Section 10.7 Parametric Equations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Section 10.7 Parametric Equations"

Transcription

1 299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x- (rcos(θ), rsin(θ)) and y-coordinates on a circle of radius r as a function of θ. The value of x and y depended on r the value of θ. Notice that the graph of the equation of the θ circle, x 2 + y 2 = r 2, is not a function since you can find a vertical line that passes through more than one point of the curve, But, by defining x = rcos(θ) and y = rsin(θ) on the interval [0, 2π), each coordinate is a function of θ. This means we can view each coordinate as a separate curve. x = rcos(θ) is the curve of the cosine function with amplitude r and y = r sin(θ) is the curve of the sine function with amplitude r. When we graph the points (x, y) on the graph to obtain the circle of radius r, the variable θ no longer is part of the equation for the plane curve. We call the variable θ in this case a parameter and the equations x = rcos(θ) and y = rsin(θ) parametric equations. Definition Let x = f(t) and y = g(t) be two functions defined on the interval I. The set of all points (x, y) = (f(t), g(t)) for all t in I is called the plane curve. The equations x = f(t) and y = g(t) for all t in I are called the parametric equations and t is called the parameter. Suppose the parametric equations are defined on the interval a t b. As we plot the points (x, y) as t goes from a to b, the successive points trace out the graph in a certain direction. The direction the graph is traced out as t goes from a to b is called the orientation. Thus, in the case of the circle of radius r discussed above, as t goes from 0 to 2π, the graph gets traced out in a counterclockwise direction so the orientation would be counterclockwise. To indicate the orientation on a graph, we draw arrows on the plane the curve showing the direction that curve is traced out.

2 Graph the curve whose parametric equations are given and show its orientation: Ex. 1 x = 0.5t, y = 0.25t 2 4 t 2 Let's begin by making a table of values and deriving a set of points (x, y) to plot: t x = 0.5t y = 0.25t 2 (x, y) ( 2, 4) ( 1.5, 2.25) ( 1, 1) ( 0.5, 0.25) (0, 0) (0.5, 0.25) (1, 1) Now, plot the points and sketch the graph: 00 To verify the graph, we can get rid of the parameter. In the equation x = 0.5t, we can solve for t to get t = 2x. Now, substitute 2x in for t in the equation y = 0.25t 2 and simplify: y = 0.25(2x) 2 y = 0.25(4x 2 ) y = x 2 Since 4 t 2, then 4 2x 2 or 2 x 1. Thus, y = x 2 for 2 x 1 Notice that it matches our graph. Objective 2: Finding a rectangular equation for a curve defined parametrically. In the last example, we took care to pay attention to the domain of the function after getting rid of the parameter. Many times, the graph of the curve after getting rid of the parameter will contain more points than the parameterized curve so we will need to pay careful attention to which piece of the curve actually belongs on the graph.

3 01 Find the rectangular equation of the curve and sketch the graph: Ex. 2 x = cos(t), y = 4sin(t); t 0 First, solve each of the parametric equations for the trigonometric function: x = cos(t) y = 4sin(t) cos(t) = x sin(t) = y 4 Recall, that cos 2 (t) + sin 2 (t) = 1, x so ( ) 2 + ( y 4 ) 2 = 1 or x2 9 + y2 16 = 1 Thus, the graph is an ellipse with center at (0, 0), major axis running along the y-axis, vertices (0, 4) and (0, 4) and b-intercepts (, 0) and (, 0), Since t starts at zero, the graph starts at (, 0) and traces out in a counterclockwise fashion. Suppose t had been restricted to a much smaller interval. Although, the process would be exactly the same as we did above, we may need to eliminate part of the graph depending on the restriction on t. If 0 t π was the restriction, we would only have the upper half of the graph. If π 2 t π 2 was the restriction, we would only have the left side. 0 t π π 2 t π 2

