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10.3 Graphing Cube Root FunctionsSseemtia& Que*tiem; How can you use transformations of parent cube root functions to graph

functions of the form f (*): o'r/(r - n) t- k or g(x) : 'lf,Q - n) * f,

ffiryffiffiK Graphing and Analyzing the ParentCube Root Function

The cube root parent function is f(x) : 1/i. To graph f (x),choose values ofx and find corresponding valuesof7. Choose both negative and positive values of x.

Graph the function f (x) - W.Identify the domain and range of the function.

Make the table of values.

Use the table to graph the function.

Identify the domain and range of the function.

The domain is the

The range is

Does the graph of f (x) : li have any symmetry?

The graph has - - ,-- --

1. Can the radicand in a cube root function be negative?

Module 10 513 Lesson 3

Ef[$$ffi Graphing Cube Root FunctionsTransformations of the Cube Root Parent Function/(r) = W

Forthefunctionf(r) - a{x-h +k,(h,k)isthegraph'spointof symmetry. Usethevalues of a,h,andkto draw each graph. For example, the point (i, t) on the graph of the parent function becomes the point(t + lr, a + k) on the graph of the given function.

{_i Graph the cube root functions.

The transformations of the graph of f (x) - W that produce the graph of g(x) are:

. a vertical stretch by afactor of2

. a translation of 3 units to the right and 5 units up

Choose points on f(x) - W and find the transformed corresponding points on g(x) : Zlfu - 3 a 5.

Graph g(x) = 2{Q - 3 * 5 using the transformed points.

@ Graphg(x):21/x-tas.

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Vertical translation f(x) + k Y:Ux+Z 3unitsuPy:Ui-q 4unitsdown

Horizontal translationy:Ux-Z 2unitsrighty:fx+1 lunitsleft

Vertica I stretch/compressiony : 6 U* vertical stretch by 6

y : !1,/x vertical compression by ]

Horizontal stretch/compression

y: Whorizontal stretch by 5

y: trhx horizontal compression byJ

y - - l* across x-axis

y - tr/-* across y-axis

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Module 10 I514 Lesson 3

@ Graphg(x) = {LJl - AThe transformations of the graph of f (x) - W that produce the graph ofg(x) are:

. a horizontal stretch by a factor of2

. a translation of 10 units to the right and 4 units up

Choose points on /(x) : Ui and find the transformed corresponding points on g(x) -"l-Graph g@) : Uif. - i0) + 4 using the transformed points.

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Graph the cube root function.

2. GraphS(x) : f vi Lr 6.

Module 10 515 Lesson 3

Writing Cube Root FunctionsGiven the graph of the transformed function g(x1 : oilT* - h) a k, you can determine the values of the parametersby using the reference points (-1, t), (0, o), and (t, t) thit you used to graphg(x) in the previous example.

For the given graphs, write a cube root function.

@ Write the function in the formg(O: oU* -i + k.

Identify the values of a, h, andk.

Identify the values of h and k from the point of symmetry.

(t,X): (t, z), so h: I andk:7.Identify the value of a from either of the other two referencepoints (-1, i) or (1, 1).

The reference point (1, 1) has general coordinates (t, + t, a + k).Substituting I for h andT for k and setting the general coordinatesequal to the actual coordinates gives this result:

(t +r,a+k)- (2,, o + 7): (2,l),so a:2.a:2 h: I k:7The function is g(x) : 2:,/x - | + 7.

@ Wit. the function in the formgl*1 : iffi I + t.Identify the values of b, h, and k.

Identify the values of h andk from the point of syrnmetry./\tt(l,t):\2, )soh:2andk:Identify the value of b from either of the other two reference points.

The rightmost reference point has general coordinates (U + t ,1 + k).Substituting 2 for h and **--*" for k and setting the general coordinatesequal to the actual coordinates gives this result:

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(o*r,,*b- .

The function isg(x) :

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Module 10 516 Lesson 3

1!ffi:sl*t3;ss*g:{*siffiffiFor the given graphq write a cube root function.

l. . Wiite the function in the formg(r) - a{i-11 q ft.

4. Wite the function in the formg(x) :

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Lesson 3Module 10

' The shoulder height h (in centimeters) of a particularelephant is modeled by the function h(t) : 62.1W + 76,

where f is the age (in years) of the elephant. Graph thefunction and examine its average rate of change over theequal r-intervals (0, 2o), (20,40), and (+0, oo). what ishappening to the average rate ofchange as the f-values ofthe intervals increase? Use the graph to find the heightwhen f : 35.

