6.3 Parametric Equations and Motion


 Katrina Taylor
 1 years ago
 Views:
Transcription
1 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 These topics can be used to model the path of an object such as a baseball or a golf ball. t = 0, = 420 t = 4, = 164 t = 5, = 20 t = 1, = 404 t = 2, = 356 t = 3, = 276 [0, 5] b [ 10, 500] FIGURE 6.23 The position of the rock at 0, 1, 2, 3, 4, and 5 seconds. 6.3 Parametric Equations and Motion Parametric Equations Imagine that a rock is dropped from a 420ft tower. The rock s height in feet above the ground t seconds later (ignoring air resistance) is modeled b = 16t as we saw in Section 2.1. Figure 6.23 shows a coordinate sstem imposed on the scene so that the line of the rock s fall is on the vertical line = 2.5. The rock s original position and its position after each of the first 5 seconds are the points (2.5, 420), (2.5, 404), (2.5, 356), (2.5, 276), (2.5, 164), (2.5, 20), which are described b the pair of equations = 2.5, = 16t , when t = 0, 1, 2, 3, 4, 5. These two equations are an eample of parametric equations with parameter t. As is often the case, the parameter t represents time. Parametric Curves In this section we stud the graphs of parametric equations and investigate motion of objects that can be modeled with parametric equations. DEFINITION Parametric Curve, Parametric Equations The graph of the ordered pairs 1, 2 where = ƒ1t2, = g1t2 are functions defined on an interval I of tvalues is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval. When we give parametric equations and a parameter interval for a curve, we have parametrized the curve. A parametrization of a curve consists of the parametric equations and the interval of tvalues. Sometimes parametric equations are used b companies in their design plans. It is then easier for the compan to make larger and smaller objects efficientl b just changing the parameter t. Graphs of parametric equations can be obtained using parametric mode on a grapher. EXAMPLE 1 Graphing Parametric Equations For the given parameter interval, graph the parametric equations = t 22, = 3t. 3 t 12 t 3 (c) 3 t 3 SOLUTION In each case, set Tmin equal to the left endpoint of the interval and Tma equal to the right endpoint of the interval. Figure 6.24 shows a graph of the parametric equations for each parameter interval. The corresponding relations are different because the parameter intervals are different. Now tr Eercise 7.
2 476 CHAPTER 6 Applications of Trigonometr [ 10, 10] b [ 10, 10] [ 10, 10] b [ 10, 10] [ 10, 10] b [ 10, 10] (c) FIGURE 6.24 Three different relations defined parametricall. (Eample 1) Eliminating the Parameter When a curve is defined parametricall it is sometimes possible to eliminate the parameter and obtain a rectangular equation in and that represents the curve. This often helps us identif the graph of the parametric curve as illustrated in Eample 2. EXAMPLE 2 Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve = 12t, = 2  t,  q 6 t 6 q. SOLUTION We solve the first equation for t: = 12t 2t = 1  t = Then we substitute this epression for t into the second equation: = 2  t [ 10, 5] b [ 5, 5] FIGURE 6.25 The graph of = (Eample 2) = = The graph of the equation = is a line with slope 0.5 and intercept 1.5 (Figure 6.25). Now tr Eercise 11. EXPLORATION 1 Graphing the Curve of Eample 2 Parametricall 1. Use the parametric mode of our grapher to reproduce the graph in Figure Use 2 for Tmin and 5.5 for Tma. 2. Prove that the point 117, 102 is on the graph of = Find the corresponding value of t that produces this point. 3. Repeat part 2 for the point 23, Assume that 1a, b2 is on the graph of = Find the corresponding value of t that produces this point. 5. How do ou have to choose Tmin and Tma so that the graph in Figure 6.25 fills the window?