4 02 Ex. x = sin(t), y = 2cos(t); π t 0 First, solve each of the parametric equations for the trigonometric function: x = sin(t) y = 2cos(t) sin(t) = x cos(t) = y 2 Recall, that cos 2 (t) + sin 2 (t) = 1, y so ( 2 ) 2 + ( x ) 2 = 1 or x2 9 + y2 4 = 1 Thus, the graph is an ellipse with center at (0, 0), major axis running along the x-axis, vertices (, 0) and (, 0) and b-intercepts (0, 2) and (0, 2), Since t starts at π, the graph starts at (0, 2) and traces out in a clockwise fashion. It ends at (0, 2). Ex. 4 x = sec(t), y = tan(t); π 6 t π Recall, that tan 2 (t) + 1 = sec 2 (t), so (y) = (x) 2 or x 2 y 2 = 1 Thus, the graph is a hyperbola with center at (0, 0), transverse axis running along the x-axis, vertices (1, 0) and ( 1, 0) and oblique asymptotes y = ± x Since t starts at π, the graph 6 starts at (sec( π 6 ), tan( π 6 )) = ( 2, ) (1.15, 0.58) and traces out in a clockwise fashion. It ends at (sec( π ), tan( π )) = (2, ) (2, 1.7).

5 0 Objective Time as a parameter. A common use of parametric equations is when t represents the time. We can describe the path of an object using time as the parameter. The advantage with using parametric equations in this situation is that not only can we determine the path of the object, but also the time that the object reaches a certain point on the path. In calculus, we can derive the parametric equations that determine the path of a projectile through the air. Parametric Equations of a Projectile Motion Let an object be propelled upward at an angle θ with the horizontal from a height h above the horizontal. If the initial speed is v o, then, neglecting for air resistance, the resulting motion is called Projectile Motion and the parametric equations for this motion are: x = (v o cos(θ))t and y = 1 2 gt2 + (v o sin(θ))t + h where t is the time in seconds and g is the acceleration due to gravity (on Earth, g 2 ft/s 2 or 9.8 m/s 2 ). Solve the following: Ex. 5 Standing on top of a hill that was six feet above the horizontal, George Johnson hits a golf ball at 125 ft/sec at an angle of 5 from the horizontal. a) Find the parametric equations that describe the position of the ball as a function of time. b) How long was the ball in the air? c) Find the horizontal distance the ball travelled before hitting the ground. d) When is the ball at its maximum height? What is that maximum height? a) The initial velocity v o = 125 ft/sec. Since the velocity is given in ft/sec, we will use g = 2 ft/sec 2. The angle θ = 5 and h is 6 ft. x = (v o cos(θ))t = 125cos(5 )t t y = 1 2 gt2 + (v o sin(θ))t + h = 1 2 2t2 + (125sin(5 ))t t t + 6 h v o θ

6 04 Thus, our parametric equations are: x t and y 16t t + 6 b) When the ball hits the ground, the height the ball above the ground is 0. To find how long the ball was in the air, we need to first find when the ball hit the ground. Setting y = 0 and solving yields: y 16t t + 6 = 0 (use the quadratic formula) t = b± b2 4ac 2a = ± ± ( )2 4( 16)(6) 2( 16) 4.56 seconds or seconds Since the negative answers does not sense, the ball was in the air for about 4.56 seconds. c) Since the ball was in the air for approximately 4.56 seconds, then the horizontal distance travelled is: x (4.56) ft. d) Notice that the equation for y is a quadratic equation with the coefficient of the squared term being negative. Thus, the maximum value (k) will occur at the vertex (h, k). h = b 2a ( 16) seconds Plugging t = into y, we get the maximum height: y 16(2.2405) (2.2405) ft Thus, the ball reached a maximum height of about ft at t seconds. If you graph the function using equations from part a, we need to remember the x-values are the horizontal distance travelled and not the time. Ex. 6 At pm, Juan gets in his car and drives towards Atlanta at an average speed of 55 mph. Two hour later, LaTonya leaves from the same location and travels along the same route to Atlanta at 77 mph. How long will it take LaTonya to catch Juan? Use a simulation of two motions to verify your answer. Let t be the time that Juan is on the road. Since LaTonya leaves two hours later, her time on the road is t 2. Since distance = rate time,

