational function form to

    determine your answer to part (a).

    x0 5

    y

    5

    10

    10

    g9(x) h9(x) k9(x)

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    10.3 Transformations of Radical Functions 797

    9. The art students want to add some clouds to the top of the logo. For the clouds, they

    will use the inverses of cubic functions. They start with the function c(x) 5 2 3 Ï·· x 1 14.

    x0 5

    y

    5

    10

    10

    a. Transform this function and write 2 more equations to create the clouds the students

    want. Graph the results.

    b. Color the graph to show the waves and the clouds on the logo.

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    In many graphic design programs, a trace path can be created. A trace

    path is an invisible line or curve that acts as the baseline of text that is

    added to the design. When you insert text on a trace path, the text follows

    the line or curve. The text shown, for example, follows the curve f(x) 5 2 x 2 .

    10. The art students are experimenting with different square root and cube root function

    graphs to use as trace paths for the sur$ng school’s name: Radical Sur$ng School.

    They have narrowed their trace paths down to 2 choices. The graphs of the functions

    are shown.

    h(x) 5 3 Ï········ 2(x 2 1) j(x) 5 2 3 Ï······ x 2 1

    x

    2

    4

    22

    1050

    210215220 25 2015

    y

    24

    26

    28

    6

    8

    a. Graph the function 3 Ï·· x and list its domain, range, and x- and y-intercepts.

    Domain:

    Range:

    x-intercept(s):

    y-intercept(s):

    b. Compare and contrast the graphs of the functions and their equations. What do

    you notice?

    c. Compare the effects of increasing the A-value with increasing the B-value in a

    radical function. What do you notice?

    Yo

    ur T

    ext H

    ere

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    d. Label each graph with the correct equation and include the domain restrictions.

    11. Choose one of the cube root functions as a trace path for the title of the sur$ng school.

    Or, write a different radical function to use as a trace path. Graph the function on the

    coordinate plane in Question 8, and write the title of the school on the trace path.

    Be prepared to share your methods and solutions.

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