3 SECTION 6.3 Parametric Equations and Motion 477 If we do not specif a parameter interval for the parametric equations = ƒ1t2, = g1t2, it is understood that the parameter t can take on all values that produce real numbers for and. We use this agreement in Eample 3. EXAMPLE 3 Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve = t 22, = 3t. Parabolas The inverse of a parabola that opens up or down is a parabola that opens left or right. We will investigate these curves in more detail in Chapter 8. SOLUTION Here t can be an real number. We solve the second equation for t, obtaining t = /3, and substitute this value for into the first equation. = t 22 = a 2 3 b  2 = = Figure 6.24c shows what the graph of these parametric equations looks like. In Chapter 8 we will call this a parabola that opens to the right. Interchanging and, we can identif this graph as the inverse of the graph of the parabola 2 = Now tr Eercise 15. EXAMPLE 4 Eliminating the Parameter Eliminate the parameter and identif the graph of the parametric curve = 2 cos t, = 2 sin t, 0 t 2p. [ 4.7, 4.7] b [ 3.1, 3.1] FIGURE 6.26 The graph of the circle of Eample 4. SOLUTION The graph of the parametric equations in the square viewing window of Figure 6.26 suggests that the graph is a circle of radius 2 centered at the origin. We confirm this result algebraicall = 4 cos 2 t + 4 sin 2 t = 41cos 2 t + sin 2 t2 = 4112 = 4 cos 2 t + sin 2 t = 1 The graph of = 4 is a circle of radius 2 centered at the origin. Increasing the length of the interval 0 t 2p will cause the grapher to trace all or part of the circle more than once. Decreasing the length of the interval will cause the grapher to onl draw a portion of the complete circle. Tr it! Now tr Eercise 23. In Eercise 65, ou will find parametric equations for an circle in the plane. Lines and Line Segments We can use vectors to help us find parametric equations for a line as illustrated in Eample 5.
4 478 CHAPTER 6 Applications of Trigonometr A( 2, 3) O 1 B(3, 6) P(, ) FIGURE 6.27 Eample 5 uses vectors to construct a parametrization of the line through A and B. EXAMPLE 5 Finding Parametric Equations for a Line Find a parametrization of the line through the points A = 12, 32 and B = 13, 62. SOLUTION Let P1, 2 be an arbitrar point on the line through A and B. As ou! can see from Figure 6.27, the vector is the tailtohead vector sum of and. You can also see that AP! OP! is a scalar multiple of AB! OA! AP. If we let the scalar be t, we have OP! OP! = OA! = OA! + AP!! + t # AB 8, 9 = 82, 39 + t , , 9 = 82, 39 + t85, 39 8, 9 = t, 3 + 3t9 This vector equation is equivalent to the parametric equations = t and = 3 + 3t. Together with the parameter interval 1 q, q2, these equations define the line. We can confirm our work numericall as follows: If t = 0, then = 2 and = 3, which gives the point A. Similarl, if t = 1, then = 3 and = 6, which gives the point B. Now tr Eercise 27. The fact that t = 0 ields point A and t = 1 ields point B in Eample 5 is no accident, as a little reflection on Figure 6.27 and the vector equation OP! = OA!! + t # AB should suggest. We use this fact in Eample 6. EXAMPLE 6 Finding Parametric Equations for a Line Segment Find a parametrization of the line segment with endpoints A = 12, 32 and B = 13, 62. SOLUTION In Eample 5 we found parametric equations for the line through A and B: = t, = 3 + 3t We also saw in Eample 5 that t = 0 produces the point A and t = 1 produces the point B. A parametrization of the line segment is given b = t, = 3 + 3t, 0 t 1. As t varies between 0 and 1 we pick up ever point on the line segment between A and B. Now tr Eercise 29. Simulating Motion with a Grapher Eample 7 illustrates several was to simulate motion along a horizontal line using parametric equations. We use the variable t for the parameter to represent time. EXAMPLE 7 Simulating Horizontal Motion Gar walks along a horizontal line (think of it as a number line) with the coordinate of his position (in meters) given b s = 0.11t 320t t where 0 t 12. Use parametric equations and a grapher to simulate his motion. Estimate the times when Gar changes direction.
