G. GRAPHING FUNCTIONS
|
|
- Crystal Horton
- 7 years ago
- Views:
Transcription
1 G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression looks. We consider these here.. Right-left translation. Let c > 0. Start with the graph of some function f(). Keep the -ais and y-ais fied, but move the graph c units to the right, or c units to the left. (See the pictures below.) You get the graphs of two new functions: { { right f( c) () Moving the f() graph c units to the gives the graph of. left f( + c) If f() is given by a formula in, then f( c) is the function obtained by replacing by c wherever it occurs in the formula. For instance, f() = 2 + f( ) = ( ) 2 + ( ) = 2, by algebra. Eample. Sketch the graph of f() = Solution. By algebra, f() = ( ) 2. Therefore by (), its graph is the one obtained by moving the graph of 2 one unit to the right, as shown. The result is a parabola touching the -ais at =. 2 ( ) 2 To see the reason for the rule (), suppose the graph of f() is moved c units to the right: it becomes then the graph of a new function g(), whose relation to f() is described by (see the picture): value of g() at 0 = value of f() at 0 c = f( 0 c). This shows that g() = f( c). The reasoning is similar if the graph is translated c units to the left. Try giving the argument yourself while referring to the picture. h() f() c c o c o g() The effect of up-down translation of the graph is much simpler to see. If c > 0, { { up f() + c (2) Moving the f() graph c units gives the graph of. down f() c since for eample moving the graph up by c units has the effect of adding c units to each function value, and therefore gives us the graph of the function f() + c Eample 2. Sketch the graph of +. Solution Combine rules () and (2). First sketch, then move its graph unit to the right to get the graph of, then unit up to get the graph of +, as shown. +
2 2 Eample 3. Sketch the curve y = Solution We complete the square : = ( ) 3 = ( + 2) 2 3, 2 so we move the graph of 2 to the left 2 units, then 3 units down, getting the graph shown Changing scale: stretching and shrinking. Let c >. To stretch the -ais by the factor c means to move the point to the position formerly occupied by c, and in general, the point 0 to the position formerly occupied by c 0. Similarly, to shrink the -ais by the factor c means to move 0 to the position previously taken by 0 /c. What happens to the graph of f() when we stretch or shrink the -ais? (3) { Stretching Shrinking the -ais by c changes the graph of f() into that of { f(/c) f(c). The picture eplains this rule; it illustrates stretching by the factor c >. The new function has the same value at 0 that f() has at 0 /c, so that it is given by the rule 0 f( 0 /c), which means that it is the function f(/c). f( ) f( c) 0 c 0 If the y-ais is stretched by the factor c >, each y-value is multiplied by c, so the new graph is that of the function cf(): (4) { Stretching Shrinking the y-ais by c changes the graph of f() into that of { c f() f()/c. Eample 4. Sketch the graph of 2. Solution. Start with the graph of /, move it unit to the right to get the graph of /( ), then shrink the -ais by the factor 2 to get the graph of the given function. See the picture. /2 3. Reflecting in the - and y-aes: even and odd functions. To reflect the graph of f() in the y-ais, just flip the plane over around the y-ais. This carries the point (, y) into the point (, y), and the graph of f() into the graph of f( ). Namely, the new function has the same y-value at 0 as f() has at 0, so it is given by the rule 0 f( 0 ) and is the function f( ). f (- ) f( ) Similarly, reflecting the y-plane in the -ais carries (, y) to the point (, y) and the graph of f() gets carried into that of f(). Finally, relecting first in the y-ais and then in the -ais carries the point (, y) into the point (, y). This is called a reflection through the origin. The graph of f() gets carried into the graph of f( ), by combining the above two results. Summarizing: - 0 f() -f(-) 0
3 G. GRAPHING FUNCTIONS 3 (5) Reflecting in the y-ais -ais origin moves the graph of f() into that of f( ) f() f( ). Of importance are those functions f() whose graphs are symmetric with respect to the y-ais that is, reflection in the y-ais doesn t change the graph; such functions are called even. Functions whose graphs are symmetric with respect to the origin are called odd. In terms of their epression in, (6) (7) f( ) = f() f( ) = f() definition of even function definition of odd function Eample 5. Show that a polynomial with only even powers, like , is an even function, and a polynomial with only odd powers, like , is an odd function this, by the way, eplains the terminology even and odd used for functions. Solution. We have to show (6) and (7) hold for polynomials with respectively only even or odd powers, but this follows immediately from the fact that for any non-negative integer n, we have { n ( ) n = ( ) n n, if n is even, = n, if n is odd. The following easily proved rules predict