Finding Equations of Sinusoidal Functions From RealWorld Data


 Hubert Dixon
 2 years ago
 Views:
Transcription
1 Finding Equations of Sinusoidal Functions From RealWorld 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 be able to use your calculator during the exam! Please make sure you have read through sections.,., and.3 carefully because there will be references made to examples and terms defined in those sections. Also, this supplement will be using Example and problems and 3 from section.3 (pages 3 and6) from your text book, so make sure you have those in front of you. In order to find the equation of a sinusoidal function from a set of given realworld data you must follow three steps. Step : Make sure the data has a sinusoidal relationship by plotting the data in a graph. Step : If the data shows a sinusoidal relationship, then begin finding the basic parameters of a sinusoidal function (i.e. amplitude, period, etc.). Step 3: Use the information from Step to find an equation that fits the data. As a check on your work plot the equation on a graphing calculator to make sure it fits the data nicely (be aware that doing this by hand will NOT give you an equation that fits the data exactly, but it should appear to fit it nearly perfectly). Steps and 3 are easy, it is Step that requires the most work. Our ultimate goal is to end up with an equation like y Asin B x C D or y Acos B x C D, thus we need to find A, B, C, and D. Let us start by working through Example on page 3. The data has already been plotted for you and it is fairly obvious that it represents a sinusoidal function. By replacing the months with numbers (January = and December = )and using them as our x values, the table of given values is: x y First find the coefficient A. Remember the amplitude A of a sinusoidal function tells us the distance along the yaxis from the graph s midline to its maximum or minimum, therefore A Max. Min.. For a pure sinusoidal wave such as y sin x, this formula gives us A, which is what we expect. For the data given above, the maximum yvalue is 9 and the minimum y
2 value is 5.5, therefore A We can use either the value 6.75 or the value 6.75 for our A. The final value we choose depends on the equation we decide to use at the end. Remember that a negative sign in front of a function reflects it across the xaxis (i.e. the graph is turned upsidedown ). Next find B (in physics and engineering B is usually called the radian frequency ). Remember that the period P is related by P, so in order to B find B we first need to find P (notice that B ). There are several ways to do P this. First we can measure the distance along the xaxis between two maxima (plural for maximum) or two minima (plural for minimum). You can visualize this by recalling the graph of y cos x, its first maximum occurs when x 0 and its next maximum occurs at x. The distance between two points on a number line is x x, therefore the period is P 0 which agrees with the formula P since B for B y cos x. What if we don t have two maxima or two minima such as with y sin x? After all, the data given above has only one maximum y 9 at x 6. If you refer back to section. (page 75), the first full paragraph emphasizes the fact that every cycle (or period) of a sinusoidal function can be divided into four arc sections of equal length. Now notice that between the maximum and minimum values there are two arc sections. Figure : Two arcsections (the solid portion) of y = sin x.
3 Since each arc section makes onefourth of one period, two of them make onehalf of one period. Therefore the distance from one maximum to the next minimum makes onehalf of a period. Thus the period is twice this distance. Let s use y sin x as an example. The first maximum occurs at x and the next minimum occurs at x 3. The distance between these two points is 3. Multiplying the result by gives us a period of, which is what we expect. Turning back to the problem at hand, our first maximum occurs at x 6 and our next minimum occurs at x, and 6 6 so our period P is which gives us a B of 6. Why we couldn t use from the start? After all, they gave us points and the months repeat themselves every year. In the realworld you can t assume that the data given to you will always represent one period. It may represent several periods, or even less than one period! Also, why not use? We can use a positive or negative value for B just like we can choose a positive or negative value for A, but using a negative value for B can sometimes complicate coming up with the appropriate equation. Thus, for the remainder of this handout and the remainder of the textbook, we will assume B 0. Next let s find the vertical shift D. D is the