5 SECTION 6.3 Parametric Equations and Motion 479 T=0 X=8.5 Y=5 Start, t = 0 SOLUTION We arbitraril choose the horizontal line = 5 to displa this motion. The graph C 1 of the parametric equations, simulates the motion. His position at an time t is given b the point 1 1 1t2, 52. Using TRACE in Figure 6.28 we see that when t = 0, Gar is 8.5 m to the right of the ais at the point (8.5, 5), and that he initiall moves left. Five seconds later he is 9 m to the left of the ais at the point (9, 5). And after 8 seconds he is onl 2.7 m to the left of the ais. Gar must have changed direction during the walk. The motion of the trace cursor simulates Gar s motion. A variation in 1t2, C 1 : 1 = 0.1(t 320t t  85), 1 = 5, 0 t 12, C 2 : 2 = 0.11t 320t t  852, 2 = t, 0 t 12, can be used to help visualize where Gar changes direction. The graph C 2 shown in Figure 6.29 suggests that Gar reverses his direction at 3.9 seconds and again at 9.5 seconds after beginning his walk. Now tr Eercise 37. T=5 X= 9 Y=5 5 sec later, t = 5 C 1 C 1 T=8 X= 2.7 Y=5 3 sec after that, t = 8 (c) FIGURE 6.28 Three views of the graph C 1 : 1 = 0.11t 320t t  852, 1 = 5, 0 t 312, in the 124 b 310, 104 viewing window. (Eample 7) Grapher Note The equation 2 = t is tpicall used in the parametric equations for the graph C 2 in Figure We have chosen 2 = t to get two curves in Figure 6.29 that do not overlap. Also notice that the coordinates of C 1 are constant 1 1 = 52, and that the coordinates of C 2 var with time t 1 2 = t2. C 2 T=3.9 X= Y= 3.9 [ 12, 12] b [ 15, 15] [ 12, 12] b [ 15, 15] Eample 8 solves a projectilemotion problem. Parametric equations are used in two was: to find a graph of the modeling equation and to simulate the motion of the projectile. C 2 T=9.5 X= Y= 9.5 FIGURE 6.29 Two views of the graph C 1 : 1 = 0.1(t 320t t  85), and the graph C 2 : 2 = 0.11t 320t 2 1 = 5, 0 t t  852, 2 = t, 0 t 12 in the 312, 124 b 315, 154 viewing window. (Eample 7) EXAMPLE 8 Simulating Projectile Motion A distress flare is shot straight up from a ship s bridge 75 ft above the water with an initial velocit of 76 ft/sec. Graph the flare s height against time, give the height of the flare above water at each time, and simulate the flare s motion for each length of time. 1 sec 2 sec (c) 4 sec (d) 5 sec SOLUTION An equation that models the flare s height above the water t seconds after launch is = 16t t A graph of the flare s height against time can be found using the parametric equations 1 = t, 1 = 16t t (continued)
6 480 CHAPTER 6 Applications of Trigonometr To simulate the flare s flight straight up and its fall to the water, use the parametric equations 2 = 5.5, 2 = 16t t (We chose 2 = 5.5 so that the two graphs would not intersect.) Figure 6.30 shows the two graphs in simultaneous graphing mode for 0 t 1, 0 t 2, (c) 0 t 4, and (d) 0 t 5. We can read that the height of the flare above the water after 1 sec is 135 ft, after 2 sec is 163 ft, after 4 sec is 123 ft, and after 5 sec is 55 ft. Now tr Eercise 39. T=1 X=5.5 Y=135 T=2 X=5.5 Y=163 T=4 X=5.5 Y=123 T=5 X=5.5 Y=55 [0, 6] b [0, 200] [0, 6] b [0, 200] [0, 6] b [0, 200] (c) [0, 6] b [0, 200] (d) FIGURE 6.30 Simultaneous graphing of (height against time) and 2 = 5.5, 2 = 16t 2 1 = t, 1 = 16t t t + 75 (the actual path of the flare). (Eample 8) 0 v 0 v 0 sin v 0 cos FIGURE 6.31 Throwing a baseball. In Eample 8 we modeled the motion of a projectile that was launched straight up. Now we investigate the motion of objects, ignoring air friction, that are launched at angles other than 90 with the horizontal. Suppose that a baseball is thrown from a point 0 feet above ground level with an initial speed of v 0 ft/sec at an angle u with the horizontal (Figure 6.31). The initial velocit can be represented b the vector The path of the object is modeled b the parametric equations The component is simpl v = 8v 0 cos u, v 0 sin u9. = 1v 0 cos u2t, = 16t 2 + 1v 0 sin u2t + 0. distance = 1component of initial velocit2 * time. The component is the familiar vertical projectilemotion equation using the component of initial velocit. [0, 450] b [0, 80] FIGURE 6.32 The fence and path of the baseball in Eample 9. See Eploration 2 for was to draw the wall. EXAMPLE 9 Hitting a Baseball Kevin hits a baseball at 3 ft above the ground with an initial speed of 150 ft/sec at an angle of 18 with the horizontal. Will the ball clear a 20ft wall that is 400 ft awa? SOLUTION The path of the ball is modeled b the parametric equations = 1150 cos 18 2t, = 16t sin 18 2t + 3. A little eperimentation will show that the ball will reach the fence in less than 3 sec. Figure 6.32 shows a graph of the path of the ball using the parameter interval 0 t 3 and the 20ft wall. The ball does not clear the wall. Now tr Eercise 43.
