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 informationSECTION 115 Parametric Equations. Parametric Equations and Plane Curves. Parametric Equations and Plane Curves Projectile Motion Cycloid
 Parametric Equations 9. A hperbola with foci (, ) and (6, ) and vertices (, ) and (, ).. An ellipse with foci (, ) and (, 6) and vertices (, ) and (, ).. A parabola with ais the ais and passing through
More information1.5 Parametric Relations and Inverses
SECTION.5 Parametric Relations and Inverses 9 What ou ll learn about Relations Defined Parametricall Inverse Relations and Inverse Functions... and wh Some functions and graphs can best be defined parametricall,
More informationAnnual rate GRAPHS OF FUNCTIONS. Linear and Constant Functions. Linear Function
.6 Graphs of Functions () 7 80. Printing costs. To determine the cost of printing a book, a printer uses a linear function of the number of pages. If the cost is $8.60 for a 00page book and $.0 for a
More information3.1 Quadratic Functions
33337_030.qp 252 2/27/06 Chapter 3 :20 PM Page 252 Polnomial and Rational Functions 3. Quadratic Functions The Graph of a Quadratic Function In this and the net section, ou will stud the graphs of polnomial
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 informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Problems to look over Ch and Section 1. Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the ellipse and locate the foci. 1) x 2 4 + y 2
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 informationChapter 10 Conics, Parametric Equations, and Polar Coordinates
Chapter 1 Conics, Parametric Equations, and Polar Coordinates Chapter Summary Section Topics 1.1 Conics and Calculus Understand the definition of a conic section. Analyze and write equations of parabolas
More information10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it
3330_00.qd /8/05 9:00 AM Page 735 Section 0. Introduction to Conics: Parabolas 735 0. Introduction to Conics: Parabolas What ou should learn Recognize a conic as the intersection of a plane and a doublenapped
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 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 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 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 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 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 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 information4.7 Inverse Trigonometric. Functions
78 CHAPTER 4 Trigonometric Functions What ou ll learn about Inverse Sine Function Inverse Cosine and Tangent Functions Composing Trigonometric and Inverse Trigonometric Functions Applications of Inverse
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers
Sllabus Objecties: 5. The student will eplore methods of ector addition and subtraction. 5. The student will deelop strategies for computing a ector s direction angle and magnitude gien its coordinates.
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 information1.2. Graphs of Equations. The Graph of an Equation. What you should learn. Why you should learn it
3330_010.qd 1 1/7/05 Chapter 1 1. 8:31 AM Page 1 Function and Their Graphs Graphs of Equations What ou should learn Sketch graphs of equations. Find  and intercepts of graphs of equations. Use smmetr
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 informationStandard Equation of a Circle
Math 370 Precalculus Sec 10.1: Conics We will study all 4 types of conic sections, which are curves that result from the intersection of a right circular cone and a plane that does not contain the vertex.
More informationQuadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it
0_00.qd 8 /7/05 Chapter. 9:0 AM Page 8 Polnomial and Rational Functions Quadratic Functions and Models What ou should learn Analze graphs of quadratic functions. Write quadratic functions in standard form
More informationRecall from Geometry that a circle can be determined by fixing a point (called the center) and a positive number (called the radius) as follows.
98 Hooked on Conics 7. Circles Recall from Geometr that a circle can be determined b fiing a point called the center) and a positive number called the radius) as follows. Definition 7.. A circle with center
More informationA2.4.B Write the equation of a parabola using given attributes, including focus, directrix. Explore Deriving the StandardForm Equation
. d d Locker LESSON 5.1 Parabolas Teas Math Standards The student is epected to: A..B Write the equation of a parabola using given attributes, including verte, focus, directri, ais of smmetr, and direction
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 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 informationWork with a partner. The following steps show a method of solving ax 2 + bx + c = 0. Explain what was done in each step.
9.5 Solving Quadratic Equations Using the Essential Question How can ou derive a formula that can be used to write the solutions of an quadratic equation in standard form? Deriving the Work with a partner.
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.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D. Review of Trigonometric Functions D7 APPENDIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving
More informationAnswers to Algebra 2 Unit 2 Practice
LESSON 71 1. a. A(l ) 0l l. C Answers to Algebra Unit Practice b. The Area of a Rectangle with Perimeter 0 Area (cm ) 00 00 00 00 0 A(l) 0 0 0 0 Length (cm) c. Yes; the length of a rectangle that has
More informationChapter 4: Linear Systems of Equations
HOSP 1107 (Business Math) Learning Centre Chapter 4: Linear Sstems of Equations An pair of linear equations (with two variables) can be solved b using algebra or graphing. To solve sstems of equations
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 informationAngles and Degree Measure. Figure D.25 Figure D.26
APPENDIX D. Review of Trigonometric Functions D7 APPE N DIX D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions
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 informationSection 14.6 Tangent planes and differentials
Section 14.6 Tangent planes and differentials (3/23/08) Overview: In this section we stud linear functions of two variables and equations of tangent planes to the graphs of functions of two variables.
