# G. GRAPHING FUNCTIONS

Save this PDF as:

Size: px
Start display at page:

## 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

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### 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:

### 4.1 Radian and Degree Measure

Date: 4.1 Radian and Degree Measure Syllabus Objective: 3.1 The student will solve problems using the unit circle. Trigonometry means the measure of triangles. Terminal side Initial side Standard Position

### 4.5 Graphing Sine and Cosine

.5 Graphing Sine and Cosine Imagine taking the circumference of the unit circle and peeling it off the circle and straightening it out so that the radian measures from 0 to π lie on the x axis. This is

### Algebra. 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

### Symmetry. A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (-x, y) is also on the graph.

Symmetry When we graphed y =, y = 2, y =, y = 3 3, y =, and y =, we mentioned some of the features of these members of the Library of Functions, the building blocks for much of the study of algebraic functions.

### 6.1 - Introduction to Periodic Functions

6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that

### M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.

M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides

### PRINCIPLES OF PROBLEM SOLVING

PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to

### D.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

### Core 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...

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### Lesson 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

### South 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

### Trigonometric 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

### Triangle Definition of sin and cos

Triangle Definition of sin and cos Then Consider the triangle ABC below. Let A be called. A HYP (hpotenuse) ADJ (side adjacent to the angle ) B C OPP (side opposite to the angle ) sin OPP HYP BC AB ADJ

### Functions 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

### Higher. Functions and Graphs. Functions and Graphs 18

hsn.uk.net Higher Mathematics UNIT UTCME Functions and Graphs Contents Functions and Graphs 8 Sets 8 Functions 9 Composite Functions 4 Inverse Functions 5 Eponential Functions 4 6 Introduction to Logarithms

### POLYNOMIAL 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

### 29 Wyner PreCalculus Fall 2016

9 Wyner PreCalculus Fall 016 CHAPTER THREE: TRIGONOMETRIC EQUATIONS Review November 8 Test November 17 Trigonometric equations can be solved graphically or algebraically. Solving algebraically involves

### SAT 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

### C3: 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

### (x) = lim. x 0 x. (2.1)

Differentiation. Derivative of function Let us fi an arbitrarily chosen point in the domain of the function y = f(). Increasing this fied value by we obtain the value of independent variable +. The value

### SOLVING 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

### Solutions 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

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

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

### MEMORANDUM. 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

### Lecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if

Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close

0 The Quadratic Function TERMINOLOGY Ais of smmetr: A line about which two parts of a graph are smmetrical. One half of the graph is a reflection of the other Coefficient: A constant multiplied b a pronumeral

### 1.7 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

### CHAPTER 13. Definite Integrals. Since integration can be used in a practical sense in many applications it is often

7 CHAPTER Definite Integrals Since integration can be used in a practical sense in many applications it is often useful to have integrals evaluated for different values of the variable of integration.

### ( ) b.! = 7" 4 has coordinates 2. ( ) d.! = has coordinates! ( ) b.! = 7" 3 has coordinates 1

Chapter 4: Circular Functions Lesson 4.. 4-. a.! b.! c. i. 0!! " radians 80! " 6 radians 4-. a. and b. ii. iii. 45!! " radians 80! " 4 radians 60!! " radians 80! " radians 4-. Possible patterns that can

### Homework 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

### Focusing properties of spherical and parabolic mirrors

Physics 5B Winter 2009 Focusing properties of spherical and parabolic mirrors 1 General considerations Consider a curved mirror surface that is constructed as follows Start with a curve, denoted by y()

### 4. Graphing and Inverse Functions

4. Graphing and Inverse Functions 4. Basic Graphs 4. Amplitude, Reflection, and Period 4.3 Vertical Translation and Phase Shifts 4.4 The Other Trigonometric Functions 4.5 Finding an Equation From Its Graph

### Ax 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

### 6.3 Inverse Trigonometric Functions

Chapter 6 Periodic Functions 863 6.3 Inverse Trigonometric Functions In this section, you will: Learning Objectives 6.3.1 Understand and use the inverse sine, cosine, and tangent functions. 6.3. Find the

### Introduction to Fourier Series

Introduction to Fourier Series We ve seen one eample so far of series of functions. The Taylor Series of a function is a series of polynomials and can be used to approimate a function at a point. Another

### Math 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

### INVERSE TRIGONOMETRIC FUNCTIONS

Mathematics, in general, is fundamentally the science of self-evident things. FELIX KLEIN. Introduction In Chapter, we have studied that the inverse of a function f, denoted by f, eists if f is one-one

### GRE 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

### Math Rational Functions

Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

### Functions: 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.

### 1.2. Mathematical Models: A Catalog of Essential Functions

1.2. Mathematical Models: A Catalog of Essential Functions Mathematical model A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon

### Dear 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

### How to add sine functions of different amplitude and phase

Physics 5B Winter 2009 How to add sine functions of different amplitude and phase In these notes I will show you how to add two sinusoidal waves each of different amplitude and phase to get a third sinusoidal

### ax 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

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

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

### Trigonometric Identities and Equations

LIALMC07_0768.QXP /6/0 0:7 AM Page 605 7 Trigonometric Identities and Equations In 8 Michael Faraday discovered that when a wire passes by a magnet, a small electric current is produced in the wire. Now

### 10.1. 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

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Unit 8 Inverse Trig & Polar Form of Complex Nums.