7 SECTION 6.3 Parametric Equations and Motion 481 EXPLORATION 2 Etending Eample 9 1. If our grapher has a line segment feature, draw the fence in Eample Describe the graph of the parametric equations = 400, = 201t/32, 0 t Repeat Eample 9 for the angles 19, 20, 21, and ft 30 ft FIGURE 6.33 The Ferris wheel of Eample 10. A In Eample 10 we see how to write parametric equations for position on a moving Ferris wheel, using time t as the parameter. EXAMPLE 10 Riding on a Ferris Wheel Jane is riding on a Ferris wheel with a radius of 30 ft. As we view it in Figure 6.33, the wheel is turning counterclockwise at the rate of one revolution ever 10 sec. Assume that the lowest point of the Ferris wheel (6 o clock) is 10 ft above the ground and that Jane is at the point marked A (3 o clock) at time t = 0. Find parametric equations to model Jane s path and use them to find Jane s position 22 sec into the ride. 40 P 30 θ A FIGURE 6.34 A model for the Ferris wheel of Eample 10. SOLUTION Figure 6.34 shows a circle with center 10, 402 and radius 30 that models the Ferris wheel. The parametric equations for this circle in terms of the parameter u, the central angle of the circle determined b the arc AP, are = 30 cos u, = sin u, 0 u 2p. To take into account the rate at which the wheel is turning we must describe u as a function of time t in seconds. The wheel is turning at the rate of 2p radians ever 10 sec, or 2p/10 = p/5 rad/sec. So, u = 1p/52t. Thus, parametric equations that model Jane s path are given b = 30 cos a p 5 tb, = sin a p 5 tb, t Ú 0. We substitute t = 22 into the parametric equations to find Jane s position at that time: = 30 cos a p 5 # 22b = sin a p 5 # 22b L 9.27 L After riding for 22 sec, Jane is approimatel 68.5 ft above the ground and approimatel 9.3 ft to the right of the ais, using the coordinate sstem of Figure Now tr Eercise 51.
8 482 CHAPTER 6 Applications of Trigonometr QUICK REVIEW 6.3 (For help, go to Sections P.2, P.4, 1.3, 4.1, and 6.1.) In Eercises 1 and 2, find the component form of the vectors OA!, OB!, and (c) AB! where O is the origin. 1. A = 13, 2), B = 14, A = 11, 3), B = 14, 32 In Eercises 3 and 4, write an equation in pointslope form for the line through the two points , 22, 14, , 32, 14, 32 In Eercises 5 and 6, find and graph the two functions defined implicitl b each given relation = = 5 In Eercises 7 and 8, write an equation for the circle with given center and radius , 02, , 52, 3 In Eercises 9 and 10, a wheel with radius r spins at the given rate. Find the angular velocit in radians per second. 9. r = 13 in., 600 rpm 10. r = 12 in., 700 rpm SECTION 6.3 EXERCISES Eercise numbers with a gra background indicate problems that the authors have designed to be solved without a calculator. In Eercises 1 4, match the parametric equations with their graph. Identif the viewing window that seems to have been used. (c) 1. = 4 cos 3 t, = 2 sin 3 t 2. = 3 cos t, = sin 2t (d) 3. = 2 cos t + 2 cos 2 t, = 2 sin t + sin 2t 4. = sin t  t cos t, = cos t + t sin t In Eercises 5 and 6, complete the table for the parametric equations and plot the corresponding points. 5. = t + 2, = 1 + 3/t t = cos t, = sin t t 0 /2 pp 3p/2 2p In Eercises 7 10, graph the parametric equations = 3  t 2, = 2t, in the specified parameter interval. Use the standard viewing window t t t t 4 In Eercises 11 26, use an algebraic method to eliminate the parameter and identif the graph of the parametric curve. Use a grapher to support our answer. 11. = 1 + t, = t 12. = 23t, = 5 + t 13. = 2t  3, = 94t, 3 t = 53t, = 2 + t, 15. = t 2, = t + 1 [Hint: Eliminate t and solve for in terms of.] 16. = t, = t = t, = t 32t = 2t 21, = t [Hint: Eliminate t and solve for in terms of.] 19. = 4  t 2, = t [Hint: Eliminate t and solve for in terms of.] 20. = 0.5t, = 2t 33, 2 t = t  3, = 2/t, 5 t = t + 2, = 4/t, t Ú = 5 cos t, = 5 sin t 24. = 4 cos t, = 4 sin t 1 t = 2 sin t, = 2 cos t, 0 t 3p/2 26. = 3 cos t, = 3 sin t, 0 t p In Eercises find a parametrization for the curve. 27. The line through the points 12, 52 and 14, The line through the points 3, 321 and 15, The line segment with endpoints 13, 42 and 16, 32
