New Functions from Old Functions
|
|
- Evan Simon
- 6 years ago
- Views:
Transcription
1 .3 New Functions rom Old Functions In this section we start with the basic unctions we discussed in Section. and obtain new unctions b shiting, stretching, and relecting their graphs. We also show how to combine pairs o unctions b the standard arithmetic operations and b composition. Transormations o Functions B appling certain transormations to the graph o a given unction we can obtain the graphs o certain related unctions. This will give us the abilit to sketch the graphs o man unctions quickl b hand. It will also enable us to write equations or given graphs. Let s irst consider translations. I c is a positive number, then the graph o c is just the graph o shited upward a distance o c units (because each -coordinate is increased b the same number c). Likewise, i t c, where c, then the value o t at is the same as the value o at c (c units to the let o ). Thereore, the graph o c is just the graph o shited c units to the right (see Figure ). Vertical and Horizontal Shits Suppose c. To obtain the graph o c, shit the graph o a distance c units upward c, shit the graph o a distance c units downward c, shit the graph o a distance c units to the right c, shit the graph o a distance c units to the let
2 SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 39 =ƒ+c =cƒ (c>) =(+c) c =ƒ =(-c) =(_) =ƒ c c = ƒ c c =ƒ-c =_ƒ FIGURE Translating the graph o ƒ FIGURE Stretching and relecting the graph o ƒ Now let s consider the stretching and relecting transormations. I c, then the graph o c is the graph o stretched b a actor o c in the vertical direction (because each -coordinate is multiplied b the same number c). The graph o is the graph o relected about the -ais because the point, is replaced b the point,. (See Figure and the ollowing chart, where the results o other stretching, compressing, and relecting transormations are also given.) Vertical and Horizontal Stretching and Relecting Suppose c. To obtain the graph o c, stretch the graph o verticall b a actor o c c, compress the graph o verticall b a actor o c c, compress the graph o horizontall b a actor o c c, stretch the graph o horizontall b a actor o c, relect the graph o about the -ais, relect the graph o about the -ais Figure 3 illustrates these stretching transormations when applied to the cosine unction with c. For instance, in order to get the graph o cos we multipl the -coordinate o each point on the graph o cos b. This means that the graph o cos gets stretched verticall b a actor o. = cos =cos = cos =cos FIGURE 3 =cos =cos
3 4 CHAPTER FUNCTIONS AND MODELS EXAMPLE Given the graph o s, use transormations to graph s, s, s, s, and s. SOLUTION The graph o the square root unction s, obtained rom Figure 3(a) in Section., is shown in Figure 4(a). In the other parts o the igure we sketch s b shiting units downward, s b shiting units to the right, s b relecting about the -ais, s b stretching verticall b a actor o, and s b relecting about the -ais. _ (a) =œ FIGURE 4 (b) =œ - EXAMPLE Sketch the graph o the unction () 6. SOLUTION (c) =œ - (d) =_œ (e) =œ () =œ _ Completing the square, we write the equation o the graph as This means we obtain the desired graph b starting with the parabola and shiting 3 units to the let and then unit upward (see Figure 5). 6 3 (_3, ) _3 _ FIGURE 5 (a) = (b) =(+3)@+ EXAMPLE 3 Sketch the graphs o the ollowing unctions. (a) sin (b) sin SOLUTION (a) We obtain the graph o sin rom that o sin b compressing horizontall b a actor o (see Figures 6 and 7). Thus, whereas the period o sin is, the period o sin is. =sin =sin π π π 4 π π FIGURE 6 FIGURE 7
4 SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 4 (b) To obtain the graph o sin, we again start with sin. We relect about the -ais to get the graph o sin and then we shit unit upward to get sin. (See Figure 8.) =-sin FIGURE 8 π π 3π π EXAMPLE 4 Figure 9 shows graphs o the number o hours o dalight as unctions o the time o the ear at several latitudes. Given that Ankara, Turke, is located at approimatel 4 N latitude, ind a unction that models the length o dalight at Ankara FIGURE 9 Graph o the length o dalight rom March through December at various latitudes Hours N 5 N 4 N 3 N N Mar. Apr. Ma June Jul Aug. Sept. Oct. Nov. Dec. Source: Lucia C. Harrison, Dalight, Twilight, Darkness and Time (New York: Silver, Burdett, 935) page 4. SOLUTION Notice that each curve resembles a shited and stretched sine unction. B looking at the blue curve we see that, at the latitude o Ankara, dalight lasts about 4.8 hours on June and 9. hours on December, so the amplitude o the curve (the actor b which we have to stretch the sine curve verticall) is B what actor do we need to stretch the sine curve horizontall i we measure the time t in das? Because there are about 365 das in a ear, the period o our model should be 365. But the period o sin t is, so the horizontal stretching actor is c 365. We also notice that the curve begins its ccle on March, the 8th da o the ear, so we have to shit the curve 8 units to the right. In addition, we shit it units upward. Thereore, we model the length o dalight in Ankara on the tth da o the ear b the unction L t.8 sin 365 t 8 Another transormation o some interest is taking the absolute value o a unction. I, then according to the deinition o absolute value, when