7 05 then the distance that Juan travels is x 1 = 55t and the distance that LaTonya travels is x 2 = 77(t 2). At the point when LaTonya catches Juan, each will have travelled the same distance, so set x 1 = x 2 and and solve for the time t: x 1 = x 2 55t = 77(t 2) (distribute) 55t = 77t 154 (subtract 77t from both sides) 22t = 154 (divide both sides by 22) t = 7 hours. LaTonya's time on the road is t 2 = 5. Thus, it takes LaTonya 5 hours to catch Juan. To verify the answer, we will use a simulation of two motions. To do this, we will graph the distance each of them travelled at different times until their positions become aligned. To make it easier to see the alignment on the graph, we will let y 1 = 2 and y 2 = 4. Thus, on our graph, the line that is generated at y = 2 will represent the distance Juan has travelled and the line generated at y = 4 will represent the distance that LaTonya travelled. t = 2, t 2 = 0 t =, t 2 = 1 t = 4, t 2 = 2 t = 5, t 2 = t = 6, t 2 = 4 t = 7, t 2 = 5

8 Notice the distance are aligned at t = 7 hours in the simulation. This verifies our answer. 06 Objective 4: Finding Parametric Equations. Given an equation in rectangular form defined as a function y = f(x), the most basic parametric equations to use is the let x = t and y = f(t) for t in the domain of f. Sometimes, the conditions of the application dictate that such a set of parametric equations will not work. In that case, our choice of parametric equations must allow for the variable x to still be able to match all the values in the domain of f. For instance, if the domain of f is [, 2], we cannot choose x = t 2 since this would only allow for values of x 0. Find two different sets of parametric equations for the following: Ex. 7 y = 2x 2 + Let x = t. Then y = 2t 2 +, so the parametric equations are: x = t, y = 2t 2 +, < t < Another pair that would work would be to let x = t. Then y = 2t 6 +, so the parametric equations are: x = t, y = 2t 6 +, < t < Find parametric equations for the following object in motion: Ex. 8 x y2 = 1 where the parameter t is on seconds and a) the motion around the ellipse is clockwise, begins at the point ( 2, 0), and requires 2 seconds to make one revolution. b) the motion around the ellipse is counterclockwise, begins at the point (0, 1), and requires 4 seconds to make one revolution. a) The equation is an ellipse with center at (0, 0), major along the x-axis, vertices of ( 2, 0) and (2, 0) and b-intercepts (0, 1) and (0, 1). Since the ellipse begins at ( 2, 0), then at t = 0, x = 2 and y = 0.

9 07 Thus, x = 2cos(ωt) and y = sin(ωt) or y = sin(ωt). Since the orientation is clockwise, then as the curve leaves the point ( 2, 0), the y-values will be positive, so y = sin(ωt). Hence, our equations are: x = 2cos(ωt) and y = sin(ωt) Since, as the graph starts to get traced out, the x-values are negative and the y-values are positive, so ω has to be greater than 0. Also, it takes 2 seconds to make one revolution, the period is 2 = 2π which implies ω = π. Therefore, our parametric equations are: x = 2cos(πt) and y = sin(πt) 0 t 2 b) The equation is an ellipse with center at (0, 0), major along the x-axis, vertices of ( 2, 0) and (2, 0) and b-intercepts (0, 1) and (0, 1). Since the ellipse begins at (0, 1), then at t = 0, x = 0 and y = 1. Thus, x = 2sin(ωt) or 2sin(ωt) and y = cos(ωt). Since the orientation is counterclockwise, then as the curve leaves the point (0, 1), the x-values will be negative, so x = 2sin(ωt). Hence, our equations are: x = 2sin(ωt) and y = cos(ωt) Since, as the graph starts to get traced out, the x-values are negative and the y-values are positive, so ω has to be greater than 0. Also, it takes 4 seconds to make one revolution, the period is 4 = 2π ω which implies ω = π. Therefore, our parametric equations are: 2 x = 2sin( π 2 t) and y = cos( π 2 t) 0 t 4 There are some curves that cannot be represented easily in terms of x and y, but are very easy to represent using parametric equations. One example is a curve called a cycloid. Its path is traced out by a point on the rim of a circle of radius a as the circle rolls along a fixed line without slipping. To derive the parametric equations, let P = (x, y) be a point on the rim of a circle of radius a such that P starts at the origin. We will allow the circle to roll along the positive x-axis: ω