Section 105 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 informationApplications of Trigonometry
5144_Demana_Ch06pp501566 01/11/06 9:31 PM Page 501 CHAPTER 6 Applications of Trigonometr 6.1 Vectors in the Plane 6. Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationYears t. Definition Anyone who has drawn a circle using a compass will not be surprised by the following definition of the circle: x 2 y 2 r 2 304
Section The Circle 65 Dollars Purchase price P Book value = f(t) Salvage value S Useful life L Years t FIGURE 3 Straightline depreciation. The Circle Definition Anone who has drawn a circle using a compass
More informationSection 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 # 317 odd, 1, 3 Given a vector function r(t ) = f (t) i +
More informationTHE PARABOLA section. Developing the Equation
80 (0) Chapter Nonlinear Sstems and the Conic Sections. THE PARABOLA In this section Developing the Equation Identifing the Verte from Standard Form Smmetr and Intercepts Graphing a Parabola Maimum or
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
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 informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More informationAlex and Morgan were asked to graph the equation y = 2x + 1
Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and intercept wa First, I made a table. I chose some values, then plugged
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationImagine a car is traveling along the highway and you look down at the situation from high above: highway
Chapter 22 Parametric Equations Imagine a car is traveling along the highway you look down at the situation from high above highway curve (static) place car moving point (dynamic) Figure 22.1 The dynamic
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review 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
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More information1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model
. Piecewise Functions YOU WILL NEED graph paper graphing calculator GOAL Understand, interpret, and graph situations that are described b piecewise functions. LEARN ABOUT the Math A cit parking lot uses
More informationC3: Functions. Learning objectives
CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms oneone and manone mappings understand the terms domain and range for a mapping understand the
More informationLesson 03: Kinematics
www.scimsacademy.com PHYSICS Lesson 3: Kinematics Translational motion (Part ) If you are not familiar with the basics of calculus and vectors, please read our freely available lessons on these topics,
More information12.1. VectorValued Functions. VectorValued Functions. Objectives. Space Curves and VectorValued Functions. Space Curves and VectorValued Functions
12 VectorValued Functions 12.1 VectorValued Functions Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. Objectives! Analyze and sketch a space curve given
More informationContents. How You May Use This Resource Guide
Contents How You Ma Use This Resource Guide ii 9 Fractional and Quadratic Equations 1 Worksheet 9.1: Similar Figures.......................... 5 Worksheet 9.: Stretch of a Spring........................
More informationAlgebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
Algebra Module A7 The Parabola Copright This publication The Northern Alberta Institute of Technolog. All Rights Reserved. LAST REVISED December, The Parabola Statement of Prerequisite Skills Complete
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More information10.2 The Unit Circle: Cosine and Sine
0. The Unit Circle: Cosine and Sine 77 0. The Unit Circle: Cosine and Sine In Section 0.., we introduced circular motion and derived a formula which describes the linear velocit of an object moving on
More informationParametric Equations
Parametric Equations by: Kim Clark (kimclark@wcc.wayne.cc.nc.us) and Hal Kilpatrick (halk@wcc.wayne.cc.nc.us) Wayne Community College 3000 Wayne Memorial Drive Goldsboro, N.C. 275338002 (919) 7355152
More informationMath 259 Winter 2009. Recitation Handout 1: Finding Formulas for Parametric Curves
Math 259 Winter 2009 Recitation Handout 1: Finding Formulas for Parametric Curves 1. The diagram given below shows an ellipse in the plane. 51 13 (a) Find equations for (t) and (t) that will describe
More informationSupplementary Lesson: Loglog and Semilog Graph Paper
Supplementar Lesson: Loglog and Semilog Graph Paper Chapter 7 looks at some elementar functions of algebra, including linear, quadratic, power, eponential, and logarithmic. The following supplementar
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationGraph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:
 Root Functions Graph each function. Compare to the parent graph. State the domain and range...5.. 5. 6 is multiplied b a value greater than, so the graph is a vertical stretch of. Another wa to identif
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationMath 21a Old Exam One Fall 2003 Solutions Spring, 2009
1 (a) Find the curvature κ(t) of the curve r(t) = cos t, sin t, t at the point corresponding to t = Hint: You ma use the two formulas for the curvature κ(t) = T (t) r (t) = r (t) r (t) r (t) 3 Solution:
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationSolutions 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 informationChapter 8. Lines and Planes. By the end of this chapter, you will
Chapter 8 Lines and Planes In this chapter, ou will revisit our knowledge of intersecting lines in two dimensions and etend those ideas into three dimensions. You will investigate the nature of planes
More informationSECTION 74 Algebraic Vectors
74 lgebraic Vectors 531 SECTIN 74 lgebraic Vectors From Geometric Vectors to lgebraic Vectors Vector ddition and Scalar Multiplication Unit Vectors lgebraic Properties Static Equilibrium Geometric vectors
More informationFamilies of Quadratics
Families of Quadratics Objectives To understand the effects of a, b, and c on the graphs of parabolas of the form a 2 b c To use quadratic equations and graphs to analze the motion of projectiles To distinguish
More information3.4 The PointSlope Form of a Line
Section 3.4 The PointSlope Form of a Line 293 3.4 The PointSlope Form of a Line In the last section, we developed the slopeintercept form of a line ( = m + b). The slopeintercept form of a line is
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More information4 NonLinear relationships
NUMBER AND ALGEBRA NonLinear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
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 informationLinear Equations in Two Variables
Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More information27.2. Multiple Integrals over Nonrectangular Regions. Introduction. Prerequisites. Learning Outcomes
Multiple Integrals over Nonrectangular Regions 7. Introduction In the previous Section we saw how to evaluate double integrals over simple rectangular regions. We now see how to etend this to nonrectangular
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationChapter 3A  Rectangular Coordinate System
 Chapter A Chapter A  Rectangular Coordinate Sstem Introduction: Rectangular Coordinate Sstem Although the use of rectangular coordinates in such geometric applications as surveing and planning has been
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More information17.1 Connecting Intercepts and Zeros
Locker LESSON 7. Connecting Intercepts and Zeros Teas Math Standards The student is epected to: A.7.A Graph quadratic functions on the coordinate plane and use the graph to identif ke attributes, if possible,
More informationShake, Rattle and Roll
00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar
More informationSpeed A B C. Time. Chapter 3: Falling Objects and Projectile Motion
Chapter 3: Falling Objects and Projectile Motion 1. Neglecting friction, if a Cadillac and Volkswagen start rolling down a hill together, the heavier Cadillac will get to the bottom A. before the Volkswagen.