5 4 CHAPTER FUNCTIONS AND MODELS and when. This tells us how to get the graph o rom the graph o : The part o the graph that lies above the -ais remains the same; the part that lies below the -ais is relected about the -ais. EXAMPLE 5 Sketch the graph o the unction. SOLUTION We irst graph the parabola in Figure (a) b shiting the parabola downward unit. We see that the graph lies below the -ais when, so we relect that part o the graph about the -ais to obtain the graph o in Figure (b). FIGURE (a) = - (b) = - Combinations o Functions Two unctions and t can be combined to orm new unctions t, t, t, and t in a manner similar to the wa we add, subtract, multipl, and divide real numbers. I we deine the sum t b the equation t t then the right side o Equation makes sense i both and t are deined, that is, i belongs to the domain o and also to the domain o t. I the domain o is A and the domain o t is B, then the domain o t is the intersection o these domains, that is, A B. Notice that the sign on the let side o Equation stands or the operation o addition o unctions, but the sign on the right side o the equation stands or addition o the numbers and t. Similarl, we can deine the dierence t and the product t, and their domains are also A B. But in deining the quotient t we must remember not to divide b. Algebra o Functions Let and t be unctions with domains A and B. Then the unctions t, t, t, and t are deined as ollows: t t t t domain A B domain A B t t domain A B t domain A B t t
6 SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 43 Another wa to solve 4 : _ EXAMPLE 6 I s and t s4, ind the unctions t, t, t, and t. SOLUTION The domain o s is,. The domain o t s4 consists o all numbers such that 4, that is, 4. Taking square roots o both sides, we get, or, so the domain o t is the interval,. The intersection o the domains o and t is,,, Thus, according to the deinitions, we have t s s4 t s s4 t s s4 s4 3 t s s4 4 Notice that the domain o t is the interval, ; we have to eclude because t. The graph o the unction t is obtained rom the graphs o and t b graphical addition. This means that we add corresponding -coordinates as in Figure. Figure shows the result o using this procedure to graph the unction t rom Eample 6. 5 =(+g)() =(+g)() 4 3 = g(a) (a)+g(a) =œ 4-.5 =ƒ (a) g(a).5 ƒ=œ a FIGURE FIGURE Composition o Functions There is another wa o combining two unctions to get a new unction. For eample, suppose that u su and u t. Since is a unction o u and u is, in turn, a unction o, it ollows that is ultimatel a unction o. We compute this b substitution: u t s
7 44 CHAPTER FUNCTIONS AND MODELS The procedure is called composition because the new unction is composed o the two given unctions and t. In general, given an two unctions and t, we start with a number in the domain o t and ind its image t. I this number t is in the domain o, then we can calculate the value o t. The result is a new unction h t obtained b substituting t into. It is called the composition (or composite) o and t and is denoted b t ( circle t ). Deinition Given two unctions and t, the composite unction t (also called the composition o and t) is deined b t t The domain o t is the set o all in the domain o t such that t is in the domain o. In other words, t is deined whenever both t and t are deined. The best wa to picture t is b either a machine diagram (Figure 3) or an arrow diagram (Figure 4). FIGURE 3 The g machine is composed o the g machine (irst) and then the machine. (input) g g() { } (output) g g FIGURE 4 Arrow diagram or g { } EXAMPLE 7 I and t 3, ind the composite unctions and t. SOLUTION We have t t t 3 3 t t t 3 NOTE You can see rom Eample 7 that, in general, t t. Remember, the notation t means that the unction t is applied irst and then is applied second. In Eample 7, t is the unction that irst subtracts 3 and then squares; t is the unction that irst squares and then subtracts 3. EXAMPLE 8 I s and t s, ind each unction and its domain. (a) t (b) t (c) (d) t t SOLUTION (a) t t (s ) ss s 4 The domain o t is,.
8 SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 45 (b) t t t(s) s s For s to be deined we must have. For s s to be deined we must have I a b, then a b. s, that is, s, or 4. Thus, we have 4, so the domain o t is the closed interval, 4. (c) (s) ss s 4 The domain o is,. (d) t t t t t(s ) s s This epression is deined when both and s. The irst inequalit means, and the second is equivalent to s, or 4, or. Thus,, so the domain o t t is the closed interval,. It is possible to take the composition o three or more unctions. For instance, the composite unction t h is ound b irst appling h, then t, and then as ollows: t h t h EXAMPLE 9 Find t h i, t, and h 3. SOLUTION t h t h t So ar we have used composition to build complicated unctions rom simpler ones. But in calculus it is oten useul to be able to decompose a complicated unction into simpler ones, as in the ollowing eample. EXAMPLE Given F cos 9, ind unctions, t, and h such that F t h. SOLUTION Since F cos 9, the ormula or F sas: First add 9, then take the cosine o the result, and inall square. So we let h 9 t cos Then t h t h t 9 cos 9 cos 9 F.3 Eercises. Suppose the graph o is given. Write equations or the graphs that are obtained rom the graph o as ollows. (a) Shit 3 units upward. (b) Shit 3 units downward. (c) Shit 3 units to the right. (d) Shit 3 units to the let. (e) Relect about the -ais. () Relect about the -ais. (g) Stretch verticall b a actor o 3. (h) Shrink verticall b a actor o 3.