10 08 Y P C t B 2a O X A Let C be the center the circle and t be the angle of how much the circle has rotated. Since the circle has rolled the distance from O to A, d(o, A), then the amount of the circle that has travelled on the ground is the arc length from A to P. Since there is no slippage, then arc length AP = d(o, A) The arc length AP = rθ = at since the radius r = a and t is the angle defined above. This implies that d(o, A) = at. Also, at = d(o, A) = d(o, X) + d(x, A) which implies d(o, X) = at d(x, A) In examining the diagram, d(x, A) = d(p, B) which in the length of the opposite side of t in triangle PCB. Thus PB = asin(t) which implies d(o, X) = at d(x, A) = at asin(t) = a(t sin(t)) Since d(o, X) = x, then x = a(t sin(t)) Now, d(a, C) = d(a, B) + d(b, C) or d(a, B) = d(a, C) d(b, C), but d(a, C) = a and d(b, C) is the adjacent side to t in triangle PCB which means d(b, C) = acos(t). Substituting these relationships in, we get: d(a, B) = a acos(t) = a(1 cos(t)) Since y = d(a, B), then y = a(1 cos(t)) Although we have proved this for t being less than or equal to π 2, these equations can be extended to any real number t. Parametric Equations for a Cycloid x = a(t sin(t)), y = a(1 cos(t)), < t <. The period will be 2πa. Suppose an object is moving without friction from a starting point P to an ending point Q that is lower than P, but not directly below P. If gravity is the only force acting on it, then the quickest way it can get to Q is along the path of an inverted cycloid. This is known as the curve of quickest decent. Another example is suppose that several objects are placed on an inverted at different locations of cycloid at the same time and began sliding at the same moment. If none of the objects were placed on the lowest point of the cycloid, they all will reach the lowest point at the same time.

Section 10.4: Motion in Space: Velocity and Acceleration

Section 10.4: Motion in Space: Velocity and Acceleration 1 Section 10.4: Motion in Space: Velocity and Acceleration Velocity and Acceleration Practice HW from Stewart Textbook (not to hand in) p. 75 # 3-17 odd, 1, 3 Given a vector function r(t ) = f (t) i +

More information

Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

More information

Calculating Areas Section 6.1

Calculating Areas Section 6.1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics Calculating Areas Section 6.1 Dr. John Ehrke Department of Mathematics Fall 2012 Measuring Area By Slicing We first defined

More information

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25

Math 115 Spring 2014 Written Homework 10-SOLUTIONS Due Friday, April 25 Math 115 Spring 014 Written Homework 10-SOLUTIONS 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 information

Graphing Quadratics using Transformations 5-1

Graphing Quadratics using Transformations 5-1 Graphing Quadratics using Transformations 5-1 5-1 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 information

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 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

More information

with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

with a, b and c representing real numbers, and a is not equal to zero. 3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

More information

BROCK UNIVERSITY MATHEMATICS MODULES

BROCK 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 information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

6.1 - Introduction to Periodic 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

More information

Conic Sections in Cartesian and Polar Coordinates

Conic Sections in Cartesian and Polar Coordinates Conic Sections in Cartesian and Polar Coordinates The conic sections are a family of curves in the plane which have the property in common that they represent all of the possible intersections of a plane

More information

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 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 information

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2. INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

More information

29 Wyner PreCalculus Fall 2016

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

More information

Graphing Quadratic Functions

Graphing 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 information

(c) What values are the input of the function? (in other words, which axis?) A: The x, or horizontal access are the input values, or the domain.

(c) What values are the input of the function? (in other words, which axis?) A: The x, or horizontal access are the input values, or the domain. Calculus Placement Exam Material For each function, graph the function by hand using point plotting. You must use a minimum of five points for each function. Plug in 0, positive and negative x-values for

More information

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts.