the odd- or even-ness of the product or quotient of two odd or even functions: (8) (9) even even = even odd odd = even odd even = odd even/even = even odd/odd = even odd/even = odd Eample is of the form odd/even, therefore it is odd; (3 + 4 ) /2 ( 3 ) has the form even odd, so it is odd. 4. The trigonometric functions. The trigonometric functions offer further illustrations of the ideas about translation, change of scale, and symmetry that we have been discussing. Your book reviews the standard facts about them in section 9., which you should refer to as needed. The graphs of sin and cos are crudely sketched below. (In calculus, the variable is always to be in radians; review radian measure in section 9. if you have forgotten it. Briefly, there are 2π radians in a 360 o angle, so that for eample a right angle is π/2 radians.) As the graphs suggest and the unit circle picture shows, (0) cos( ) = cos (even function) sin( ) = sin (odd function). From the standard triangle at the right, one sees that cos(π/2 ) = sin, π/2 -
4 4 and since cos is an even function, this shows that () cos( π/2) = sin. From (), we see that moving the graph of cos to the right by π/2 units turns it into the graph of sin. (See picture.) The trigonometric function cos cos - cos( - ) (2) tan = sin cos is also important; its graph is sketched at the right. It is an odd function, by (9) and (0), since it has the form odd/even. sin sin π - sin (- ) π 2π π Periodicity An important property of the trigonometric functions is that they repeat their values: (3) sin( + 2π) = sin, cos( + 2π) = cos. This is so because + 2π and represent in radians the same angle. From the graphical point of view, equations (3) say that if we move the graph of sin or cos to the left by 2π units, it will coincide with itself. From the function viewpoint, equations (3) say that sin and cos are periodic functions, with period 2π. In general, let c > 0; we say that f() is periodic, with period c, if (4) (4 ) f( + c) = f() c for all, and is the smallest positive number for which (4) is true. By rule (), the graph of a periodic function having period c coincides with itself when it is translated c units to the left. If we replace by c in (4), we see that the graph will also coincide with itself if it is moved to the right by c units. But beware: if a function is made by combining other periodic functions, you cannot always predict the period. For eample, although it is true that tan( + 2π) = tan and cos 2 ( + 2π) = cos 2, the period of both tan and cos 2 is actually π, as the above figure suggests for tan. The general sinusoidal wave. The graph of sin is referred to as a pure wave or a sinusoidal oscillation. We now consider to what etent we can change how it looks by applying the geometric operations of translation and scale change discussed earlier. a) Start with sin, which has period 2π and oscillates between ±. b) Stretch the y ais by the factor A > 0; by (4) this gives Asin, which has period 2π and oscillates between ±A. c) Shrink the -ais by the factor k > 0; by (3), this gives Asin k, which has period 2π/k, since Asin k( + 2π ) = Asin(k + 2π) = Asin k. k d) Move the graph φ units to the right; by (), this gives
5 G. GRAPHING FUNCTIONS 5 (5) A sin k( φ), A, k > 0, φ 0, general sinusoidal wave which has period 2π/k angular frequency k amplitude A phase angle φ (the wave repeats itself every 2π/k units); (has k complete cycles as goes from 0 to 2π); (the wave oscillates between A and A); (the midpoint of the wave is at = φ). Notice that the function (5) depends on three constants: k, A, and φ. We call such constants parameters; their value determines the shape and position of the wave. By using trigonometric identities, it is possible to write (5) in another form, which also has three parameters: A φ - A π/k (6) a sink + b cosk The relation between the parameters in the two forms is: (7) a = Acos kφ, b = Asin kφ; A = a 2 + b 2, tan kφ = b a. Proof of the equivalence of (5) and (6). (5) (6): from the identity sin(α + β) = sin α cosβ + cosαsin β, we get Asin(k( φ)) = Asin(k kφ) = Acoskφsin k Asin kφcosk which has the form of (6), with the values for a and b given in (7). (6) (5): square the two equations on the left of (7) and add them; this gives a 2 + b 2 = A 2 (cos 2 kφ + sin 2 kφ) = A 2, showing that A = a 2 + b 2. If instead we take the ratio of the two equations on the left of (7), we get b/a = tankφ, as promised. Eample 7. Find the period, frequency, amplitude, and phase angle of the wave represented by the functions a) 2 sin(3 π/6) b) 2 cos(2 π/2) Solution. a) Writing the function in the form (5), we get 2 sin3( π/8), which shows it has period 2π/3, frequency 3, amplitude 2, and phase angle π/8 (or 0 o ). b) We get rid of the sign by using cos = cos( π) translating the cosine curve π units to the right is the same as reflecting it in the -ais (this is the best way to remember such relations). We get then 2 cos(2 π/2) = 2 cos(2 π/2 π) = 2 sin(2 π), by (); = 2 sin2( π/2).