Graph h(t) : 62.r:'/i + 76.

The graph is the graph of f (x) - W translated up 76and stretched vertically by a factor of 62.1. Graph thetransformed points (o,zo), (a, zoo.z), (zz,zez.z), and,(A+, n+.+). Connect the points with a smooth curve.

First interval:

Average Rate ofchange =:8.43

Modeling with Cube Root Functionsfunctions to model real-world situations.You can use cube root

Second interval:

Average Rate ofchange =:

Third interval:

Average Rate of change =

325.

260.

195 .

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The average rate of change is becoming less.

Drawing a vertical line up from 35 gives a value of about 280 cm.

Module 10 518 Lesson 3

@ The velocity of a l40O-kilogram car.at the end of a 400-meter run is modeled bythe function v: I5.2W, where v is the velocity in kilometers per hour andp isthe power of its engine in horsepower. Graph the function and examine its averagerate of change over rhe equalp-intervals (0, 60), (60, 120), and (tZO, tSO). What ishappening to the average rate of change as thep-values of the intervals increase? Usethe function to find the velocity when p is 100 horsepower.

Graph V:15.2W.

The graph is the graph of f (x) - W stretched -.-*--*-.-.- by a

factor of 15.2. Graph the transformed points (0, O), (S, **-),(zz,

-), (o+,

--) , (125, ), and, (2rc,-).

Connect the points with a smooth curve.

The rate of change over the interval (0, oO) is

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60-0 which is about 20 60 100 140 180Power (hp)

::ilThe rate of change over the interval (OO, tZO) is ffi whichis about

:-;.iThe rate of change over the interval (tZO, tSO) " - ,rO _ f ZO- which is about

The average rate of change is becoming

Substitute p = 100 in the function.

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The velocity is about ""kmih.

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5. The fetch is the length of water over a wind that is blowing in the same direction. The functiontA : I .t i,ff, nt^t""the speed of the wind s in kilometers per hour to the fetch/in.kilometers. Graph thefuirction and examine its average rate of change over the intervals (zo, ao), (ao, t+o), and (tao, 200). Whatis happening to the average rate of change as thefvalues of the intervals increase? Use the function to findthe speed of the wind whenJ: ${.

20 60 100 140 180Fetch (km)

5. Discussion Why is the dornain of f (x) - W all real numbers?

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7. Identift rryhich transformations (stretches or compressions, reflections, and translations) of f(x) : x3change the following attributes of the function.

a. tocation of the point of symmetry

b. Symmetry about a point

Hs*ertial egg$1{}$ Check-ln How do_pelqqq[ers a, b, h, and k effect the graphs off (x) : ,'r/ (* - n) * k nd g!x) : 'rf tS* .n) + *t

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Module 10 520 Lesson 3

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1. Graph the function g(x) : Ui + 3. Identifythe domain and range of the function.

s. s(x): !V=

7. g(x): -2U-x * 3

g(x) : Vi - 5. Identify the ' Extra Practice

domain and range of the function.

Predict the effect of parameters on the graphs of cube root functions.3. g(x): Ui + e 4. g(x): i/x - s

2. Graph the function

6. g($: ffi

. Online Homework

. Hints and Help

Module 10 521

8. g(x):{x1+-z

Lesson 3

Graph the cube root functions.

g. g(x) : 3i/* + 4

10. g(x) :21./i * 3

tl. g(r):{x-t+Z

Module 10 522 Lesson 3

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For the given graphs, write a cube root function.

12. Write the function in the form g (x) : a'{x - h + k.

13. Write the function in the form g(x) : oi/* - n + *.

14. Write the function in the formg(x) : It. - nl

Module l0 523 Lesson 3

15.

16 24 32 40 48Volume

The radius of a stainless steel ball, in centimeters, can bemodeled by r(m): 0.31 Um,where z is the mass of thein grams. Use the function to find r when m : 125.

17. Describe the steps for graphing g(x): l,/* + S - tt.

Modeling Write a situation that can be modeled by a cube rootfunction. Give the function.