The x-intercepts of the graph are the x-values for the points where the graph intersects the x-axis. A parabola may have one, two, or no x-intercepts. Chapter 10-1 Identify Quadratics and their graphs A parabola is the graph of a quadratic function. A quadratic function is a function that can be written in the form, f(x) = ax 2 + bx + c, a 0 or y = ax

More information

5.2 Unit Circle: Sine and Cosine Functions

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

More information

POLAR COORDINATES DEFINITION OF POLAR COORDINATES

POLAR 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 information

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to:

FUNCTIONS. Introduction to Functions. Overview of Objectives, students should be able to: FUNCTIONS Introduction to Functions Overview of Objectives, students should be able to: 1. Find the domain and range of a relation 2. Determine whether a relation is a function 3. Evaluate a function 4.

More information

Section 10-5 Parametric Equations

Section 10-5 Parametric Equations 88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative

More information

Trigonometric Functions

Trigonometric Functions Trigonometric Functions MATH 10, Precalculus J. Robert Buchanan Department of Mathematics Fall 011 Objectives In this lesson we will learn to: identify a unit circle and describe its relationship to real

More information

Applications of Integration Day 1

Applications of Integration Day 1 Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

More information

Test Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of

Test Bank Exercises in. 7. Find the intercepts, the vertical asymptote, and the slant asymptote of the graph of Test Bank Exercises in CHAPTER 5 Exercise Set 5.1 1. Find the intercepts, the vertical asymptote, and the horizontal asymptote of the graph of 2x 1 x 1. 2. Find the intercepts, the vertical asymptote,

More information

Math 4 Review Problems

Math 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 information

MPE Review Section II: Trigonometry

MPE Review Section II: Trigonometry MPE Review Section II: Trigonometry Review similar triangles, right triangles, and the definition of the sine, cosine and tangent functions of angles of a right triangle In particular, recall that the

More information

Final Exam Practice--Problems

Final Exam Practice--Problems Name: Class: Date: ID: A Final Exam Practice--Problems Problem 1. Consider the function f(x) = 2( x 1) 2 3. a) Determine the equation of each function. i) f(x) ii) f( x) iii) f( x) b) Graph all four functions

More information

Prime Coe cients Formula for Rotating Conics

Prime Coe cients Formula for Rotating Conics Prime Coe cients Formula for Rotating Conics David Rose Polk State College David Rose (Institute) Prime Coe cients Formula for Rotating Conics 1 / 18 It is known that the graph of a conic section in the

More information

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement)

y = a sin ωt or y = a cos ωt then the object is said to be in simple harmonic motion. In this case, Amplitude = a (maximum displacement) 5.5 Modelling Harmonic Motion Periodic behaviour happens a lot in nature. Examples of things that oscillate periodically are daytime temperature, the position of a weight on a spring, and tide level. If

More information

Parametric Equations

Parametric Equations Parametric Equations Part 1: Vector-Valued Functions Now that we have introduced and developed the concept of a vector, we are ready to use vectors to de ne functions. To begin with, a vector-valued function

More information

The quest to find how x(t) and y(t) depend on t is greatly simplified by the following facts, first discovered by Galileo:

The quest to find how x(t) and y(t) depend on t is greatly simplified by the following facts, first discovered by Galileo: Team: Projectile Motion So far you have focused on motion in one dimension: x(t). In this lab, you will study motion in two dimensions: x(t), y(t). This 2D motion, called projectile motion, consists of

More information

Physics Unit 2: Projectile Motion

Physics Unit 2: Projectile Motion Physics Unit 2: Projectile Motion Jan 31 10:07 AM What is a projectile? A projectile is an object that is launched, or projected, by some means and continues on its own inertia. The path of the projectile

More information

Rotational Mechanics - 1

Rotational Mechanics - 1 Rotational Mechanics - 1 The Radian The radian is a unit of angular measure. The radian can be defined as the arc length s along a circle divided by the radius r. s r Comparing degrees and radians 360

