Section 5.2 Families of Functions · Families of Functions provide a convenient way to analyze various types of functions that occur frequently in the real world. This workbook presents

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Section 5.2 Families of FunctionsTERMINOLOGY 5.2

Previously Used:

Function Function Notation Graphing Window

New Terms to Learn:

Absolutely Value Functions

Your definition

Formal definition

Example

Aymptotes (Horizontal and Vertial)

Your definition

Formal definition

Example

Exponential Functions

Your definition

Formal definition

Example

206

Chapter 5 — Functions

206

Family of Functions

Your definition

Formal definition

Example

Linear Functions

Your definition

Formal definition

Example

Logarithmic Functions

Your definition

Formal definition

Example

Logistic Functions

Your definition

Formal definition

Example

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Polynomial Functions

Your definition

Formal definition

Example

Power Functions

Your definition

Formal definition

Example

Quadratic Functions

Your definition

Formal definition

Example

Radical Functions

Your definition

Formal definition

Example

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208

Rate of Change

Your definition

Formal definition

Example

Rational Function

Your definition

Formal definition

Example

Function Behavior

Your definition

Formal definition

Example

READINGASSIGNMENT 5.2 Sections 11.1, 11.2 and 12.2 through 12.4

READINGANDSELF-DISCOVERYQUESTIONS 5.21. Whatareexamplesofcommonlyoccurringfunctions?

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2. Whataresomekeycharacteristicsofafunction?

3. Whatarecomponentsofafamilyoffunctionscard?

KEYCONCEPTS 5.2Families of Functionsprovideaconvenientwaytoanalyzevarioustypesoffunctionsthatoccurfrequentlyintherealworld.Thisworkbookpresentsacomprehensiveanalysislaidoutlikeabaseballcardinkeycomponentsforeachof14differentfamiliesoffunctionswhichwecallafamilyoffunctionscard.Thepurposeofthecardistoprovideaconcisewayofanalyzingaspecificfamilyoffunctions.

Limitation/Caution Althoughcomprehensive,thesecardsdonotprofileeveryaspectofthatfamilyoffunctions.

Inanalyzing a function,youidentifykeycharacteristicsof thatfunctionincludingdomainandrange,trends(increasingordecreasingbehavior),intercepts,andasymptotes.

Limitation/Caution: Caremustbetakentoensurethatyouidentifyalltheimportantbehaviorsofthefunction.

Bygraphingfunctionsonthesamedomain and range,youcanmoreeasilycompareandcontrast thebehaviorsoffunctions.

Limitation/Caution: Youneednotuse thecomplete rangeof the functions.Also, the rangecanbedifficulttodeterminebeforeyougraphthefunction.

Youcanobservethetrendsoffunctions(whentheyareincreasinganddecreasing)fromtheirgraphs.

Limitation/Caution: The functionsmay have kinks where the function briefly decreases and thencontinuestoincreasesorviceversa)thataredifficulttodiscern.

Theintercepts(wherefunctionscrossthex-ory-axes)areimportantfeaturesofgraphsoffunctionsastheyarepointsofreferenceforthegraphsandgiveyouinformationaboutwhenthefunctionvalueis0andwhatthefunctionvalueiswhenthedomainvalueis0.

Limitation/Caution: x-interceptscanbedifficulttodetermineexactly.

Ifyouknowthegeneralshape ofafamilyoffunctions,thenyouknowthegeneralshapeofanyfunctionofthatfamily.Suchknowledgecanguideyouingraphingthefunctionandindeterminingthebehaviorofthefunction.

Limitation/Caution: Aswith trends, the general shapemay have slight variations that are hard todetermine.

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You use horizontal asymptotes in comparing the long-term behaviors of functions and the vertical asymptotestodetermineatwhatpointsfunctionbehaviorbecomesarbitrarilylarge(inabsolutevalue).Usuallyasymptotesarefoundalgebraically.

Limitation/Caution: Asymptotescanbeinappropriatelygraphedongraphingcalculators.

Functions that are built from members of families of functions can have interesting features orcharacteristics. Suchfeaturescanbecombinationsoffeaturesfromthecomponentfamilies.

Limitation/Caution: Interestingfeaturesaresometimesintheeyeofthebeholder.

METHODOLOGY 5.2INTERPRETINGAFUNCTIONUSINGAFAMILYOFFUNCTIONCARD

Aswehaveseen,graphsoffunctionsareapowerfulwaytounderstandthebehaviorofafunction.

Limitation/Caution:Therearemanyfunctionsthatdonotfallintoanyfamily.