More informationEssential Question How can you derive a general formula for solving a quadratic equation? ax 2 + bx = c. c a. b a x = x 2 + c a + ( b 2 = b a x + (
COMMON CORE Learning Standards HSNCN.C.7 HSAREI.B.4b 3.4 Using the Quadratic Formula Essential Question How can ou derive a general formula for solving a quadratic equation? Deriving the Quadratic Formula
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 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 information9.4 PLANNING. and r 2
LESSON 9.4 Mathematicians assume the right to choose, within the limits of logical contradiction, what path the please in reaching their results. HENRY ADAMS You will need graph paper Factored Form So
More information10.2 The Unit Circle: Cosine and Sine
Foundations of Trigonometr 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 a circular path
More informationF818 Finding the yintercept from Ordered Pairs
F88 Finding the intercept from Ordered Pairs Pages 5 Standards: 8.F.A., 8.F.B. Goals: Students will find the intercept of a line from a set of ordered pairs. Prior Knowledge Required: Can add, subtract,
More informationQUADRATIC RELATIONS AND FUNCTIONS
CHAPTER 3 CHAPTER TABLE OF CONTENTS 3 Solving Quadratic Equations 32 The Graph of a Quadratic Function 33 Finding Roots from a Graph 34 Graphic Solution of a QuadraticLinear Sstem 35 Algebraic Solution
More information8 FURTHER CALCULUS. 8.0 Introduction. 8.1 Implicit functions. Objectives. Activity 1
8 FURTHER CALCULUS Chapter 8 Further Calculus Objectives After studing this chapter ou should be able to differentiate epressions defined implicitl; be able to use approimate methods for integration such
More informationChapter 5 Graphing Linear Equations and Inequalities
.1 The Rectangular Coordinate Sstem (Page 1 of 28) Chapter Graphing Linear Equations and Inequalities.1 The Rectangular Coordinate Sstem The rectangular coordinate sstem (figure 1) has four quadrants created
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 informationGraphs and Functions in the Cartesian Coordinate System
C H A P T E R Graphs and Functions in the Cartesian Coordinate Sstem List price (in thousands of dollars) 0 0 0 (99, 0,) (98,,67) (00, 7,) he first selfpropelled automobile to carr passengers was built
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 information1.1 Graphing Equations
Chapter 1: Functions and Their Graphs (Page 1 of 46) 1.1 Graphing Equations Solution, Graph, Dependent & Independent Variables For an equation in two variables and y, a solution, or a solution point, is
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 informationCh 5 Projectile Motion. Projectile motion has both horizontal and. motion.
Projectile motion has both horizontal and vertical components of motion. In Chapter 4 we studied simple straightline motion linear motion. Now we'll look at nonlinear motion motion along a curved path.
More informationPrecalculus Chapter 9 Summary Sec 9.3: Conic Sections Parabola
Precalculus Chapter 9 Summary Sec 9.3: Conic Sections Parabola Definition: Focal length: y axis P(x, y) Focal chord: focus Vertex xaxis directrix Focal width/ Latus Rectum: Derivation of equation of
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 information3.3. section. 140 (320) Chapter 3 Graphs and Functions in the Cartesian Coordinate System FIGURE FOR EXERCISE 52 MISCELLANEOUS
0 (0) Chapter Graphs and Functions in the Cartesian Coordinate Sstem Selling price (in thousands of dollars) 0 a) Use the graph on the net page to estimate the average retail price of a earold car in
More informationPolar and Parametric Equations
Polar and Parametric Equations CK12 Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content,
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 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 information4.1. Interpreting Graphs CONDENSED LESSON
CONDENSED LESSON.1 Interpreting Graphs In this lesson ou will interpret graphs that show information about realworld situations make a graph that reflects the information in a stor invent a stor that
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 informationSECTION 26 Inverse Functions
182 2 Graphs and Functions SECTION 26 Inverse Functions OnetoOne Functions Inverse Functions Man important mathematical relationships can be epressed in terms of functions. For eample, C d f(d) V s
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 information4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant
SECTION 4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant 6 What ou ll learn about The Tangent Function The Cotangent Function The Secant Function The Cosecant Function... and wh This will give us
More informationPreCalculus 40 Final Outline/Review:
20152016 PreCalculus 40 Final Outline/Review: NonCalculator Section: 13 multiple choice and 8 open ended. Calculator Section: 7 multiple choice and 13 open ended. First Semester Topics: o Logarithmics
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 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 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 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 informationEssential Question How can you use a quadratic function to model a reallife situation?