More information

Mathematics. GSE Pre-Calculus Unit 2: Trigonometric Functions

Mathematics. GSE Pre-Calculus Unit 2: Trigonometric Functions Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Pre-Calculus Unit : Trigonometric Functions These materials are for nonprofit educational purposes only. Any other use may constitute

More information

θ. The angle is denoted in two ways: angle θ

θ. The angle is denoted in two ways: angle θ 1.1 Angles, Degrees and Special Triangles (1 of 24) 1.1 Angles, Degrees and Special Triangles Definitions An angle is formed by two rays with the same end point. The common endpoint is called the vertex

More information

Physics 1120: 2D Kinematics Solutions

Physics 1120: 2D Kinematics Solutions Questions: 1 2 3 4 5 6 7 8 9 10 11 Physics 1120: 2D Kinematics Solutions 1. In the diagrams below, a ball is on a flat horizontal surface. The inital velocity and the constant acceleration of the ball

More information

5-1. Lesson Objective. Lesson Presentation Lesson Review

5-1. Lesson Objective. Lesson Presentation Lesson Review 5-1 Using Transformations to Graph Quadratic Functions Lesson Objective Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x h) 2 + k. Lesson Presentation Lesson

More information

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13

3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13 Contents 1 Limits 2 2 Derivatives 3 2.1 Difference Quotients......................................... 3 2.2 Average Rate of Change...................................... 4 2.3 Derivative Rules...........................................

More information

6.3 Parametric Equations and Motion

6.3 Parametric Equations and Motion SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh

More information

Projectile motion simulator. http://www.walter-fendt.de/ph11e/projectile.htm

Projectile motion simulator. http://www.walter-fendt.de/ph11e/projectile.htm More Chapter 3 Projectile motion simulator http://www.walter-fendt.de/ph11e/projectile.htm The equations of motion for constant acceleration from chapter 2 are valid separately for both motion in the x

More information

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

MA Lesson 19 Summer 2016 Angles and Trigonometric Functions DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common

More information

Motion in Two Dimensions

Motion in Two Dimensions Motion in Two Dimensions 1. The position vector at t i is r i and the position vector at t f is r f. The average velocity of the particle during the time interval is a.!!! ri + rf v = 2 b.!!! ri rf v =

More information

x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

x 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 information

Tangent and normal lines to conics

Tangent and normal lines to conics 4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

More information

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved.

2.1 QUADRATIC FUNCTIONS AND MODELS. Copyright Cengage Learning. All rights reserved. 2.1 QUADRATIC FUNCTIONS AND MODELS Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic functions. Write quadratic functions in standard form and use the results

More information

9-3 Polar and Rectangular Forms of Equations

9-3 Polar and Rectangular Forms of Equations 9-3 Polar and Rectangular Forms of Equations Find the rectangular coordinates for each point with the given polar coordinates Round to the nearest hundredth, if necessary are The rectangular coordinates

More information

Mechanics 1. Revision Notes

Mechanics 1. Revision Notes Mechanics 1 Revision Notes July 2012 MECHANICS 1... 2 1. Mathematical Models in Mechanics... 2 Assumptions and approximations often used to simplify the mathematics involved:... 2 2. Vectors in Mechanics....

More information

Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Angles 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 information

Form of a linear function: Form of a quadratic function:

Form of a linear function: Form of a quadratic function: Algebra IIA Unit II: Quadratic Functions Foundational Material o Graphing and transforming linear functions o Solving linear equations and inequalities o Fit data using linear models Goal o Graph and transform

More information

4-6 Inverse Trigonometric Functions

4-6 Inverse Trigonometric Functions Find the exact value of each expression, if it exists. 1. sin 1 0 Find a point on the unit circle on the interval with a y-coordinate of 0. 3. arcsin Find a point on the unit circle on the interval with

More information

Yimin Math Centre. 2/3 Unit Math Homework for Year Motion Part Simple Harmonic Motion The Differential Equation...