Example 1 Example 2

Interpretthefunctiongivenby26 2 4y x x+ = − Interpretthefunctiongivenby 23 4 12x y x= − +

Steps Discussion1 Rewrite in

functional notation

If the function is given in equation form, solve for the dependent variable and present in functional notation

Ex 1

2

2

2

6 2 42 4 6

( ) 2 4 6

y x xy x xf x x x

+ = −

= − −

= − −

Ex 2

2 Determine family of functions

Compare the symbolic structure of the function to each family of functions to determine which is the appropriate family.

Ex 1 QuandraticFunctionFamily

Ex 2

3 Rewrite the function in the standard form

Use the standard forms that appear on the family of functions card Ex

1 2( ) 2 4 6f x x x= − −

Thisisthestandardformdisplayedonthecard.

Ex 2

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Steps Discussion4 Graph the

function based on the key characteristics of this family of functions

Use the key characteristics of this family to determine the intercepts, key ordered pairs, and its basic behaviors.

Exam

ple

1

2

2

2 4 6 02 3 0

( 1)( 3) 01, 3

x xx xx x

x

− − =

− − =+ − == −

2

4 12 2(2)

2(1) 4(1) 62 4 68

bxa

yyy

−= − = − =

= − −= − −= −

Vertex: Zeros:

-6 6

y2

1

-11

x

(1, –8)

f(x) = 2x2 – 4x – 6

-1 3

Example 2

y

x

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212

Steps Discussion5 Validate your

graph.Choose 5 key ordered pairs from the graph to make sure those ordered pairs satisfy the symbolic representation of the function.

Exam

ple

1

x y

–1 0

3 0

1 –8

0 –6

2 –6

2

2

2

2

2

( 1) 2( 1) 4( 1) 6 0(3) 2(3) 4(3) 6 0(1) 2(1) 4(1) 6 8(0) 2(0) 4(0) 6 6(2) 2(2) 4(2) 6 6

fffff

− = − − − − =

= − − =

= − − = −

= − − = −

= − − = −

Exam

ple

2

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MODEL 5.2

Interpretthefunctiongivenby:

2

19

xyx−

=−

Step 1 Rewrite in function notation 2

1( )9

xg xx−

=−

Step 2 Determine which family of functions

RationalFunctionFamily

Step 3 Rewrite the function in the standard form

1( )( 3)( 3)

xg xx x

−=

+ −

Step 4 Graph the function based on the characteristics of this family of functions

10-10

-7

7

1 1

y

x

g(x) = x – 1x2 – 9

Step 5 Validate your graph Choose5keypoints.

x y1 03 undefined–3 undefined0 0.1–1 0.3

2

2

2

2

2

(1) 1( ) 0(1) 9(3) 1(3) (3) 9( 3) 1( 3) ( 3) 9(0) 1 1(0) 0.11(0) 9 9( 1) 1 2( 1) 0.25( 1) 9 8

g x

g undefined

g undefined

g

g

−= =

−−

= =−− −

− = =− −− −

= = =− −

− − −− = = = −

− − −

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Linear f(x) = mx + b

-10 10

y

1

1

6

-6

x

f(x) = 2x + 3

-10 10

y

1

1

6

-6

x

f(x) = –½ x + 3

Logarithm f(x) = logb x (andb>0)

-4

-6

2

1

15

7

f(x) = log2(x)

y

x

-4

-6

0.5

1

15

7 y

x

f(x) = log (x)12

Logarithm f(x) = logb x (andb>0)

Domain All positive real numbersRange All real numbers

ChaRaCteRistiC 1 There is a vertical asymptote at y = 0

ChaRaCteRistiC 2 There is no horizontal asymptote

appeaRanCe

Always increasing if b > 1 and always decreasing if b < 1.

inteResting FeatuRes

These function increase very slowly when b > 5 and x > 100 but do not have horizontal

asymptotes.

Real-WoRlD appliCation

The loudness of sound is measured with a logarithmic function.

Linear f(x) = mx + b

Domain All real numbersRange All reals except when m = 0 and

f(x) = b. Then the range is b.

ChaRaCteRistiC 1 (0, b) is the y-intercept ChaRaCteRistiC 2 m is the slope

appeaRanCe

A straight line that is increasing, decreasing, or constant depending on whether m is positive,

negative or zero, respectively.


These functions have a constant rate of change and the algebraic definition of these functions

can be determined from two ordered pairs.


The manufacturing cost of a cell phone is a linear function of the number of cell phones

produced.

