8.7 The Parabola. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix.

Size: px
Start display at page:

Download "8.7 The Parabola. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix."

Transcription

1 8.7 The Parabola The Hubble Space Telescope orbits the Earth at an altitude of approimatel 600 km. The telescope takes about ninet minutes to complete one orbit. Since it orbits above the Earth s atmosphere, the telescope can perform its scientific work without the negative effects of the atmosphere. The primar reflector of the Hubble Space Telescope is parabolic. A parabola is the set or locus of points P in the plane such that the distance from P to a fied point F equals the distance from P to a fied line l. Focus F PF = PD The fied point F is called the focus. The fied line l is called the directri. Verte P The verte of a parabola is located midwa between the focus and the directri. The parabola never crosses the directri. We will use this information to help sketch parabolas. Directi D l INVESTIGATE & INQUIRE You will need two clear plastic rulers, a sheet of paper, and a pencil. Step Draw a 5-cm line segment near the bottom of the piece of paper. Label the line l. Mark a point about 4 cm above the middle of the segment. Label this point F. F l 8.7 The Parabola MHR 653

2 Step 2 Choose a length, k cm, that is greater than or equal to half the distance from the point F to the line l. You ma want to make k less than 8 cm. Step 3 To locate a point P that is k cm from line l and k cm from point F, place one ruler so that its 0 mark is on line l and the ruler is perpendicular to line l. Place the other ruler so that its 0 mark is on point F. Adjust the positions of the rulers to locate P that is k cm from l and k cm from F. k k F l Step 4 Mark a second point that is also k cm from line l and k cm from point F. Step 5 Repeat steps 3 and 4 using different values for k until ou have marked enough points to define a complete curve. Step 6 Draw a smooth curve through the points. The curve is an eample of a parabola.. How man aes of smmetr does the parabola have? 2. Steps 3 and 4 instructed ou to mark two points for the chosen distance, k. Are there an values of k for which onl one point can be marked? If so, describe the location of the point on the parabola. 3. What is the relationship between the verte of the parabola, line l, and point F? 4. In Step 2, wh must k be greater than or equal to half the distance from line l to the point F? 654 MHR Chapter 8

3 EXAMPLE Finding the Equation of a Parabola From its Locus Definition Use the locus definition of the parabola to find an equation of the parabola with focus F(0, 3) and directri = 3. SOLUTION Draw a diagram. The verte is located midwa between the focus and directri, so V(0, 0) is the verte. Since the parabola never crosses the directri, the parabola must open up. Let P(, ) be an point on the parabola. 2 The focus is F(0, 3). Let D(, 3) be an point on the directri. The locus definition of the parabola can be stated algebraicall as PF = PD. Use the formula for the length of a line segment, l = ( 2 ) 2 + ( 2, ) 2 to rewrite PF and PD. PF = ( 0) 2 + ( 3) 2 = 2 + ( 3) 2 PD = ( ) 2 + ( ( 3)) 2 = ( + 3) 2 Substitute: 2 + ( 3) 2 = ( + 3) Square both sides: 2 + ( 3) 2 = ( + 3) 2 Epand: = Solve for : 2 = 2 2 = 2 An equation of the parabola is = F(0, 3) V(0, 0) = 3 In Eample, note that, when the focus is (0, 3) and the directri is = 3, the equation of the parabola is = 2, and that 2 = In general, if the focus is F(0, p) and the directri is = p, then the equation of the parabola is = The Parabola MHR 655

4 The standard form of the equation of a parabola with its verte at the origin and a horizontal directri is = 2 The verte is V(0, 0). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(0, p). The equation of the directri is = p. The ais of smmetr is the -ais. The standard form of the equation of a parabola with its verte at the origin and a vertical directri is = 2 The verte is V(0, 0). If p > 0, the parabola opens right. If p < 0, the parabola opens left. The focus is F(p, 0). The equation of the directri is = p. The ais of smmetr is the -ais. = 4p 2 F(0, p) P(, ) ( p, ) P(, ) 0 V(0, 0) V(0, 0) 0 F(p, 0) = p (, p) = p = 4p 2 EXAMPLE 2 Writing an Equation of a Parabola With Verte (0, 0) Write an equation in standard form for the parabola with focus (4, 0) and directri = 4. SOLUTION The verte is located midwa between the focus and the directri, so the verte is V(0, 0). Since the parabola never crosses the directri, the parabola opens right. 656 MHR Chapter 8

5 The standard form of the equation of a parabola opening right with verte at the origin is = 2. The directri is = p, so p = 4. Since p = 4, the equation is = 2. 6 The equation can be modelled graphicall. 4 2 V(0, 0) F(4, 0) = 4 4 = 6 2 EXAMPLE 3 Determining the Characteristics of a Parabola From its Equation For the parabola = 8 2, determine the direction of the opening, the coordinates of the verte and the focus, and the equation of the directri. SOLUTION The equation is in the form =, so the graph opens up or down. Find the value of p. Rewrite = 8 2 as =. 4( 2) So, p = 2. Since p < 0, the graph opens down. 2 V(0, 0) The verte is V(0, 0). The focus is F(0, p), or F(0, 2). The equation of the directri is = p, or = The equation can be modelled graphicall. = F(0, 2) = 2 8 Recall that we can translate a parabola = a 2, with verte at the origin, to = a( h) 2 + k b translating h units to the left or right and k units up or down. This translation results in a parabola with verte (h, k). The equation of the resulting parabola can be epressed in standard form. 8.7 The Parabola MHR 657