Yimin Math Centre. 2/3 Unit Math Homework for Year Motion Part Simple Harmonic Motion The Differential Equation... 2/3 Unit Math Homework for Year 12 Student Name: Grade: Date: Score: Table of contents 9 Motion Part 3 1 9.1 Simple Harmonic Motion The Differential Equation................... 1 9.2 Projectile Motion

More information

How to Graph Trigonometric Functions

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 information

Quadratic. Functions

Quadratic. Functions Quadratic Functions 5A Quadratic Functions and Complex Numbers Lab Explore Parameter Changes 5-1 Using Transformations to Graph Quadratic Functions 5- Properties of Quadratic Functions in Standard Form

More information

Lecture 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 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 information

12.1. Vector-Valued Functions. Vector-Valued Functions. Objectives. Space Curves and Vector-Valued Functions. Space Curves and Vector-Valued Functions

12.1. Vector-Valued Functions. Vector-Valued Functions. Objectives. Space Curves and Vector-Valued Functions. Space Curves and Vector-Valued Functions 12 Vector-Valued Functions 12.1 Vector-Valued Functions Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Analyze and sketch a space curve given

More information

Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

More information

Simple Harmonic Motion Concepts

Simple Harmonic Motion Concepts Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called

More information

This is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0).

This is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0). This is Conic Sections, chapter 8 from the book Advanced Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/)

More information

Worksheet for Week 1: Circles and lines

Worksheet for Week 1: Circles and lines Worksheet Math 124 Week 1 Worksheet for Week 1: Circles and lines This worksheet is a review of circles and lines, and will give you some practice with algebra and with graphing. Also, this worksheet introduces

More information

Polynomial Degree and Finite Differences

Polynomial Degree and Finite Differences CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

More information

10 Polar Coordinates, Parametric Equations

10 Polar Coordinates, Parametric Equations Polar Coordinates, Parametric Equations ½¼º½ ÈÓÐ Ö ÓÓÖ Ò Ø Coordinate systems are tools that let us use algebraic methods to understand geometry While the rectangular (also called Cartesian) coordinates

More information

Physics 201 Homework 5

Physics 201 Homework 5 Physics 201 Homework 5 Feb 6, 2013 1. The (non-conservative) force propelling a 1500-kilogram car up a mountain -1.21 10 6 joules road does 4.70 10 6 joules of work on the car. The car starts from rest

More information

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots.

9.1 Solving Quadratic Equations by Finding Square Roots Objectives 1. Evaluate and approximate square roots. 9.1 Solving Quadratic Equations by Finding Square Roots 1. Evaluate and approximate square roots. 2. Solve a quadratic equation by finding square roots. Key Terms Square Root Radicand Perfect Squares Irrational

More information

People s Physics book 3e Ch 25-1

People s Physics book 3e Ch 25-1 The Big Idea: In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate

More information

3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models

3.1. Quadratic Equations and Models. Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1 Quadratic Equations and Models Quadratic Equations Graphing Techniques Completing the Square The Vertex Formula Quadratic Models 3.1-1 Polynomial Function A polynomial function of degree n, where n

More information

PREREQUISITE/PRE-CALCULUS REVIEW

PREREQUISITE/PRE-CALCULUS REVIEW PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which

More information

4.1 Radian and Degree Measure

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

More information

9.1 Trigonometric Identities

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)

More information

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS

CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS CHAPTER 3: GRAPHS OF QUADRATIC RELATIONS Specific Expectations Addressed in the Chapter Collect data that can be represented as a quadratic relation, from experiments using appropriate equipment and technology

More information

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

More information

4. Graphing and Inverse Functions

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

More information

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.

MATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4. MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin

More information

Solutions to Homework 10

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

More information

Chapter 5: Trigonometric Functions of Real Numbers

Chapter 5: Trigonometric Functions of Real Numbers Chapter 5: Trigonometric Functions of Real Numbers 5.1 The Unit Circle The unit circle is the circle of radius 1 centered at the origin. Its equation is x + y = 1 Example: The point P (x, 1 ) is on the

More information

Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

More information

8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections 8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

More information

Extra Problems for Midterm 2

Extra Problems for Midterm 2 Extra Problems for Midterm Sudesh Kalyanswamy Exercise (Surfaces). Find the equation of, and classify, the surface S consisting of all points equidistant from (0,, 0) and (,, ). Solution. Let P (x, y,

More information

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out:

Trigonometry (Chapters 4 5) Sample Test #1 First, a couple of things to help out: First, a couple of things to help out: Page 1 of 24 Use periodic properties of the trigonometric functions to find the exact value of the expression. 1. cos 2. sin cos sin 2cos 4sin 3. cot cot 2 cot Sin

More information

Section 3.2. Graphing linear equations

Section 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 information

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}.