19. Find the y-intercept for the function y : 6l/16 - 7 1 1r.

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Module l0 524 Lesson 3

20. Find the x-intercept for the function y : s{x - h I lc.

21. Describe the translation(s) used to get g(x) -Select all that apply.

A. translated 9 units right E.

B. translated 9 units left F.

C. translated 9 units up G.

D. translated 9 units down H.

Vx-9*12fromf(x):1/7.

translated 12 units right

translated 12 units left

translated 12 units up

translated 12 units down

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22. Explain tha Errcr Tim says that to graph g(x) - Vx - 6 f 3, you need totranslate the graph of f (x) : yi 6 units to the left and then 3 units up. Whatmistake did he make?

eammunicate Mathematica! Ideas Why does the square root function have a restricted domain butthe cube root function does not?

Justify Reascning Does a horizontal translation and a vertical translation of the function f (x) : {Vafifect the functions domain or range? Explain.

Module l0 525 Lesson 3

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Lesson Ferforrman(e Task

The side length of a 243-gram copper cube is 3 centimeters. Use this information to write amodel for the radius of a copper sphere as a function of its mass. Then, find the radius of acopper sphere with a mass of 50 grams. How would changing the material affect the function?

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Module 10 526 Lesson 3

Xse.cmti*$ Qa;estis*r: How can you use radical functions to solvereal-world problems?

Find the inverse functionf '(x) for f(x) :2x3 + 10.

Replaceflx) with y.

Solve for x3.

Take the cube root.

Switch x andy.Replace 7 withf 1(x).

Y :2x3 + lov- 10/-- +J2 -^

KeyVocabularycube root function

{funcion de rsiz tubica)index (indice)inverse function

{funcion inversa)square root function

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Graph 7 : - G- f 2. Describe the domain and range.

Sketch the graph of y - - ,,C.It begins at the origin and passes through (t, -t).For y - -!/x - 3 + 2,h: 3 and k : 2.Shift the graph of y : -Ji right 3 units and up2 units. The graph begins at (2, z) and passesthrough (4, 1).

Domain: {x:x} 3} Range: {lry <2}

GraphT:Y*aZ_ +.

Sketch the graph of y - Ui.It passes through (-t, -t), (0, o), ana (t, i).For y : i/x + 2 - 4, h : -2 and k : -4.Shift the graph of7 : W left 2 units and down4 units. The graph passes through (-:, -S),(-2,-+), and (-t, -:).

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Module 10 527 Study Guide Review

EXgRCtSESFind the inverse of each function. Restrict the domain where necessary. {i*ss** ?#. ?}

1. f(x): t6x' 2. f(x):x'-20

Identify the transformations of the graph f (x) :function. &sss*rt f$"$

3. g(x): -J4.

*G that produce the graph of the

,l4. h(x) : ),/x + 1

Identify the transformations of the graph/(*) - Vi that produce the graph of thefunction. iless*n ?CI.3j

s. g(x) :4lx 6. h(x) -- i/v - 5 a 3

X &"X *'B &.& ffimd&sm& ffiaxrxe*&wms

Find the inverse of each function. State any restrictions on the domain.{**sson ?#"?}

1. f(x): x' + 9 2. f (x): -7x'

3. f(x): -2x3 + 7 4. f(*): 5x2 + 3

Identify the transformations of the graph f (x) : Ji or h(x) : Ui thatproduce the graph of the function. ffl*sscyrs ?#.3, f #.3j

5. g(x):!'t._ s_ + 6. g(x):fs*+z

7. g(x):{x-4-l 8. g(x):inx + ro

E$SENTIEt Q['ESTISIU9. How do you use a parent square root or cube root function to graph a transformation of the

function? fless*xs ?0.4 f S"3,,

. Online Homework

. Hints and Help

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Module l0 529 Study Guide Review

MODULE 1Oe,l*xcr &EVtrw

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1. Look at each equation' ls it the inverse of f(x) : x3 - 16? select Yes or No for A-c'Assessment Readiness

A. f-1(x1:{x- 16

B. f-'1x1:{ x+ 16

C. f-l(x):i/x+t02. Consider the graphed function. Choose True or False for each statement.

A. The equation for the function is Q Ves

y:t/y-1-2.B. The function has exactly one y-intercept. Q Ves

C. The range of the function is y < -2. Q Ves

ONo

A plane's average speed when flying from one city to another is 550 mi/hand is 430 mi/h on the return flight. To the nearest mile per hour, what is

the plane's average speed for the entire trip? Explain your answer.

The kinetic energy E (in joules) of a 1250-kilogram compact car is givenby the equation E : 625s2, where s is the speed of the car (in meters per second).Write an inverse model that gives the speed of the car as a function of itskinetic energy. lf the kinetic energy doubles, will the speed double?Explain why or why not.

Module 10 530 Study Guide Review