6.3 Parametric Equations and Motion

Size: px
Start display at page:

Download "6.3 Parametric Equations and Motion"

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 420-ft 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 t-values 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 t-values. 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 2-2, = 3t. -3 t 1-2 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 = 1-2t, = 2 - t, - q 6 t 6 q. SOLUTION We solve the first equation for t: = 1-2t 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 2-2, = 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 2-2 = 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 = 1-2, 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 tail-to-head 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 = 8-2, 39 + t , , 9 = 8-2, 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 = 1-2, 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 3-20t 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 3-20t t - 85), 1 = 5, 0 t 12, C 2 : 2 = -0.11t 3-20t 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 3-20t t - 852, 1 = 5, 0 t 3-12, in the 124 b 3-10, 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 projectile-motion 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 3-20t t - 85), and the graph C 2 : 2 = -0.11t 3-20t 2 1 = 5, 0 t t - 852, 2 = -t, 0 t 12 in the 3-12, 124 b 3-15, 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 = 1-component of initial velocit2 * time. The -component is the familiar vertical projectile-motion 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 20-ft 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 20-ft 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 = 1-3, -2), B = 14, A = 1-1, 3), B = 14, -32 In Eercises 3 and 4, write an equation in point-slope 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. = 2-3t, = 5 + t 13. = 2t - 3, = 9-4t, 3 t = 5-3t, = 2 + t, 15. = t 2, = t + 1 [Hint: Eliminate t and solve for in terms of.] 16. = t, = t = t, = t 3-2t = 2t 2-1, = 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 3-3, -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 1-2, 52 and 14, The line through the points -3, -321 and 15, The line segment with endpoints 13, 42 and 16, -32

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

SECTION 11-5 Parametric Equations. Parametric Equations and Plane Curves. Parametric Equations and Plane Curves Projectile Motion Cycloid

SECTION 11-5 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 information

1.5 Parametric Relations and Inverses

1.5 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 information

Annual rate GRAPHS OF FUNCTIONS. Linear and Constant Functions. Linear Function

Annual 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 00-page book and $.0 for a

More information

3.1 Quadratic Functions

3.1 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 information

Applications of Trigonometry

Applications of Trigonometry 5144_Demana_Ch06pp501-566 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 information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE 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 information

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: 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 information

Chapter 10 Conics, Parametric Equations, and Polar Coordinates

Chapter 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 information

10.2. Introduction to Conics: Parabolas. Conics. What you should learn. Why you should learn it

10.2. 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 double-napped

More information

Years 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

Years 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 Straight-line depreciation. The Circle Definition Anone who has drawn a circle using a compass

More information

DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

DISTANCE, 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 information

SECTION 2.2. Distance and Midpoint Formulas; Circles

SECTION 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 information

Solving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form

Solving 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 information

THE PARABOLA section. Developing the Equation

THE 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 information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex 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 information

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

4.7 Inverse Trigonometric. Functions

4.7 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 information

Precalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers

Precalculus 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 information

Chapter 6 Quadratic Functions

Chapter 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 information

1.2. Graphs of Equations. The Graph of an Equation. What you should learn. Why you should learn it

1.2. 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 information

Section 10.7 Parametric Equations

Section 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 y-coordinates on a circle of radius r as a function of

More information

Standard Equation of a Circle

Standard 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 information

Quadratic Functions and Models. The Graph of a Quadratic Function. These functions are examples of polynomial functions. Why you should learn it

Quadratic 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 information

Recall from Geometry that a circle can be determined by fixing a point (called the center) and a positive number (called the radius) as follows.

Recall 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 information

A2.4.B Write the equation of a parabola using given attributes, including focus, directrix. Explore Deriving the Standard-Form Equation

A2.4.B Write the equation of a parabola using given attributes, including focus, directrix. Explore Deriving the Standard-Form 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 information

Imagine a car is traveling along the highway and you look down at the situation from high above: highway

Imagine 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 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

Work with a partner. The following steps show a method of solving ax 2 + bx + c = 0. Explain what was done in each step.