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Polynomial p(x) = anxn + ... + a1x

1 + a0 (andnapositiveinteger)

-6 6

y

10

1

87

-87

x

f(x) = 3x3 + 4x2 – 27x + 16

-6 6

y

20

1

140

-140

x

f(x) = x4 – 6x3 – 3x2 + 56x – 48

Exponential f(x) = b • g x

(andg>0,b>0)

12

y

1

11

-20.5 -7

x

f(x) = 3•2x

12

y

1

11

-20.5 -7

x

f(x) = 5 •

12

x

Exponential f(x) = b • g x

(andg>0,b>0)

Domain All real numbersRange All positive real numbers

ChaRaCteRistiC 1 (0, b) is the y-interceptChaRaCteRistiC 2 g is the growth rate

appeaRanCe

Always increasing if g > 1 and always decreasing if g < 1.


The algebraic definition of these functions can be determined from two ordered pairs.


The value of an investment as it grows at a constant, compounded interest rate is an

exponential function of time. In an animal bone, the amount of radioactive carbon-14 as it decays

is an exponential function of time.

Polynomial p(x) = anxn + ... + a1x

1 + a0 (andnapositiveinteger)

Domain All real numbersRange If n is odd, all reals. If n is even and an < 0,

then all reals less than the maximum value of the function. If n is even and an > 0, then all reals greater than the maximum value of the function.

ChaRaCteRistiC 1 Crosses the x-axis no more than n times

ChaRaCteRistiC 2 If n is odd, it will cross the x-axis at least once.

appeaRanCe

As x gets farther from 0, the function approximates the behavior of its term of highest degree (anx

n).


To handicap races between dragsters and “funny cars,” the National Hot Rod Association uses a

polynomial: f(x) = 71.682x -60.427x2 + 84.710x3 -27.769x4 +

4.296x5 - 0.262x6

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216

Quadratic f(x) = ax2 + bx + c

-10 10

y

1

1

6

-6

x

(–3, –4)

f(x) = 2(x –(–3))2 + (–4)

-10 10

y

1

1

7

-7

x

(3, 2)f(x) = –1(x – 3)2 + 2

Rational f(x) = p(x) q(x)

(inreducedform)

12-12

-8

8

1

1

y

x

f(x) = (x – 2)(x + 3)(x + 1)(x – 5)

Quadratic f(x) = ax2 + bx + c

Domain All real numbersRange If a > 0, then the range is all real

numbers greater than k. If a < 0, then the range is all real

numbers less than k.

ChaRaCteRistiC 1 The vertex (h, k) is on the line of symmetry

ChaRaCteRistiC 2 The line of symmetry is x = h

appeaRanCe

Looks like╰╯if a > 0 or╭╮if a < 0.


y − k = a(x − h)2 is another symbolic representation of the function


The rate at which a ball falls in the Earth’s gravitational field is

a quadratic function of time.

Rational f(x) = p(x) q(x)

(inreducedform)

Domain All numbers for which q(x) ≠ 0Range Can be all real numbers

ChaRaCteRistiC Has the same number of vertical asymptotes as the solutions to q(x) = 0

appeaRanCe

Crosses the x-axis the same number of times as the number of solutions to p(x) = 0


These functions have both vertical and horizontal asymptotes when q is not a constant.


A rational function can be used to determine how much pure water needs to be added to dilute a 20% saline solution to

a 5% saline solution.

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Power (q is odd) f(x) = a • x p/q

(andp,qpositiveintegers,reducedform, q≠1)

-10 10

y

2

1

8

-2

x

f(x) = 2 • x23

-10

y

2

1

20

-20

x10

f(x) = 2 • x53

Power (q is even) f(x) = a • x p/q

(andp,qpositiveintegers,reducedform,q≠1)

13

y

1

14

-1 1 x

f(x) = 3 • x½

-4 10

y

2

1

20

-20

x

f(x) = 2 • x54

Power (a > 0, q is odd) f(x) = a • x p/q


Domain All non-negative real numbersRange All non-negative real numbers

ChaRaCteRistiC The point (0, 0) is on the graph

appeaRanCe

Touches the x-axis at (0, 0)


These functions are either symmetric in the origin or symmetric in the y-axis.


The surface area of a human being can be modeled by the power function

S(h) = 327h17/40

where h is the height in inches.

Power (a > 0, q is even) f(x) = a • x p/q


Domain All real numbersRange If p is even, the range is all non-

negative real numbers. If p is odd, the range is all reals.

ChaRaCteRistiC The point (0, 0) is on the graph

appeaRanCe

If p is even, touches the x-axis at (0, 0).If p is odd, crosses the x-axis at (0, 0).