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value, or {y y 1.25}. Use a table of values to graph each equation. State the domain and range. 1. y = x 2 + 3x + 1 x y 3 1 2 1 1 1 0 1 1 5 2 11 Graph the ordered pairs, and connect them to create a smooth curve. The parabola

More information

Spherical coordinates 1

Spherical coordinates 1 Spherical coordinates 1 Spherical coordinates Both the earth s surface and the celestial sphere have long been modeled as perfect spheres. In fact, neither is really a sphere! The earth is close to spherical,

More information

MAC 1114: Trigonometry Notes

MAC 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 information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 110 B) 120 C) 60 D) 150 Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. ) 56

More information

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle.

Pre-Calculus II. where 1 is the radius of the circle and t is the radian measure of the central angle. Pre-Calculus II 4.2 Trigonometric Functions: The Unit Circle The unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle

More information

POLAR COORDINATES: WHAT THEY ARE AND HOW TO USE THEM

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

More information

Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

More information

SECTION 4.5: GRAPHS OF SINE AND COSINE FUNCTIONS

SECTION 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 information

Lesson 36 MA 152, Section 3.1

Lesson 36 MA 152, Section 3.1 Lesson 36 MA 5, Section 3. I Quadratic Functions A quadratic function of the form y = f ( x) = ax + bx + c, where a, b, and c are real numbers (general form) has the shape of a parabola when graphed. The

More information

The graph of. horizontal line between-1 and 1 that the sine function is not 1-1 and therefore does not have an inverse.

The graph of. horizontal line between-1 and 1 that the sine function is not 1-1 and therefore does not have an inverse. Inverse Trigonometric Functions The graph of If we look at the graph of we can see that if you draw a horizontal line between-1 and 1 that the sine function is not 1-1 and therefore does not have an inverse.

More information

1. [20 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Exist.

1. [20 Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Exist. Answer Key, Math, Final Eamination, December 9, 9. [ Points] Evaluate each of the following limits. Please justify your answers. Be clear if the limit equals a value, + or, or Does Not Eist. (a lim + 6

More information

Angles and Their Measure

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

More information

Chapter 7 Quadratic Functions & Applications I

Chapter 7 Quadratic Functions & Applications I Chapter 7 Quadratic Functions & Applications I In physics the study of the motion of objects is known as Kinematics. In kinematics there are four key measures to be analyzed. These are work, potential,

More information

Example 1. Example 1 Plot the points whose polar coordinates are given by

Example 1. Example 1 Plot the points whose polar coordinates are given by Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

More information

The graphs of quadratic functions are so popular that they were given their own name. They are called parabolas.

The graphs of quadratic functions are so popular that they were given their own name. They are called parabolas. DETAILED SOLUTIONS AND CONCEPTS - QUADRATIC 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 information

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole

More information

D.3. Angles and Degree Measure. Review of Trigonometric Functions

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

More information

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

More information

1. AREA BETWEEN the CURVES

1. AREA BETWEEN the CURVES 1 The area between two curves The Volume of the Solid of revolution (by slicing) 1. AREA BETWEEN the CURVES da = {( outer function ) ( inner )} dx function b b A = da = [y 1 (x) y (x)]dx a a d d A = da

More information

IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed. Name: Date: Class:

IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed. Name: Date: Class: IB Math Standard Level Year 2: May 11, Paper 2, TZ 1 IB Practice Exam: 11 Paper 2 Zone 1 90 min, Calculator Allowed Name: Date: Class: 1. The following diagram shows triangle ABC. AB = 7 cm, BC = 9 cm

More information