vertical midpoint between the maximum and minimum of a sinusoidal function. Remember that the midpoint y y between two points is their average. For example, for either y sin xor y cos x, the constant D is: D 0, which is what we expect. For our problem, the maximum is 9 and the minimum is 5.5, therefore: D.5. Lastly, let s find the phase shift (horizontal shift) C. This is the trickiest parameter to find sometimes, and unlike A, B, and D there are an infinite number of choices possible for C! Of course we ll only be focusing on finding a few of them. Again let s refer back to page 75. We can interpret C as the starting point (along the xaxis) for our graph. For y sin x this is the critical point onefourth of a period behind the next (or any) maximum (see Figure ), alternatively
4 if we extend the graph backwards for negative values of x, it is also the point onefourth of a period after the previous (or any) minimum (see Figure 3). Figure : The critical point before a maximum. Figure 3: The critical point after a minimum. Thus, for y sin x C 0, but we could also use,,, and so on. To see for yourself, try graphing y sin x, ysin x, ysin x, and ysin x on a graphing calculator to see that they all yield the same graph. Note that this is only true if A 0. If A is negative, then C is located onefourth of a period before any minimum or onefourth of a period after any maximum (this is simply the opposite of the case when A 0 ). Graph y sin x to verify this. For y cos x the problem is much easier. Let the starting point of the graph of y cos x be its first maximum. Thus, we can find C by finding the first (or any) maximum and its horizontal distance from the origin x 0. For y cos x C 0 again, and we can also use the values,, and just like we did for y sin x (test this on a graphing calculator). Again, this is only true when A 0. If A is negative, then C is the horizontal distance to any minimum. We ll hold off on choosing a value for C for our problem for now. The reason for this is that an appropriate value for C depends on whether we want to use a sine or cosine function and whether or not there are any restrictions such as ( A 0 and C 0 ). So let s recap the information we ve determined:
5 A 6.75 B 6 C? D.5 Now let s go along with the book s decision to use the form sin y A B x C D. The book decides to use A = 6.75 so we ll go along with that for now. Next we find C. Remember that since we re using the sine function and we re using A > 0, so C must be located onefourth of a period behind any maximum. The period of our function is, and onefourth of is 3. The maximum occurs at x 6, which gives us: C Max. P Therefore our final answer is: y 6.75sin x This agrees with the book s answer on page. Alternatively we could have used the fact that C is onefourth of a period after any minimum. The minimum occurs at x, thus C Min. P 3 5. In this case our answer would have been: y 6.75sin x Let s find one more alternative using a cosine function cos y A B x C D. In this case, C is the horizontal distance from the origin to the maximum and therefore C 6. Thus our answer would be: y 6.75cos x Graph all three answers to see that they all yield the same graph. Stop here and try working out problems and 3 on page 6 on your own before moving on to the next page. Don t be discouraged if you answers differ slightly in your choice of C from the answers given in the back of the book. Remember, there are an infinite number of choices for C. You can verify that your answer works by plotting your answer and the book s answer in a graphing calculator. Once you re done working out and 3 see the next page for a detailed explanation of the book s solution and alternative solutions
6 y Solution(s) to Problem from section.3 (page 6): Problem 0 8, 5, 9 7, 6 5, 5, 6, 3, , 0.5.5, x We want to find an equation in the form sin list some important points: y A B x C D where A 0. Let s There are two maxima at the points 7,. There are two minima at the points 9.5,0.5 and.5,0.5. We can see that the data seems to follow a sinusoidal relationship. See the next page for a summary of the information we can gather from the data., and
7 From the given information we can find our parameters A, P (in order to find B and C), B, C, and D. A So A = 5.5 (because the given restriction says A 0 ) P 7 5 P (we need this value to find C) B P 5 C Max. P D 5.75 Therefore, the answer is: y 5.5sin x Because the book gives no restriction on the phase shift C, we can use an alternative value by using the point onefourth of a period after the first minimum. C Min. P Using this value, we come up with the alternative answer: y 5.5sin x Now, just as some extra practice, let s assume the restrictions were A 0 and 0 C. In this case, P, B and D are the same, but A and C must change. A 5.5 A 5.5 (because of our restriction) P 5 and 3.75 P B 5 D 5.75