9 46 CHAPTER FUNCTIONS AND MODELS. Eplain how the ollowing graphs are obtained rom the graph o. (a) 5 (b) 5 (c) (d) 5 (e) 5 () The graph o is given. Match each equation with its graph and give reasons or our choices. (a) 4 (b) 3 (c) 3 (d) 4 (e) 6! 7. 5 _4 3 # _.5 $ _6 _3 3 6 % _3 4. The graph o is given. Draw the graphs o the ollowing unctions. (a) 4 (b) 4 (c) (d) 3 8. (a) How is the graph o sin related to the graph o sin? Use our answer and Figure 6 to sketch the graph o sin. (b) How is the graph o s related to the graph o s? Use our answer and Figure 4(a) to sketch the graph o s. 9 4 Graph the unction b hand, not b plotting points, but b starting with the graph o one o the standard unctions given in Section., and then appling the appropriate transormations The graph o is given. Use it to graph the ollowing unctions. (a) (b) ( ) (c) (d) 3. cos 4. 4 sin 3 5. sin 7. s s tan sin The graph o s3 is given. Use transormations to create a unction whose graph is as shown..5 =œ The cit o New Delhi, India, is located near latitude 3 N. Use Figure 9 to ind a unction that models the number o hours o dalight at New Delhi as a unction o the time o ear. To check the accurac o our model, use the act that on March 3 the Sun rises at 6:3 A.M. and sets at 6:39 P.M. in New Delhi. 6. A variable star is one whose brightness alternatel increases and decreases. For the most visible variable star, Delta Cephei, the time between periods o maimum brightness is 5.4 das, the average brightness (or magnitude) o the star
10 SECTION.3 NEW FUNCTIONS FROM OLD FUNCTIONS 47 is 4., and its brightness varies b.35 magnitude. Find a unction that models the brightness o Delta Cephei as a unction o time. 7. (a) How is the graph o ( ) related to the graph o? (b) Sketch the graph o sin. (c) Sketch the graph o. 8. Use the given graph o to sketch the graph o. Which eatures o are the most important in sketching? Eplain how the are used. s 36. 3, t 37. sin, t s 38. 3, t , t 4. s 3, t 4 4 Find t h. 4. s, t, h 3 4., t cos, h s Use graphical addition to sketch the graph o t Epress the unction in the orm t. 43. F 44. F sin(s) 45. u t scos t u t tan t tan t 46. g Epress the unction in the orm t h. 47. H H sec 4 (s) H s Use the table to evaluate each epression. (a) t (b) t (c) (d) t t (e) t 3 () t 6 g t Find t, t, t, and t and state their domains. 3. 3, t 3 3. s, t s 5. Use the given graphs o and t to evaluate each epression, or eplain wh it is undeined. (a) t (b) t (c) t (d) t 6 (e) t t () Use the graphs o and t and the method o graphical addition to sketch the graph o t. 33., t g 34. 3, t 35 4 Find the unctions (a) t, (b) t, (c), and (d) t t and their domains. 35., t
11 48 CHAPTER FUNCTIONS AND MODELS 5. Use the given graphs o and t to estimate the value o t or 5, 4, 3,..., 5. Use these estimates to sketch a rough graph o t. 53. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed o 6 cm s. (a) Epress the radius r o this circle as a unction o the time t (in seconds). (b) I A is the area o this circle as a unction o the radius, ind A r and interpret it. 54. A spherical balloon is being inlated and the radius o the balloon is increasing at a rate o cm s. (a) Epress the radius r o the balloon as a unction o the time t (in seconds). (b) I V is the volume o the balloon as a unction o the radius, ind V r and interpret it. 55. A ship is moving at a speed o 3 km h parallel to a straight shoreline. The ship is 6 km rom shore and it passes a lighthouse at noon. (a) Epress the distance s between the lighthouse and the ship as a unction o d, the distance the ship has traveled since noon; that is, ind so that s d. (b) Epress d as a unction o t, the time elapsed since noon; that is, ind t so that d t t. (c) Find t. What does this unction represent? 56. An airplane is ling at a speed o 35 km h at an altitude o one mile and passes directl over a radar station at time t. (a) Epress the horizontal distance d (in kilometres) that the plane has lown as a unction o t. (b) Epress the distance s between the plane and the radar station as a unction o d. (c) Use composition to epress s as a unction o t. 57. The Heaviside unction H is deined b H t i t i t It is used in the stud o electric circuits to represent the sudden surge o electric current, or voltage, when a switch is instantaneousl turned on. (a) Sketch the graph o the Heaviside unction. g (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 5 seconds and 4 volts are applied instantaneousl to the circuit. Write a ormula or V t in terms o H t. (Note that starting at t 5 corresponds to a translation.) 58. The Heaviside unction deined in Eercise 57 can also be used to deine the ramp unction cth t, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph o the ramp unction th t. (b) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t and the voltage is graduall increased to volts over a 6-second time interval. Write a ormula or V t in terms o H t or t 6. (c) Sketch the graph o the voltage V t in a circuit i the switch is turned on at time t 7 seconds and the voltage is graduall increased to volts over a period o 5 seconds. Write a ormula or V t in terms o H t or t Let and t be linear unctions with equations m b and t m b. Is t also a linear unction? I so, what is the slope o its graph? 6. I ou invest dollars at 4% interest compounded annuall, then the amount A o the investment ater one ear is A.4. Find A A, A A A, and A A A A. What do these compositions represent? Find a ormula or the composition o n copies o A. 6. (a) I t and h 4 4 7, ind a unction such that t h. (Think about what operations ou would have to perorm on the ormula or t to end up with the ormula or h.) (b) I 3 5 and h 3 3, ind a unction t such that t h. 6. I 4 and h 4, ind a unction t such that t h. 63. (a) Suppose and t are even unctions. What can ou sa about t and t? (b) What i and t are both odd? 64. Suppose is even and t is odd. What can ou sa about t? 65. Suppose t is an even unction and let h t. Is h alwas an even unction? 66. Suppose t is an odd unction and let h t. Is h alwas an odd unction? What i is odd? What i is even?