More informationNet Force and TwoDimensional Motion
5 Net Force and TwoDimensional Motion In 1922, one of the Zacchinis, a famous famil of circus performers, was the first human cannon ball to be shot across an arena into a net. To increase the ecitement,
More informationAngles Between Intersecting Lines (The Construct Intersection Command)
Angles Between Intersecting Lines (The Construct Intersection Command) Introduction... 2 Angles Defined... 2 Vectors Defined... 3 Two Factors That Determine Included Angle... 4 Eample... 7 Eercise... 9
More information10 Quadratic Equations and
www.ck1.org Chapter 10. Quadratic Equations and Quadratic Functions CHAPTER 10 Quadratic Equations and Quadratic Functions Chapter Outline 10.1 GRAPHS OF QUADRATIC FUNCTIONS 10. QUADRATIC EQUATIONS BY
More information5. Equations of Lines: slope intercept & point slope
5. Equations of Lines: slope intercept & point slope Slope of the line m rise run SlopeIntercept Form m + b m is slope; b is intercept PointSlope Form m( + or m( Slope of parallel lines m m (slopes
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More information8.7 Systems of NonLinear Equations and Inequalities
8.7 Sstems of NonLinear Equations and Inequalities 67 8.7 Sstems of NonLinear Equations and Inequalities In this section, we stud sstems of nonlinear equations and inequalities. Unlike the sstems of
More informationTRIGONOMETRY. circular functions. These functions, especially the sine and
TRIGONOMETRY Trigonometry has its origins in the study of triangle measurement. Natural generalizations of the ratios of righttriangle trigonometry give rise to both trigonometric and circular functions.
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More informationProject: OUTFIELD FENCES
1 Project: OUTFIELD FENCES DESCRIPTION: In this project you will work with the equations of projectile motion and use mathematical models to analyze a design problem. Two softball fields in Rolla, Missouri
More informationUsing the data above Height range: 6 1to 74 inches Weight range: 95 to 205
Plotting When plotting data, ou will normall be using two numbers, one for the coordinate, another for the coordinate. In some cases, like the first assignment, ou ma have onl one value. There, the second
More informationExponential and Logarithmic Functions
Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Course
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationReadings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.12.2. Professor Christopher Hoffman Math 124
Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.12.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/
More informationInverse Trigonometric Functions. Inverse Sine Function 4.71 FIGURE. Definition of Inverse Sine Function. The inverse sine function is defined by
0_007.qd /7/05 :0 AM Page Section.7.7 Inverse Trigonometric Functions Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse
More informationSection 59 Inverse Trigonometric Functions
46 5 TRIGONOMETRIC FUNCTIONS Section 59 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions
More informationSlope and Rate of Change
Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationLesson 6: Linear Functions and their Slope
Lesson 6: Linear Functions and their Slope A linear function is represented b a line when graph, and represented in an where the variables have no whole number eponent higher than. Forms of a Linear Equation
More informationCartesian Coordinate System. Also called rectangular coordinate system x and y axes intersect at the origin Points are labeled (x,y)
Physics 1 Vectors Cartesian Coordinate System Also called rectangular coordinate system x and y axes intersect at the origin Points are labeled (x,y) Polar Coordinate System Origin and reference line
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationPolynomial 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 informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
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 information5.1 Vector and Scalar Quantities. A vector quantity includes both magnitude and direction, but a scalar quantity includes only magnitude.
Projectile motion can be described by the horizontal ontal and vertical components of motion. In the previous chapter we studied simple straightline motion linear motion. Now we extend these ideas to
More information6.3 Polar Coordinates
6 Polar Coordinates Section 6 Notes Page 1 In this section we will learn a new coordinate sstem In this sstem we plot a point in the form r, As shown in the picture below ou first draw angle in standard
More informationDownloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x
Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its
More informationSTRAND: ALGEBRA Unit 3 Solving Equations
CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic
More information4.9 Graph and Solve Quadratic
4.9 Graph and Solve Quadratic Inequalities Goal p Graph and solve quadratic inequalities. Your Notes VOCABULARY Quadratic inequalit in two variables Quadratic inequalit in one variable GRAPHING A QUADRATIC
More informationSL Calculus Practice Problems
Alei  Desert Academ SL Calculus Practice Problems. The point P (, ) lies on the graph of the curve of = sin ( ). Find the gradient of the tangent to the curve at P. Working:... (Total marks). The diagram
More information3.1. 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 3.11 Polynomial Function A polynomial function of degree n, where n
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More information3 Unit Circle Trigonometry
0606_CH0_78.QXP //0 :6 AM Page Unit Circle Trigonometr In This Chapter. The Circular Functions. Graphs of Sine and Cosine Functions. Graphs of Other Trigonometric Functions. Special Identities.5 Inverse
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More informationSymmetry. A graph is symmetric with respect to the yaxis 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.
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More information