. Modeling with Quadratic Functions COMMON CORE Learning Standards HSACED.A. HSFIF.B.6 HSFBF.A.1a HSSID.B.6a Essential Question How can ou use a quadratic function to model a reallife situation? Work
More information2.2 Solving Equations Graphically
71_00.qp 176 1/7/06 11:0 AM Chapter Page 176 Solving Equations and Inequalities. Solving Equations Graphicall What ou should learn Intercepts, Zeros, and Solutions In Section 1.1, ou learned that the intercepts
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 informationApplications of Trigonometric and Circular Functions
Applications of Trigonometric and Circular Functions CHAPTER 3 Stresses in the earth compress rock formations and cause them to buckle into sinusoidal shapes. It is important for geologists to be able
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 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 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 informationPre Calculus Graphing Trig. Functions Day 1
Pre Calculus Graphing Trig. Functions Da Draw a graph of the given trigonometric function with the listed amplitude and period.. cos x. sin x Amp: ½ Period: 4 Amp: 5 Period: 0π Draw a graph of the given
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 informationMessiah College Calculus 1 Placement Exam Topics and Review: Key
Messiah College Calculus Placement Eam Topics and Review: Key Answers to problems in the tet are listed in the back of the course tetbook: Calculus, 9th edition by Larson, Hostetler, and Edwards. Solutions
More information22 Distance in the Plane
180 CHAPTER GRAPHS  Distance in the Plane Z Distance Between Two Points Z Midpoint of a Line Segment Z Circles Two basic problems studied in analtic geometr are 1. Given an equation, find its graph..
More informationCourse 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook
Course Answer Ke. Rational & Irrational Numbers Defining Real Numbers. integers; 0. terminates; repeats 3. two; number 4. ratio; integers 5. terminating; repeating 6. rational; irrational 7. real 8. root
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 informationInvestigate Slopes of Parallel and Perpendicular Lines
7. Parallel and Perpendicular Lines Focus on identifing whether two lines are parallel, perpendicular, or neither writing the equation of a line using the coordinates of a point on the line and the equation
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 information3.1 Quadratic Functions and Models
50 CHAPTER 3 Polnomial and Rational Functions 3. Quadratic Functions and Models PREPARING FOR THIS SECTION Intercepts (Section.2, pp. 5 7) Quadratic Equations (Appendi, Section A.5, pp. 988 995) Now work
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 informationInequalities and Linear Programming
4CH_PHCalter_TMSETE_949118 3//007 1:38 PM Page 1 Inequalities and Linear Programming OBJECTIVES When ou have completed this chapter, ou should be able to: Graph linear inequalities on the number line.
More informationLogarithmic Functions and Their Graphs. Logarithmic Functions. The function given by. is called the logarithmic function with base a.
0_00.qd /7/05 0:8 AM Page 9 Section. Logarithmic Functions and Their Graphs 9. Logarithmic Functions and Their Graphs What ou should learn Recognize and evaluate logarithmic functions with base a. Graph
More informationINTRODUCTION TO FUNCTIONS
4.6 Introduction to Functions (4 47) 0 with a height of 60 cm, B is a linear function of the person s weight w (in kilograms). For a weight of 45 kg, B is 00 calories. For a weight of 50 kg, B is 65 calories.
More informationSection 3.2 Drawing graphs using firstderivative tests
Section Drawing graphs using firstderivative tests Overview: In this section we sketch graphs of functions constructed from powers, linear combinations, products, and quotients b studing formulas for
More informationSystems of Linear Equations in Two Variables
5.1 Sstems of Linear Equations in Two Variables 5.1 OBJECTIVES 1. Find ordered pairs associated with two equations 2. Solve a sstem b graphing 3. Solve a sstem b the addition method 4. Solve a sstem b
More informationCharacteristics of Quadratic Functions
. Characteristics of Quadratic Functions Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete
More informationThe Slope of a Line 4.2. On the coordinate system below, plot a point, any point.
.2 The Slope of a Line.2 OBJECTIVES 1. Find the slope of a line 2. Find the slopes of parallel and perpendicular lines 3. Find the slope of a line given an equation. Find the slope given a graph 5. Graph
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 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 informationExample 1: The distance formula is derived directly from the Pythagorean Theorem. Create a right triangle with the segment below, and solve for d.
P. Cartesian Coordinate System PreCalculus P. CARTESIAN COORDINATE SYSTEM Learning Targets for P. 1. Know and be able to use the distance formula. Know and be able to use the midpoint formula. Be able
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 Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More information1. 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 information