CHAPTER 1 FIGURE 22 increasing decreasing FIGURE 23
. EXERCISES. The graph o a unction is given. (a) State the value o. (b) Estimate the value o 2. (c) For what values o is 2? (d) Estimate the values o such that. (e) State the domain and range o. () On
More information9.1 The Square Root Function
Section 9.1 The Square Root Function 869 9.1 The Square Root Function In this section we turn our attention to the square root unction, the unction deined b the equation () =. (1) We begin the section
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More informationRATIONAL FUNCTIONS. Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions
RATIONAL FUNCTIONS Finding Asymptotes..347 The Domain....350 Finding Intercepts.....35 Graphing Rational Functions... 35 345 Objectives The ollowing is a list o objectives or this section o the workbook.
More informationTangent Line Approximations
60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,.
More information(2) Find the domain of f (x) = 2x3 5 x 2 + x 6
CHAPTER FUNCTIONS AND MODELS () Determine whether the curve is the graph of a function of. If it is state the domain and the range of the function. 5 8 Determine whether the curve is the graph of a function
More informationSyllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.
Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in
More information0,0 B 5,0 C 0, 4 3,5. y x. Recitation Worksheet 1A. 1. Plot these points in the xy plane: A
Math 13 Recitation Worksheet 1A 1 Plot these points in the y plane: A 0,0 B 5,0 C 0, 4 D 3,5 Without using a calculator, sketch a graph o each o these in the y plane: A y B 3 Consider the unction a Evaluate
More information. This is the Basic Chain Rule. x dt y dt z dt Chain Rule in this context.
Math 18.0A Gradients, Chain Rule, Implicit Dierentiation, igher Order Derivatives These notes ocus on our things: (a) the application o gradients to ind normal vectors to curves suraces; (b) the generaliation
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
FUNCTIONS Quadratic Functions......8 Absolute Value Functions.....48 Translations o Functions..57 Radical Functions...61 Eponential Functions...7 Logarithmic Functions......8 Cubic Functions......91 Piece-Wise
More informationMath Review and Lessons in Calculus
Math Review and Lessons in Calculus Agenda Rules o Eponents Functions Inverses Limits Calculus Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative
More information6.4 graphs OF logarithmic FUnCTIOnS
SECTION 6. graphs of logarithmic functions 9 9 learning ObjeCTIveS In this section, ou will: Identif the domain of a logarithmic function. Graph logarithmic functions. 6. graphs OF logarithmic FUnCTIOnS
More informationMHF 4U Unit 7: Combining Functions May 29, Review Solutions
MHF 4U Unit 7: Combining Functions May 9, 008. Review Solutions Use the ollowing unctions to answer questions 5, ( ) g( ), h( ) sin, w {(, ), (3, ), (4, 7)}, r, and l ) log ( ) + (, ) Determine: a) + w
More informationFunctions. Essential Question What are some of the characteristics of the graph of a logarithmic function?