HARTFIELD PRECALCULUS UNIT 8 NOTES PAGE 1 Unit 8 Inverse Trig & Polar Form of Complex Nums. This is a SCIENTIFIC OR GRAPHING CALCULATORS ALLOWED unit. () Inverse Functions (3) Invertibility of Trigonometric

### Unit Overview. Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks

Content Area: Math Unit Title: Functions and Their Graphs Target Course/Grade Level: Advanced Math Duration: 4 Weeks Unit Overview Description In this unit the students will examine groups of common functions

### Power 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

### Core 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

### Higher Mathematics Homework A

Non calcuator section: Higher Mathematics Homework A 1. Find the equation of the perpendicular bisector of the line joining the points A(-3,1) and B(5,-3) 2. Find the equation of the tangent to the circle

### Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

### Midterm 1. Solutions

Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

### 9.1 Trigonometric Identities

9.1 Trigonometric Identities r y x θ x -y -θ r sin (θ) = y and sin (-θ) = -y r r so, sin (-θ) = - sin (θ) and cos (θ) = x and cos (-θ) = x r r so, cos (-θ) = cos (θ) And, Tan (-θ) = sin (-θ) = - sin (θ)

### ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

GENERAL COMMENTS Grade 12 Pre-Calculus Mathematics Achievement Test (January 2015) Student Performance Observations The following observations are based on local marking results and on comments made by

### The 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,

### Chapter 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

### Translating Points. Subtract 2 from the y-coordinates

CONDENSED L E S S O N 9. Translating Points In this lesson ou will translate figures on the coordinate plane define a translation b describing how it affects a general point (, ) A mathematical rule that

### Chapter 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

### Example #1: f(x) = x 2. Sketch the graph of f(x) and determine if it passes VLT and HLT. Is the inverse of f(x) a function?

Unit 3 Eploring Inverse trig. functions Standards: F.BF. Find inverse functions. F.BF.d (+) Produce an invertible function from a non invertible function by restricting the domain. F.TF.6 (+) Understand

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

### Geometry 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

### Objectives. By the time the student is finished with this section of the workbook, he/she should be able

QUADRATIC FUNCTIONS Completing the Square..95 The Quadratic Formula....99 The Discriminant... 0 Equations in Quadratic Form.. 04 The Standard Form of a Parabola...06 Working with the Standard Form of a

### APPLICATIONS OF DIFFERENTIATION

4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

### Graphs 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

### POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM

POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM HEMANT D. TAGARE. Introduction. This note is about polar coordinates. I want to explain what they are and how to use them. Many different coordinate

### Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson

Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent

### Period 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

### 1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

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.

### 5.2 Unit Circle: Sine and Cosine Functions

Chapter 5 Trigonometric Functions 75 5. Unit Circle: Sine and Cosine Functions In this section, you will: Learning Objectives 5..1 Find function values for the sine and cosine of 0 or π 6, 45 or π 4 and

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### 1 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

### Inverse Trigonometric Functions - Trigonometric Equations

Inverse Trigonometric Functions - Trigonometric Equations Dr. Philippe B. Laval Kennesaw STate University April 0, 005 Abstract This handout defines the inverse of the sine, cosine and tangent functions.

### 1. 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

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Warm up Factoring Remember: a 2 b 2 = (a b)(a + b)

Warm up Factoring Remember: a 2 b 2 = (a b)(a + b) 1. x 2 16 2. x 2 + 10x + 25 3. 81y 2 4 4. 3x 3 15x 2 + 18x 5. 16h 4 81 6. x 2 + 2xh + h 2 7. x 2 +3x 4 8. x 2 3x + 4 9. 2x 2 11x + 5 10. x 4 + x 2 20

4.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch

### www.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

### Core 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...

### Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

### Introduction 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

### 2.1 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.

### Angles and Their Measure

Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

### Inverse Circular Function and Trigonometric Equation

Inverse Circular Function and Trigonometric Equation 1 2 Caution The 1 in f 1 is not an exponent. 3 Inverse Sine Function 4 Inverse Cosine Function 5 Inverse Tangent Function 6 Domain and Range of Inverse

### D.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 APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

### Higher 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

### Week 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

### Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions. cosθ a. = cosθ a.

Section 7.5 Unit Circle Approach; Properties of the Trigonometric Functions A unit circle is a circle with radius = 1 whose center is at the origin. Since we know that the formula for the circumference