6 6 Thus the period is π, the frequency 2, the amplitude 2, and the phase angle π/2. (Note that the first three could have been read off immediately without making the above transformation.) Eample 8. Sketch the curve sin2 + cos2. Solution Transforming it into the form (5), we can get A and φ by using (9): A = 2; tan 2φ = 2φ = 35 o = 3π/4, φ = 3π/8. So the function is also representable as 2 sin2( 3π/8); it is a wave of amplitude 2, period π, frequncy 2, and phase angle 3π/8, and can be sketched using this data. 5. Reflection in the diagonal line; inverse functions. As our final geometric operation on graphs, we consider the effect of reflecting a graph in the diagonal line y =. This reflection can be carried out by flipping the plane over about the diagonal line. Each point of the diagonal stays fied; the -and y-aes are interchanged. The points (a, b) and (b, a) are interchanged, as the picture shows, because the two rectangles are interchanged. To see the effect of this on the function, let s consider first a simple eample. (b,a) y= Eample 9. If the graph of f() = 2, 0 is reflected in the diagonal, what function corresponds to the reflected graph? Solution. The original curve is the graph of the equation: y = 2, 0. Reflection corresponds to interchanging the two aes; thus the reflected curve is the graph of the equation: = y 2, y 0. To find the corresponding function, we have to epress y eplicitly in terms of, which we do by solving the equation for y: y =, 0 ; the restriction on follows because if = y 2 and y 0, then 0 also. y y y a,b) reflect no change 2 y=, >0 =y 2, y>0 y=, >0 Remarks.. When we flip the curve about the diagonal line, we do not interchange the labels on the - and y-aes. The coordinate aes remain the same it is only the curve that is moved (imagine it drawn on an overhead-projector transparency, and the transparency flipped over). This is analogous to our discussion in section of translation, where the curve was moved to the right, but the coordinate aes themselves remained unchanged. 2. It was necessary in the previous eample to restrict the domain of in the original function 2, so that after being flipped, its graph was still the graph of a function. If we
7 G. GRAPHING FUNCTIONS 7 hadn t, the flipped curve would have been a parabola lying on its side; this is not the graph of a function, since it has two y-values over each -value. The function having the reflected graph, y =, 0 is called the inverse function to the original function y = 2, 0. The general procedure may be represented schematically by: y = f() = f(y) y = g() original graph switch and y reflected graph solve for y reflected graph In this scheme, the equations = f(y) and y = g() have the same graph; all that has been done is to transform the equation algebraically, so that y appears as an eplicit function of. This function g() is called the inverse function to f() over the given interval; in general it will be necessary to restrict the domain of f() to an interval, so that the reflected graph will be the graph of a function. To summarize: f() and g() are inverse functions if (i) geometrically, the graphs of f() and g() are reflections of each other in the diagonal line y = ; (ii) analytically, = f(y) and y = g() are equivalent equations, either arising from the other by solving eplicitly for the relevant variable. Eample 0. Find the inverse function to, >. Solution. We introduce a dependent variable y, then interchange and y, getting We solve this algebraically for y, getting = y, y >. (20) y = +, > 0. (The domain is restricted because if y >, then equation (20) implies that > 0.) The right side of (20) is the desired inverse function. The graphs are sketched. It often happens that in determining the inverse to f(), the equation (2) = f(y) /(-) +/ cannot be solved eplicitly in terms of previously known functions. In that case, the corresponding equation (22) y = g() is viewed as defining the inverse function to f(), when taken with (2). Once again, care must be taken to restrict the domain of f() as necessary to ensure that the relected will indeed define a function g(), i.e., will not be multiple-valued. A typical eample is the following.