5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment As Advanced placement students, our irst assignment or the 07-08 school ear is to come to class the ver irst da in top mathematical orm. Calculus is a world o change. While
More informationSaturday X-tra X-Sheet: 8. Inverses and Functions
Saturda X-tra X-Sheet: 8 Inverses and Functions Ke Concepts In this session we will ocus on summarising what ou need to know about: How to ind an inverse. How to sketch the inverse o a graph. How to restrict
More informationRolle s Theorem and the Mean Value Theorem. Rolle s Theorem
0_00qd //0 0:50 AM Page 7 7 CHAPTER Applications o Dierentiation Section ROLLE S THEOREM French mathematician Michel Rolle irst published the theorem that bears his name in 9 Beore this time, however,
More informationMath 123 Summary of Important Algebra & Trigonometry Concepts Chapter 1 & Appendix D, Stewart, Calculus Early Transcendentals
Math Summar of Important Algebra & Trigonometr Concepts Chapter & Appendi D, Stewart, Calculus Earl Transcendentals Function a rule that assigns to each element in a set D eactl one element, called f (
More informationIncreasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video
SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl
More informationSTUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs
STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic
More informationSec 3.1. lim and lim e 0. Exponential Functions. f x 9, write the equation of the graph that results from: A. Limit Rules
Sec 3. Eponential Functions A. Limit Rules. r lim a a r. I a, then lim a and lim a 0 3. I 0 a, then lim a 0 and lim a 4. lim e 0 5. e lim and lim e 0 Eamples:. Starting with the graph o a.) Shiting 9 units
More information9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
More information4.1 & 4.2 Student Notes Using the First and Second Derivatives. for all x in D, where D is the domain of f. The number f()
4.1 & 4. Student Notes Using the First and Second Derivatives Deinition A unction has an absolute maimum (or global maimum) at c i ( c) ( ) or all in D, where D is the domain o. The number () c is called
More informationThis is only a list of questions use a separate sheet to work out the problems. 1. (1.2 and 1.4) Use the given graph to answer each question.
Mth Calculus Practice Eam Questions NOTE: These questions should not be taken as a complete list o possible problems. The are merel intended to be eamples o the diicult level o the regular eam questions.
More informationQuadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.
Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go
More informationSTRAIGHT LINE GRAPHS. Lesson. Overview. Learning Outcomes and Assessment Standards
STRAIGHT LINE GRAPHS Learning Outcomes and Assessment Standards Lesson 15 Learning Outcome : Functions and Algebra The learner is able to investigate, analse, describe and represent a wide range o unctions
More information2.5 CONTINUITY. a x. Notice that Definition l implicitly requires three things if f is continuous at a:
SECTION.5 CONTINUITY 9.5 CONTINUITY We noticed in Section.3 that the it of a function as approaches a can often be found simpl b calculating the value of the function at a. Functions with this propert
More informationInverse of a Function
. Inverse o a Function Essential Question How can ou sketch the graph o the inverse o a unction? Graphing Functions and Their Inverses CONSTRUCTING VIABLE ARGUMENTS To be proicient in math, ou need to
More informationTrigonometric Functions
Trigonometric Functions This section reviews radian measure and the basic trigonometric functions. C ' θ r s ' ngles ngles are measured in degrees or radians. The number of radians in the central angle
More informationReview of Essential Skills and Knowledge
Review of Essential Skills and Knowledge R Eponent Laws...50 R Epanding and Simplifing Polnomial Epressions...5 R 3 Factoring Polnomial Epressions...5 R Working with Rational Epressions...55 R 5 Slope
More informationChapter 2 Analysis of Graphs of Functions
Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..
More information1.2. Click here for answers. Click here for solutions. A CATALOG OF ESSENTIAL FUNCTIONS. t x x 1. t x 1 sx. 2x 1. x 2. 1 sx. t x x 2 4x.
SECTION. A CATALOG OF ESSENTIAL FUNCTIONS. A CATALOG OF ESSENTIAL FUNCTIONS A Click here for answers. S Click here for solutions. Match each equation with its graph. Eplain our choices. (Don t use a computer
More informationExponential, Logistic, and Logarithmic Functions
CHAPTER 3 Eponential, Logistic, and Logarithmic Functions 3.1 Eponential and Logistic Functions 3.2 Eponential and Logistic Modeling 3.3 Logarithmic Functions and Their Graphs 3.4 Properties of Logarithmic
More information1.1 Laws of exponents Conversion between exponents and logarithms Logarithm laws Exponential and logarithmic equations 10
CNTENTS Algebra Chapter Chapter Chapter Eponents and logarithms. Laws of eponents. Conversion between eponents and logarithms 6. Logarithm laws 8. Eponential and logarithmic equations 0 Sequences and series.
More informationReady To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 5-1 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 5-1 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationFunctions 3: Compositions of Functions
Functions : Compositions of Functions 7 Functions : Compositions of Functions Model : Word Machines SIGN SIGNS ONK KNO COW COWS RT TR HI HIS KYK KYK Construct Your Understanding Questions (to do in class).
More informationExtreme Values of Functions
Extreme Values o Functions When we are using mathematics to model the physical world in which we live, we oten express observed physical quantities in terms o variables. Then, unctions are used to describe
More informationExponential and Logarithmic Functions
Eponential and Logarithmic Functions 6 Figure Electron micrograph of E. Coli bacteria (credit: Mattosaurus, Wikimedia Commons) CHAPTER OUTLINE 6. Eponential Functions 6. Logarithmic Properties 6. Graphs
More informationMath-3 Lesson 1-4. Review: Cube, Cube Root, and Exponential Functions
Math- Lesson -4 Review: Cube, Cube Root, and Eponential Functions Quiz - Graph (no calculator):. y. y ( ) 4. y What is a power? vocabulary Power: An epression ormed by repeated Multiplication o the same
More informationExample: When describing where a function is increasing, decreasing or constant we use the x- axis values.