Work 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 information

INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1

INVESTIGATIONS 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 information

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

D.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 information

Answers to Algebra 2 Unit 2 Practice

Answers to Algebra 2 Unit 2 Practice LESSON 7-1 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 information

Chapter 4: Linear Systems of Equations

Chapter 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 information

The Graph of a Linear Equation

The 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 information

Angles and Degree Measure. Figure D.25 Figure D.26

Angles 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 information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b

More information

Section 14.6 Tangent planes and differentials

Section 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 information

Essential 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 + (

Essential 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 HSN-CN.C.7 HSA-REI.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 information

1.6. Piecewise Functions. LEARN ABOUT the Math. Representing the problem using a graphical model

1.6. 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 information

Algebra Module A47. The Parabola. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Algebra 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 information

9.4 PLANNING. and r 2

9.4 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 information

10.2 The Unit Circle: Cosine and Sine

10.2 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 information

F8-18 Finding the y-intercept from Ordered Pairs

F8-18 Finding the y-intercept from Ordered Pairs F8-8 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 information

QUADRATIC RELATIONS AND FUNCTIONS

QUADRATIC RELATIONS AND FUNCTIONS CHAPTER 3 CHAPTER TABLE OF CONTENTS 3- Solving Quadratic Equations 3-2 The Graph of a Quadratic Function 3-3 Finding Roots from a Graph 3-4 Graphic Solution of a Quadratic-Linear Sstem 3-5 Algebraic Solution

More information

8 FURTHER CALCULUS. 8.0 Introduction. 8.1 Implicit functions. Objectives. Activity 1

8 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 information

Chapter 5 Graphing Linear Equations and Inequalities

Chapter 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 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

Graphs and Functions in the Cartesian Coordinate System

Graphs 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 self-propelled automobile to carr passengers was built

More information

Click here for answers.

Click 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 information

1.1 Graphing Equations

1.1 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 information

2.4 Inequalities with Absolute Value and Quadratic Functions

2.4 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 information

Ch 5 Projectile Motion. Projectile motion has both horizontal and. motion.

Ch 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 straight-line motion linear motion. Now we'll look at nonlinear motion motion along a curved path.

More information

Precalculus Chapter 9 Summary Sec 9.3: Conic Sections- Parabola

Precalculus 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 x-axis directrix Focal width/ Latus Rectum: Derivation of equation of

More information

C3: Functions. Learning objectives

C3: Functions. Learning objectives CHAPTER C3: Functions Learning objectives After studing this chapter ou should: be familiar with the terms one-one and man-one mappings understand the terms domain and range for a mapping understand the

More information

3.3. section. 140 (3-20) Chapter 3 Graphs and Functions in the Cartesian Coordinate System FIGURE FOR EXERCISE 52 MISCELLANEOUS

3.3. section. 140 (3-20) 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 -ear-old car in

More information

Polar and Parametric Equations

Polar and Parametric Equations Polar and Parametric Equations CK-12 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 information

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

More information

10.2 The Unit Circle: Cosine and Sine

10.2 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 information

4.1. Interpreting Graphs CONDENSED LESSON

4.1. Interpreting Graphs CONDENSED LESSON CONDENSED LESSON.1 Interpreting Graphs In this lesson ou will interpret graphs that show information about real-world situations make a graph that reflects the information in a stor invent a stor that

More information

THE PARABOLA 13.2. section

THE 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 information

SECTION 2-6 Inverse Functions

SECTION 2-6 Inverse Functions 182 2 Graphs and Functions SECTION 2-6 Inverse Functions One-to-One Functions Inverse Functions Man important mathematical relationships can be epressed in terms of functions. For eample, C d f(d) V s

More information

Lesson 03: Kinematics

Lesson 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 information

4.5 Graphs of Tangent, Cotangent, Secant, and Cosecant

4.5 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 information

Pre-Calculus 40 Final Outline/Review:

Pre-Calculus 40 Final Outline/Review: 2015-2016 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 13 multiple choice and 8 open ended. Calculator Section: 7 multiple choice and 13 open ended. First Semester Topics: o Logarithmics

More information

Graph each function. Compare to the parent graph. State the domain and range. 1. SOLUTION:

Graph 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 information

Contents. How You May Use This Resource Guide

Contents. 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 information

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review

D.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review D0 APPENDIX D Precalculus Review 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 information

MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60

MATH 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 information

Essential Question How can you use a quadratic function to model a real-life situation?

