radical functions 10 - hhs algebra ii -...

763

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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.


764

10


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


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?


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.



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






10

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


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?



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




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.



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


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



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?



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.



10



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



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


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


• 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



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


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?



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?


10


7.

Brent

f –1 (x) = 1 ___

f (x)

Explain why Brent’s equation is incorrect.


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

Domain: Domain:

Range: Range:



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


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).



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


10


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



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


10


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.



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)?


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

Domain: Domain:

Range: Range:



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.


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 3

Ï··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

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.


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.





10

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?


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.


791

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?


• 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

10.3



10

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.


10

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?


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!



10

? 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 4

5 √__ 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.


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:


10


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.

x05

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?


You canuse the prime symbol (’) to

indicate that a functionis a transformation of

another function.



10

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)


10


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.

x05

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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radical functions 10 - hhs algebra ii -...

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