Business Calculus Lecture Notes (also Calculus With Applications or Business Math II) Chapter 3 Applications o Derivatives 31 Increasing and Decreasing Functions Inormal Deinition: A unction is increasing
More information( x) f = where P and Q are polynomials.
9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational
More informationLesson 9.1 Using the Distance Formula
Lesson. Using the Distance Formula. Find the eact distance between each pair of points. a. (0, 0) and (, ) b. (0, 0) and (7, ) c. (, 8) and (, ) d. (, ) and (, 7) e. (, 7) and (8, ) f. (8, ) and (, 0)
More information1.2 Functions and Their Properties PreCalculus
1. Functions and Their Properties PreCalculus 1. FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1. 1. Determine whether a set of numbers or a graph is a function. Find the domain of a function given
More informationc) domain {x R, x 3}, range {y R}
Answers Chapter 1 Functions 1.1 Functions, Domain, and Range 1. a) Yes, no vertical line will pass through more than one point. b) No, an vertical line between = 6 and = 6 will pass through two points..
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationPower Functions. A polynomial expression is an expression of the form a n. x n 2... a 3. ,..., a n. , a 1. A polynomial function has the form f(x) a n
1.1 Power Functions A rock that is tossed into the water of a calm lake creates ripples that move outward in a circular pattern. The area, A, spanned b the ripples can be modelled b the function A(r) πr,
More informationTransformations of Quadratic Functions
.1 Transormations o Quadratic Functions Essential Question How do the constants a, h, and k aect the raph o the quadratic unction () = a( h) + k? The parent unction o the quadratic amil is. A transormation
More informationy = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is
Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (,
More informationHigher. Functions and Graphs. Functions and Graphs 15
Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 5 Set Theor 5 Functions 6 Inverse Functions 9 4 Eponential Functions 0 5 Introduction to Logarithms 0 6 Radians 7 Eact Values
More informationDifferentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve.
Dierentiation The main problem o dierential calculus deals with inding the slope o the tangent line at a point on a curve. deinition() : The slope o a curve at a point p is the slope, i it eists, o the
More informationPolynomials, Linear Factors, and Zeros. Factor theorem, multiple zero, multiplicity, relative maximum, relative minimum
Polynomials, Linear Factors, and Zeros To analyze the actored orm o a polynomial. To write a polynomial unction rom its zeros. Describe the relationship among solutions, zeros, - intercept, and actors.
More informationQuadratic Functions Objective: To be able to graph a quadratic function and identify the vertex and the roots.
Name: Quadratic Functions Objective: To be able to graph a quadratic function and identif the verte and the roots. Period: Quadratic Function Function of degree. Usuall in the form: We are now going to
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. Write the equation of the line that goes through the points ( 3, 7) and (4, 5)
More informationLearning Outcomes and Assessment Standards
Lesson 5 CALCULUS (8) Rate of change Learning Outcomes and Assessment Standards Learning Outcome : Functions and Algebra Assessment standard 1..7(e) Solve practical problems involving optimisation and
More information= x. Algebra II Notes Quadratic Functions Unit Graphing Quadratic Functions. Math Background
Algebra II Notes Quadratic Functions Unit 3.1 3. Graphing Quadratic Functions Math Background Previousl, ou Identified and graphed linear functions Applied transformations to parent functions Graphed quadratic
More informationSystems of Linear Equations: Solving by Graphing
8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From
More informationMath 121. Practice Questions Chapters 2 and 3 Fall Find the other endpoint of the line segment that has the given endpoint and midpoint.
Math 11. Practice Questions Chapters and 3 Fall 01 1. Find the other endpoint of the line segment that has the given endpoint and midpoint. Endpoint ( 7, ), Midpoint (, ). Solution: Let (, ) denote the
More informationSection 3.4: Concavity and the second Derivative Test. Find any points of inflection of the graph of a function.
Unit 3: Applications o Dierentiation Section 3.4: Concavity and the second Derivative Test Determine intervals on which a unction is concave upward or concave downward. Find any points o inlection o the
More informationKEY IDEAS. Chapter 1 Function Transformations. 1.1 Horizontal and Vertical Translations Pre-Calculus 12 Student Workbook MHR 1
Chapter Function Transformations. Horizontal and Vertical Translations A translation can move the graph of a function up or down (vertical translation) and right or left (horizontal translation). A translation
More informationPolynomial Functions of Higher Degree
SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona
More information3. Several Random Variables
. Several Random Variables. Two Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation between Random Variables. Densit unction o the Sum o Two Random Variables. Probabilit
More informationPath of the Horse s Jump y 3. transformation of the graph of the parent quadratic function, y 5 x 2.