Always positive and increasing


The time required for a planet to make one complete revolution about the sun is approximated by a power function of the

planet’s distance from the sun:T(d) = k • d3/2

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218

Logistic f(x) = a

1 + c · b−x (and b>1,aandcgreaterthan0)

11

y

1

0.5

12

x

f(x) = 101 + 100 • (2.5)–x

, x > 0

Absolute Value f(x) = a • | x − h | + k

10

y

1

7

-2 1 -10x

f(x) = 2 • │x – 3│+ 4 (3, 4)

9

y

1

11

-1 1 -10x

f(x) = –2 • │x – 3│+ 4

Absolute Value f(x) = a • | x − h | + k

Domain All real numbersRange x ≥ k or x ≤ k, depending on whether

a > 0 or a < 0.

appeaRanCe

Looks like or


Has a vertex at (h, k)


The absolute value of your velocity is your speed.

Logistic f(x) = a

1 + c · b−x (andb>1,aandcgreaterthan0)

Domain All real numbersRange 0 < y < a

appeaRanCe

Looks like an S and is sometimesreferred to as a sigmoid.


Has horizontal asymptotes at y = 0 and y = a.


Logistic functions are used in determining limits on catching fish that will maintain fish populations.

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Cyclical f(x) = a • sin (x − h) + k

7-7

-5

4

1

1

y

xy = 1

y = –3

f(x) =2 • sin(x – 5) –1

Radical f(x) = ⋅ − +na x h k (anda>0)

10

y

2

1

14

-2

x(–4, 2)

-4

g(x) = 3 • √ x – (–4) + 22

Square Root

-10 10

y

1

1

7

-7

x

h(x) = –2 • √ x + 33

Cube Root

Cyclical f(x) = a • sin (x − h) + k

Domain All real numbersRange − |a| + k < y < |a| + k

appeaRanCe

Looks like a repeating wave


Maximum and minimum values are infinitely recurring


Analysis of alternating current electrical circuits requires cyclical functions.

Radical f(x) = ⋅ − +na x h k (anda>0)

DomainIndex even: All numbers for which x ≥ h.Index odd: All real numbers

Range Index even: −a ≥ 0y ≥ k and a ≤ 0y ≤ kIndex odd: All real numbers

ChaRaCteRistiC 1Any even index (n) will result in a graph similar to the square root graph.

ChaRaCteRistiC 2Any odd index (n) will result in a graph similar to the cube root graph.


The hang time in football (the time elapsed between the time a punt is kicked and when

it is caught) is modeled by the function

( )=h

T h2

where T is in seconds and h is in feet.

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220

CRITICALTHINKINGQUESTIONS 5.21. Whydowegraphtoanalyzeafunction?

2. Whatarefourcommonkeycharacteristicsofthefamilyoffunctionstoconsiderwhengraphing?

3. Inanalyzingafunctionfromitsstandardsymbolicformonafamilyoffunctionscard,whatimpactdoestheexponentplayinthebehaviorofthefunction?

4. Whatarethreeexamplesofasymptotesinthefamilyoffunctionscards?

5. Whyarewedevelopingtheskillofanalyzingafunction?

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DEMONSTRATEYOURUNDERSTANDING 5.21. Identifythefamilyoffunctionsforeachofthefollowingfunctions.

a. ( ) 3 4f x x= − b. 4 2

3 5( )2 5 4 7

xg xx x x

−=

− − +c. 8 3( ) 3h x x= −

2.Interpretthefunctiongivenby22 4x y x− = + .Besuretohighlightthefourkeycharacteristics.

y

x

3.Interpretthefunctiongivenby ( )( )2

3 6( )4 5xg x

x x+

=− +

.Besuretoclearlyidentifyanyasymptotes.

y

x

222


222

4.Interpretthefunctiongivenby 3 2xy = .Besuretohighlightthefourkeycharacteristics.

y

x

IDENTIFYANDCORRECTTHEERRORS 5.2Inthesecondcolumn,identifytheerroryoufindineachofthefollowingworkedsolutionsanddescribetheerrormade.Solvetheproblemcorrectlyinthethirdcolumn.

Problem Describe Error Correct Process

1. Identifythefamilyoffunctions:2/3( ) 5g x x= −

Worked Solution (What is wrong here?)

Thisfunctionisamemberofthepolynomial family of functions.

2. Identifythefamilyoffunctions:

27 1 3( )4 2 5

h x x x= − −


Thisfunctionisamemberoftherational family of functions.

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Problem Describe Error Correct Process

3. Identifytheverticalasymptoteofthefollowingfunction:

3( )5

xh xx−

=+


Theverticalasymptoteisatx=3

4.Identifythedomainofthefollowingfunction:

3( ) 3 4 7g x x= − +


Thedomainisx≥4

Documents

Section 5.2 Families of Functions · Families of Functions provide a convenient way to analyze various types of functions that occur frequently in the real world. This workbook presents