8 8 Eample. Find the inverse function to sin. Solution. Considering its graph, we see that for the reflected graph to define a function, we have to restrict the domain. The most natural choice is to consider the restricted function (23) y = sin, π/2 π/2. The inverse function is then denoted sin, or sometimes Arcsin ; it is defined by the pair of equivalent equations (24) = sin y, π/2 π/2 y = sin,. The domain [, ] of sin is evident from the picture it is the same as the range of sin over [ π/2, π/2]. As eamples of its values, sin = π/2, since sin π/2 = ; similarly, sin /2 = π/6. Care is needed in handling this function. For eample, substituting the left equation in (24) into the right equation says that (25) sin (sin y) = y, π/2 y π/2. It is common to see the restriction on y carelessly omitted, since the equation by itself seems obvious. But without the restriction, it is not even true; for eample if y = π, sin (sin π) = 0. π /2 - - y=sin π /2 Eercises: Section A
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus
Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:
More informationAlgebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123
Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationSouth Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be real-valued functions in a single variable. A function is real-valued if the input and output are real numbers
More information1.7 Graphs of Functions
64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the
More informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationPower functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd
5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationFunctions: Piecewise, Even and Odd.
Functions: Piecewise, Even and Odd. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway September 21-22, 2015 Tutorials, Online Homework.
More informationPeriod of Trigonometric Functions
Period of Trigonometric Functions In previous lessons we have learned how to translate any primary trigonometric function horizontally or vertically, and how to Stretch Vertically (change Amplitude). In
More informationGraphs of Polar Equations
Graphs of Polar Equations In the last section, we learned how to graph a point with polar coordinates (r, θ). We will now look at graphing polar equations. Just as a quick review, the polar coordinate
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More information2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationIntroduction to Complex Numbers in Physics/Engineering
Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information1. Introduction sine, cosine, tangent, cotangent, secant, and cosecant periodic
1. Introduction There are six trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant; abbreviated as sin, cos, tan, cot, sec, and csc respectively. These are functions of a single
More informationPRE-CALCULUS GRADE 12
PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More informationExponential Functions, Logarithms, and e
chapter 3 Starry Night, painted by Vincent Van Gogh in 889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Eponential Functions, Logarithms, and e This chapter focuses
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationEvaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More information3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?
Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using
More informationHow to Graph Trigonometric Functions
How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle
More informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
More informationFinding Equations of Sinusoidal Functions From Real-World Data
Finding Equations of Sinusoidal Functions From Real-World Data **Note: Throughout this handout you will be asked to use your graphing calculator to verify certain results, but be aware that you will NOT
More informationLecture L3 - Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A-0 A.1 Vectors A-2 A.1.1 Introduction A-2 A.1.2 Properties of a Vector A-2 A.1.3 Application of Vectors
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More information2 Session Two - Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationn 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1
. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More information1. Introduction circular definition Remark 1 inverse trigonometric functions
1. Introduction In Lesson 2 the six trigonometric functions were defined using angles determined by points on the unit circle. This is frequently referred to as the circular definition of the trigonometric
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationRoots, Linear Factors, and Sign Charts review of background material for Math 163A (Barsamian)
Roots, Linear Factors, and Sign Charts review of background material for Math 16A (Barsamian) Contents 1. Introduction 1. Roots 1. Linear Factors 4. Sign Charts 5 5. Eercises 8 1. Introduction The sign
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES NQF LEVEL 3 OUTCOME 4 - ALTERNATING CURRENT 4 Understand single-phase alternating current (ac) theory Single phase AC
More informationSection 5.0A Factoring Part 1
Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)( + 5) (
More informationMPE Review Section III: Logarithmic & Exponential Functions
MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
More information15.1. Exact Differential Equations. Exact First-Order Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact First-Order Equations 09 SECTION 5. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More information7.7 Solving Rational Equations
Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More informationA CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION
A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION Abstract. The present work is intended as a classroom note on the topic of parabolas. We present several real world applications of parabolas, outline
More informationTrigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
More information