8 , 73, 7 y 3, 7, 7, 79 5, 8, 79 6, 85 7, 87 9, 87 0, 83 8, 90 C Max. P (remember that we must reverse our formula for C when A 0). Thus, our answer is: y 5.5sin x Another pair of equivalent answers are: y 5.5sin x and (using a cosine function) y 5.5cos x Do some extra practice by figuring out how to find these last two answers. Solution(s) to Problem 3 from section.3 (page 6): The data given in the book is: x y Graphing this by hand should yield a graph similar to this one: Problem x
9 We want to find an equation in the form sin list some important points: y A B x C D where A 0. Let s There is one maximum at the point 8,90. There is one minimum at the point,7. We can see that the data seems to follow a sinusoidal relationship. From the given information we can find our parameters A, P (in order to find B and C), B, C, and D. A A 9.5 (because of the restriction on A) P 8 6 (our max. and min. occur at x = 8 and x = ) P 3 B P 6 C Max. P D sin Therefore, the answer is: y x Extra practice: Do problem 3 again but this time assume the restriction is A < 0. See if you can come up with these answers: 9.5sin y 9.75sin x Answer : y x Answer : Extra practice: Do problem 3 again but this time use a cosine function and find one answer where A > 0 and another where A < 0 and see if you can come up with these answers. 9.5cos Answer : y x
10 Answer : y 9.5cos x **Optional: Here s a neat trick to quickly find another phase shift C for any given problem (that is after you have already found one value). Now think about the graph of sine or cosine, remember that we considered C to be our starting point (on the xaxis) for the graph. Notice that if we extend the graph in either direction, these points repeat themselves every cycle/period of the graph Thus we can use the formula: Cnew Cold k P where C old is the first value of C you found the traditional way, C new is your new value for C, k is any integer (remember the integers are the numbers...,, 0,,... ) and P is the period of your function. Try this out with any of the problems worked out in this handout find a family of equivalent functions with different values of C. Remember in order to check if they re equivalent you can use your graphing calculator to graph them. If all the functions have the same graph, then they are equivalent.
How 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 informationTrigonometry 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 information29 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
More information4.5 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
More information6.1  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
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 informationSECTION 4.5: GRAPHS OF SINE AND COSINE FUNCTIONS
(Section 4.5: Graphs of Sine and Cosine Functions) 4.33 SECTION 4.5: GRAPHS OF SINE AND COSINE FUNCTIONS PART A : GRAPH f ( θ ) = sinθ Note: We will use θ and f ( θ) for now, because we would like to reserve
More informationHow to roughly sketch a sinusoidal graph
34 CHAPTER 17. SINUSOIDAL FUNCTIONS Definition 17.1.1 (The Sinusoidal Function). Let A,, C and D be fixed constants, where A and are both positive. Then we can form the new function ( ) π y = A sin (x
More informationPolynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005
Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More information6.1 Graphs of the Sine and Cosine Functions
810 Chapter 6 Periodic Functions 6.1 Graphs of the Sine and Cosine Functions In this section, you will: Learning Objectives 6.1.1 Graph variations of y=sin( x ) and y=cos( x ). 6.1. Use phase shifts of
More informationPRECALCULUS GRADE 12
PRECALCULUS 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 informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationExamples of Tasks from CCSS Edition Course 3, Unit 5
Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can
More informationLesson 3 Using the Sine Function to Model Periodic Graphs
Lesson 3 Using the Sine Function to Model Periodic Graphs Objectives After completing this lesson you should 1. Know that the sine function is one of a family of functions which is used to model periodic
More informationObjective: Use calculator to comprehend transformations.
math111 (Bradford) Worksheet #1 Due Date: Objective: Use calculator to comprehend transformations. Here is a warm up for exploring manipulations of functions. specific formula for a function, say, Given
More information4.5 GRAPHS OF SINE AND COSINE FUNCTIONS. Copyright Cengage Learning. All rights reserved.
