Section 5.2 Families of Functions · Families of Functions provide a convenient way to analyze...
Transcript of Section 5.2 Families of Functions · Families of Functions provide a convenient way to analyze...
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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
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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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Chapter 5 — Functions
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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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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.
inteResting FeatuRes
These functions have a constant rate of change and the algebraic definition of these functions
can be determined from two ordered pairs.
Real-WoRlD appliCation
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.
inteResting FeatuRes
The algebraic definition of these functions can be determined from two ordered pairs.
Real-WoRlD appliCation
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).
Real-WoRlD appliCation
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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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.
inteResting FeatuRes
y − k = a(x − h)2 is another symbolic representation of the function
Real-WoRlD appliCation
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
inteResting FeatuRes
These functions have both vertical and horizontal asymptotes when q is not a constant.
Real-WoRlD appliCation
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
(andp,qpositiveintegers,reducedform,q≠1)
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)
inteResting FeatuRes
These functions are either symmetric in the origin or symmetric in the y-axis.
Real-WoRlD appliCation
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
(andp,qpositiveintegers,reducedform,q≠1)
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).
inteResting FeatuRes
Always positive and increasing
Real-WoRlD appliCation
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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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
inteResting FeatuRes
Has a vertex at (h, k)
Real-WoRlD appliCation
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.
inteResting FeatuRes
Has horizontal asymptotes at y = 0 and y = a.
Real-WoRlD appliCation
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
inteResting FeatuRes
Maximum and minimum values are infinitely recurring
Real-WoRlD appliCation
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.
Real-WoRlD appliCation
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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Chapter 5 — Functions
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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
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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= − −
Worked Solution (What is wrong here?)
Thisfunctionisamemberoftherational family of functions.
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Problem Describe Error Correct Process
3. Identifytheverticalasymptoteofthefollowingfunction:
3( )5
xh xx−
=+
Worked Solution (What is wrong here?)
Theverticalasymptoteisatx=3
4.Identifythedomainofthefollowingfunction:
3( ) 3 4 7g x x= − +
Worked Solution (What is wrong here?)
Thedomainisx≥4