- Quadratic Functions and Transformations Content Standards F.BF. Identif the effect on the graph of replacing f() b f() k, k f(), f(k), and f( k) for specific values of k (both positive and negative)
More informationChapter 18 Quadratic Function 2
Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are
More informationNumerical Methods - Lecture 2. Numerical Methods. Lecture 2. Analysis of errors in numerical methods
Numerical Methods - Lecture 1 Numerical Methods Lecture. Analysis o errors in numerical methods Numerical Methods - Lecture Why represent numbers in loating point ormat? Eample 1. How a number 56.78 can
More informationTrigonometric Functions
TrigonometricReview.nb Trigonometric Functions The trigonometric (or trig) functions are ver important in our stud of calculus because the are periodic (meaning these functions repeat their values in a
More informationA function from a set D to a set R is a rule that assigns a unique element in R to each element in D.
1.2 Functions and Their Properties PreCalculus 1.2 FUNCTIONS AND THEIR PROPERTIES Learning Targets for 1.2 1. Determine whether a set of numbers or a graph is a function 2. Find the domain of a function
More informationZETA MATHS. Higher Mathematics Revision Checklist
ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions
More information) 4. Rational and Radical Functions. Solutions Key. (3 x 2 y) Holt McDougal Algebra 2. variation functions. check it out!
CHAPTER 5 Rational and Radical Functions Solutions Ke Are ou read? 1. D. A. B. F 5. C. 11 5 7 11-5 - 7 7-7. ( ) - (-) - 1 10. ( - ) ( )( - ) -1 1 7. ( ) z ( ) z 1 z 9. ( ) ( 5 ) ( )( 5 ) 1 11. 1 0 1(1)
More informationHomework Assignments Math /02 Spring 2015
Homework Assignments Math 1-01/0 Spring 015 Assignment 1 Due date : Frida, Januar Section 5.1, Page 159: #1-, 10, 11, 1; Section 5., Page 16: Find the slope and -intercept, and then plot the line in problems:
More informationUnit 10 - Graphing Quadratic Functions
Unit - Graphing Quadratic Functions PREREQUISITE SKILLS: students should be able to add, subtract and multipl polnomials students should be able to factor polnomials students should be able to identif
More information1.1 Exercises. (b) For what values of x is f x t x?
CHAPTER FUNCTIONS AND MODELS. Eercises. The graph of a function f is given. (a) State the value of f. Estimate the value of f. (c) For what values of is f? (d) Estimate the values of such that f. (e) State
More information7-3. Sum and Difference Identities. Look Back. OBJECTIVE Use the sum and difference identities for the sine, cosine, and tangent functions.
7-3 OJECTIVE Use the sum and difference identities for the sine, cosine, and tangent functions. Sum and Difference Identities ROADCASTING Have you ever had trouble tuning in your favorite radio station?
More informationSection 4.1 Increasing and Decreasing Functions
Section.1 Increasing and Decreasing Functions The graph of the quadratic function f 1 is a parabola. If we imagine a particle moving along this parabola from left to right, we can see that, while the -coordinates
More information4-10. Modeling with Trigonometric Functions. Vocabulary. Lesson. Mental Math. build an equation that models real-world periodic data.
Chapter 4 Lesson 4-0 Modeling with Trigonometric Functions Vocabular simple harmonic motion BIG IDEA The Graph-Standardization Theorem can be used to build an equation that models real-world periodic data.
More informationAnswers. Chapter 4 A33. + as. 4.1 Start Thinking
. + 7i. 0 i 7. 9 i. 79 + i 9. 7 i 0. 7 i. + i. 0 + i. a. lb b. $0. c. about $0.9 Chapter. Start Thinkin () = () = The raph o = is a curv line that is movin upward rom let to riht as increases. The raph
More informationA11.1 Areas under curves
Applications 11.1 Areas under curves A11.1 Areas under curves Before ou start You should be able to: calculate the value of given the value of in algebraic equations of curves calculate the area of a trapezium.
More informationReview of Prerequisite Skills for Unit # 2 (Derivatives) U2L2: Sec.2.1 The Derivative Function
UL1: Review o Prerequisite Skills or Unit # (Derivatives) Working with the properties o exponents Simpliying radical expressions Finding the slopes o parallel and perpendicular lines Simpliying rational
More informationMATH GRADE 8 UNIT 4 LINEAR RELATIONSHIPS EXERCISES
MATH GRADE 8 UNIT LINEAR RELATIONSHIPS Copright 01 Pearson Education, Inc., or its affiliate(s). All Rights Reserved. Printed in the United States of America. This publication is protected b copright,
More informationReview Exercises. lim 5 x. lim. x x 9 x. lim. 4 x. sin 2. ln cos. x sin x
MATHEMATICS 0-0-RE Dierential Calculus Martin Huard Winter 08 Review Eercises. Find the ollowing its. (Do not use l Hôpital s Rul. a) b) 0 6 6 g) j) m) sin 0 9 9 h) k) n) cos 0 sin. Find the ollowing its.