4.5 GRAPHS OF SINE AND COSINE FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of basic sine and cosine functions. Use amplitude and period to help sketch
More informationSection 3.2. Graphing linear equations
Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
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 informationWith the Tan function, you can calculate the angle of a triangle with one corner of 90 degrees, when the smallest sides of the triangle are given:
Page 1 In game development, there are a lot of situations where you need to use the trigonometric functions. The functions are used to calculate an angle of a triangle with one corner of 90 degrees. By
More informationMath 115 Spring 2014 Written Homework 10SOLUTIONS Due Friday, April 25
Math 115 Spring 014 Written Homework 10SOLUTIONS Due Friday, April 5 1. Use the following graph of y = g(x to answer the questions below (this is NOT the graph of a rational function: (a State the domain
More informationTriangle 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
More informationMultiple Reflections. What You Need For each 23 students: 1 set of hinged mirrors 1 protractor Copy of The Pirate Handout
Multiple Reflections Overview We know that when light reflects off a plane mirror, the image appears left/right reversed. Once you bring in another mirror and change the angle between them, it is much
More informationAngles 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
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.84.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.84.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More information2.2 Derivative as a Function
2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an xvalue, why don t we just use x
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus 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 information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
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 informationNSM100 Introduction to Algebra Chapter 5 Notes Factoring
Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the
More informationAn Application of Analytic Geometry to Designing Machine Partsand Dresses
Electronic Proceedings of Undergraduate Mathematics Day, Vol. 3 (008), No. 5 An Application of Analytic Geometry to Designing Machine Partsand Dresses Karl Hess Sinclair Community College Dayton, OH
More informationG. GRAPHING FUNCTIONS
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
More informationNotes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions.
Notes for EER #4 Graph transformations (vertical & horizontal shifts, vertical stretching & compression, and reflections) of basic functions. Basic Functions In several sections you will be applying shifts
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 informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationVertical Translations of Sine θ and Cosine θ
Vertical Translations of Sine θ and Cosine θ 1. Complete the following table. You will be plotting 360 of the functions. Use one decimal place. sin(θ) 0.0 0.7 sin(θ) + 2 2.0 2.7 2. Plot the values of y=sin(θ)
More informationEVALUATING ACADEMIC READINESS FOR APPRENTICESHIP TRAINING Revised For ACCESS TO APPRENTICESHIP
EVALUATING ACADEMIC READINESS FOR APPRENTICESHIP TRAINING For ACCESS TO APPRENTICESHIP MATHEMATICS SKILL OPERATIONS WITH INTEGERS AN ACADEMIC SKILLS MANUAL for The Precision Machining And Tooling Trades
More informationTransformations and Sinusoidal Functions
Section 3. Curriculum Outcomes Periodic Behaviour Demonstrate an understanding of realworld relationships by translating between graphs, tables, and written descriptions C8 Identify periodic relations
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 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 informationExam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.
Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the
More informationHigher. 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
More information9.1 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)
More informationPositive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.
1. Introduction To describe position we need a fixed reference (start) point and a way to measure direction and distance. In Mathematics we use Cartesian coordinates, named after the Mathematician and
More information55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim
Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of
More informationMath 4 Review Problems
Topics for Review #1 Functions function concept [section 1. of textbook] function representations: graph, table, f(x) formula domain and range Vertical Line Test (for whether a graph is a function) evaluating
More informationGraphing Quadratics using Transformations 51
Graphing Quadratics using Transformations 51 51 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point ( 2, 5), give the coordinates of the translated point. 1.
More information3.1 Solving Systems Using Tables and Graphs
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationPOLAR COORDINATES DEFINITION OF POLAR COORDINATES
POLAR COORDINATES DEFINITION OF POLAR COORDINATES Before we can start working with polar coordinates, we must define what we will be talking about. So let us first set us a diagram that will help us understand
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationGraphing Linear Equations in Two Variables
Math 123 Section 3.2  Graphing Linear Equations Using Intercepts  Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the
More informationBROCK UNIVERSITY MATHEMATICS MODULES
BROCK UNIVERSITY MATHEMATICS MODULES 11A.4: Maximum or Minimum Values for Quadratic Functions Author: Kristina Wamboldt WWW What it is: Maximum or minimum values for a quadratic function are the largest
More informationREVIEW EXERCISES DAVID J LOWRY
REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and
More informationOrdered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.
Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationActually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is
QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.
More informationMAC 1114: Trigonometry Notes
MAC 1114: Trigonometry Notes Instructor: Brooke Quinlan Hillsborough Community College Section 7.1 Angles and Their Measure Greek Letters Commonly Used in Trigonometry Quadrant II Quadrant III Quadrant
More informationGraphing Quadratic Functions
Problem 1 The Parabola Examine the data in L 1 and L to the right. Let L 1 be the x value and L be the yvalues for a graph. 1. How are the x and yvalues related? What pattern do you see? To enter the
More information4.1 Transformations NOTES PARENT FUNCTION TRANSLATION SCALE REFLECTIONS. Given parent function. Given parent function. Describe translation
4.1 Transformations NOTES Write your questions here! PARENT FUNCTION TRANSLATION Vertical shift Horizontal shift Describe translation Write function with vertical shift up of 5 and horizontal shift left
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 informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More information6.3 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
More informationThis assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the
More information3.6. The factor theorem
3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the
More informationFourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees
Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This
More informationEXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS
To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires
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 informationChapter 7. Functions and onto. 7.1 Functions
Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers
More informationWhat are the place values to the left of the decimal point and their associated powers of ten?
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
More informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More informationBy reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
SECTION 5.4 Special Factoring Techniques 317 5.4 Special Factoring Techniques OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor
More informationComputer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction
Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper
More informationA synonym is a word that has the same or almost the same definition of
SlopeIntercept Form Determining the Rate of Change and yintercept Learning Goals In this lesson, you will: Graph lines using the slope and yintercept. Calculate the yintercept of a line when given
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationPHYSICS 151 Notes for Online Lecture #6
PHYSICS 151 Notes for Online Lecture #6 Vectors  A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities
More information2.5 Transformations of Functions
2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [
More informationQuadratic Functions. Teachers Teaching with Technology. Scotland T 3. Symmetry of Graphs. Teachers Teaching with Technology (Scotland)
Teachers Teaching with Technology (Scotland) Teachers Teaching with Technology T 3 Scotland Quadratic Functions Symmetry of Graphs Teachers Teaching with Technology (Scotland) QUADRATIC FUNCTION Aim To
More informationFactoring Patterns in the Gaussian Plane
Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood
More informationSection 3.2 Polynomial Functions and Their Graphs
Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)
More informationDraft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then
CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,
More information2009 Chicago Area AllStar Math Team Tryouts Solutions
1. 2009 Chicago Area AllStar Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
More informationGraphing Quadratic Functions
Graphing Quadratic Functions In our consideration of polynomial functions, we first studied linear functions. Now we will consider polynomial functions of order or degree (i.e., the highest power of x
More informationMathematics. GSE PreCalculus Unit 2: Trigonometric Functions
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE PreCalculus Unit : Trigonometric Functions These materials are for nonprofit educational purposes only. Any other use may constitute
More information6.4 Special Factoring Rules
6.4 Special Factoring Rules OBJECTIVES 1 Factor a difference of squares. 2 Factor a perfect square trinomial. 3 Factor a difference of cubes. 4 Factor a sum of cubes. By reversing the rules for multiplication
More informationTrigonometry Hard Problems
Solve the problem. This problem is very difficult to understand. Let s see if we can make sense of it. Note that there are multiple interpretations of the problem and that they are all unsatisfactory.
More information4.1 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
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 informationStraightening Data in a Scatterplot Selecting a Good ReExpression Model
Straightening Data in a Scatterplot Selecting a Good ReExpression What Is All This Stuff? Here s what is included: Page 3: Graphs of the three main patterns of data points that the student is likely to
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
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 xaxis points out
More informationMCR3U  Practice Test  Periodic Functions  W2012
Name: Date: May 25, 2012 ID: A MCR3U  Practice Test  Periodic Functions  W2012 1. One cycle of the graph of a periodic function is shown below. State the period and amplitude. 2. One cycle of the graph
More informationALGEBRA & 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:
More informationBasic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.
Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More information