Essential Question How can you use a quadratic function to model a real-life situation? . Modeling with Quadratic Functions COMMON CORE Learning Standards HSA-CED.A. HSF-IF.B.6 HSF-BF.A.1a HSS-ID.B.6a Essential Question How can ou use a quadratic function to model a real-life situation? Work

More information

2.2 Solving Equations Graphically

2.2 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 information

Math 259 Winter 2009. Recitation Handout 1: Finding Formulas for Parametric Curves

Math 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. -5-1 -1-3 (a) Find equations for (t) and (t) that will describe

More information

Applications of Trigonometric and Circular Functions

Applications 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 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

Parametric Equations

Parametric 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. 27533-8002 (919) 735-5152

More information

3.4 The Point-Slope Form of a Line

3.4 The Point-Slope Form of a Line Section 3.4 The Point-Slope Form of a Line 293 3.4 The Point-Slope Form of a Line In the last section, we developed the slope-intercept form of a line ( = m + b). The slope-intercept form of a line is

More information

Pre Calculus Graphing Trig. Functions Day 1

Pre 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 information

10 Quadratic Equations and

10 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 information

Messiah College Calculus 1 Placement Exam Topics and Review: Key

Messiah 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 information

2-2 Distance in the Plane

2-2 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 information

Course 2 Answer Key. 1.1 Rational & Irrational Numbers. Defining Real Numbers Student Logbook. The Square Root Function Student Logbook

Course 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 information

Graphing Linear Equations

Graphing 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 information

Investigate Slopes of Parallel and Perpendicular Lines

Investigate 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 information

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )

ax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 ) SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as

More information

3.1 Quadratic Functions and Models

3.1 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 information

4 Non-Linear relationships

4 Non-Linear relationships NUMBER AND ALGEBRA Non-Linear relationships A Solving quadratic equations B Plotting quadratic relationships C Parabolas and transformations D Sketching parabolas using transformations E Sketching parabolas

More information

Inequalities and Linear Programming

Inequalities 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 information

Logarithmic Functions and Their Graphs. Logarithmic Functions. The function given by. is called the logarithmic function with base a.

Logarithmic 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 information

INTRODUCTION TO FUNCTIONS

INTRODUCTION 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 information

Section 3.2 Drawing graphs using first-derivative tests

Section 3.2 Drawing graphs using first-derivative tests Section Drawing graphs using first-derivative tests Overview: In this section we sketch graphs of functions constructed from powers, linear combinations, products, and quotients b studing formulas for

More information

Systems of Linear Equations in Two Variables

Systems 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 information

Characteristics of Quadratic Functions

Characteristics 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 information

The Slope of a Line 4.2. On the coordinate system below, plot a point, any point.

The 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 information

Linear Inequality in Two Variables

Linear 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 information

Math 21a Old Exam One Fall 2003 Solutions Spring, 2009

Math 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 information

Example 1: The distance formula is derived directly from the Pythagorean Theorem. Create a right triangle with the segment below, and solve for d.

Example 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 information

Supplementary Lesson: Log-log and Semilog Graph Paper

Supplementary Lesson: Log-log and Semilog Graph Paper Supplementar Lesson: Log-log and Semilog Graph Paper Chapter 7 looks at some elementar functions of algebra, including linear, quadratic, power, eponential, and logarithmic. The following supplementar

More information

The Distance Formula and the Circle

The Distance Formula and the Circle 10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,

More information

1. 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

1. 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