More informationDiagnostic Tests. (c) (sa sb )(sa sb ) Diagnostic Test: Algebra
Diagnostic Tests Success in calculus depends to a large etent on knowledge of the mathematics that precedes calculus: algebra, analtic geometr, functions, and trigonometr. The following tests are intended
More informationChapter 1 Graph of Functions
Graph of Functions Chapter Graph of Functions. Rectangular Coordinate Sstem and Plotting points The Coordinate Plane Quadrant II Quadrant I (0,0) Quadrant III Quadrant IV Figure. The aes divide the plane
More informationOne of the most common applications of Calculus involves determining maximum or minimum values.
8 LESSON 5- MAX/MIN APPLICATIONS (OPTIMIZATION) One of the most common applications of Calculus involves determining maimum or minimum values. Procedure:. Choose variables and/or draw a labeled figure..
More informationReview Topics for MATH 1400 Elements of Calculus Table of Contents
Math 1400 - Mano Table of Contents - Review - page 1 of 2 Review Topics for MATH 1400 Elements of Calculus Table of Contents MATH 1400 Elements of Calculus is one of the Marquette Core Courses for Mathematical
More informationy2 = 0. Show that u = e2xsin(2y) satisfies Laplace's equation.
Review 1 1) State the largest possible domain o deinition or the unction (, ) = 3 - ) Determine the largest set o points in the -plane on which (, ) = sin-1( - ) deines a continuous unction 3) Find the
More informationThe region enclosed by the curve of f and the x-axis is rotated 360 about the x-axis. Find the volume of the solid formed.
Section A ln. Let g() =, for > 0. ln Use the quotient rule to show that g ( ). 3 (b) The graph of g has a maimum point at A. Find the -coordinate of A. (Total 7 marks) 6. Let h() =. Find h (0). cos 3.
More informationCurve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
More informationI. Degrees and Radians minutes equal 1 degree seconds equal 1 minute. 3. Also, 3600 seconds equal 1 degree. 3.
0//0 I. Degrees and Radians A. A degree is a unit of angular measure equal to /80 th of a straight angle. B. A degree is broken up into minutes and seconds (in the DMS degree minute second sstem) as follows:.
More informationREVIEW, pages
REVIEW, pages 5 5.. Determine the value of each trigonometric ratio. Use eact values where possible; otherwise write the value to the nearest thousandth. a) tan (5 ) b) cos c) sec ( ) cos º cos ( ) cos
More informationUNCORRECTED. To recognise the rules of a number of common algebraic relations: y = x 1 y 2 = x
5A galler of graphs Objectives To recognise the rules of a number of common algebraic relations: = = = (rectangular hperbola) + = (circle). To be able to sketch the graphs of these relations. To be able
More information8.4. If we let x denote the number of gallons pumped, then the price y in dollars can $ $1.70 $ $1.70 $ $1.70 $ $1.
8.4 An Introduction to Functions: Linear Functions, Applications, and Models We often describe one quantit in terms of another; for eample, the growth of a plant is related to the amount of light it receives,
More informationWe would now like to turn our attention to a specific family of functions, the one to one functions.
9.6 Inverse Functions We would now like to turn our attention to a speciic amily o unctions, the one to one unctions. Deinition: One to One unction ( a) (b A unction is called - i, or any a and b in the
More informationMath-Essentials Unit 3 Review. Equations and Transformations of the Linear, Quadratic, Absolute Value, Square Root, and Cube Functions
Math-Essentials Unit Review Equations and Transormations o the Linear, Quadratic, Absolute Value, Square Root, and Cube Functions Vocabulary Relation: A mapping or pairing o input values to output values.
More informationAnswers to All Exercises
Answers to All Eercises CHAPTER 5 CHAPTER 5 CHAPTER 5 CHAPTER REFRESHING YOUR SKILLS FOR CHAPTER 5 1a. between 3 and 4 (about 3.3) 1b. between 6 and 7 (about 6.9) 1c. between 7 and 8 (about 7.4) 1d. between
More informationDIAGNOSTIC TESTS. (c) (sa sb )(sa sb )
DIAGNOSTIC TESTS Success in calculus depends to a large etent on knowledge of the mathematics that precedes calculus: algebra, analtic geometr, functions, and trigonometr. The following tests are intended
More informationAP Calculus Notes: Unit 1 Limits & Continuity. Syllabus Objective: 1.1 The student will calculate limits using the basic limit theorems.
Syllabus Objective:. The student will calculate its using the basic it theorems. LIMITS how the outputs o a unction behave as the inputs approach some value Finding a Limit Notation: The it as approaches
More information(C), 5 5, (B) 5, (C) (D), 20 20,
Reg. Pre-Calculus Multiple Choice. An epression is given. Evaluate it at the given value, (A) 0 (B) 9 9 (D) 0 (E). Simplif the epression. (A) (B) (D) (E) 0. Simplif the epression. (A) (B) (D) ( + ) (E).
More information3.1 Exponential Functions and Their Graphs
.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More information1.7 Inverse Functions
71_0107.qd 1/7/0 10: AM Page 17 Section 1.7 Inverse Functions 17 1.7 Inverse Functions Inverse Functions Recall from Section 1. that a function can be represented b a set of ordered pairs. For instance,
More information