6 The standard form of the equation of a parabola with verte V(h, k) and a horizontal directri is k = ( h) 2 The verte is V(h, k). If p > 0, the parabola opens up. If p < 0, the parabola opens down. The focus is F(h, k + p). The equation of the directri is = k p. The equation of the ais of smmetr is = h. The standard form of the equation of a parabola with verte V(h, k) and a vertical directri is h = ( k) 2 The verte is V(h, k). If p > 0, the parabola opens right. If p < 0, the parabola opens left. The focus is F(h + p, k). The equation of the directri is = h p The equation of the ais of smmetr is = k. = h h = ( k) 2 4p F(h, k + p) V(h, k) k = ( h) 2 4p V(h, k) F(h + p, k) = k = k p 0 0 = h p EXAMPLE 4 Writing an Equation of a Parabola With Verte (h, k) Write an equation in standard form for a parabola with focus F( 2, 6) and directri = 4. SOLUTION Since the parabola never crosses the directri, the parabola opens left. The standard form of the equation is h = ( k) 2. The verte is located midwa between the focus and the directri. The coordinates of the verte are V(, 6). Since the verte is V(, 6), h = and k = MHR Chapter 8

7 Use the focus to find the value of p. The focus F(h + p, k) is ( 2, 6). So, ( + p, 6) = ( 2, 6) + p = 2 p = 3 Now substitute known values into the standard form of the equation. h = ( k) 2 = ( 6) 2 4( 3) = ( 6) 2 2 The equation can be modelled graphicall. 2 0 = ( 6) = 4 F( 2, 6) 6 4 V(, 6) A equation of the parabola in standard form is = ( 6) 2. 2 Recall that parabolas can be graphed using a graphing calculator. If the equation of a parabola that opens left or right is given in standard form, first solve the equation for. For eample, solving = 4 2 for results in =± 4. Enter both of the resulting equations in the Y= editor. Y = 4 Y2 = 4. Adjust the window variables if necessar, and use the Zsquare instruction. 8.7 The Parabola MHR 659

8 Ke Concepts A parabola is the set or locus of points P in the plane such that the distance from P to a fied point F equals the distance from P to a fied line l. PF = PD The fied point F is called the focus. The fied line l is called the directri. The verte of a parabola is located midwa between the focus and the directri. The standard form of a parabola with verte at the origin is = 2 (opens up if p > 0, or down if p < 0), or = 2 (opens right if p > 0, or left if p < 0). The standard form of a parabola with verte (h, k) is k = ( h) 2 (opens up if p > 0, or down if p < 0), or h = ( k) 2 (opens right if p > 0, or left if p < 0). Communicate Your Understanding. In our own words, define the following terms. a) verte b) directri c) focus 2. Describe how ou would use the locus definition to find the equation of a parabola with focus F(2, 0) and directri = Describe the relationship between the ais of smmetr and the directri of an parabola. 4. Describe the similarities and differences between the parabolas 3 = 8 ( + 2) 2 and 3 = 8 ( + 2) Describe how ou would determine an equation in standard form for a parabola with focus F(, 3) and directri =. 660 MHR Chapter 8

9 Practise A. Use the locus definition of the parabola to write an equation for each of the following parabolas. a) focus (0, 2), directri = 2 b) focus (, 3), directri = 4 2. Determine the coordinates of the verte for each of the following parabolas. a) focus (6, 3), directri = 2 b) focus (3, 0), directri = 3 c) focus ( 4, 2), directri = d) focus ( 3, 4), directri = 2 3. Write an equation in standard form for the parabola with the given focus and directri. Sketch the parabola. a) focus (0, 6), directri = 6 b) focus (0, 4), directri = 4 d) focus ( 8, 0), directri = 8 e) focus (0, 2), directri = 2 f) focus (, 0), directri = g) focus (0, 3), directri = 3 h) focus ( 5, 0), directri = 5 Appl, Solve, Communicate 4. Write an equation in standard form for the parabola with the given focus and directri. Sketch the parabola. a) focus (6, 2), directri = 0 b) focus (0, 4), directri = 5 c) focus (2, 2), directri = 5 d) focus (, 4), directri = 2 e) focus ( 3, 5), directri = 5. For each of the following parabolas, determine the direction of the opening, the coordinates of the verte and the focus, and the equation of the directri. Sketch the graph, and determine the domain and range. a) = 2 b) = c) = 8 2 d) = 2 6 e) 3 = 4 ( + 2) 2 f) + 2 = ( 5) 2 0 g) = 5 ( + ) 2 h) + 3 = ( 2) i) 2 = ( + 6) Headlights Automobile headlights contain a parabolic reflector. A bulb with two filaments is used to produce low and high beams. The filament at the focus of the parabola produces the high beam. Light from the filament at the focus is reflected from the parabolic reflector to produce parallel light ras, projecting the light a greater distance. Suppose the filament for the high beams is 5 cm from the verte of the reflector. Write an equation in standard form that models the parabola. Assume that the verte is at the origin and that the filament is 5 units to the right of the origin on the -ais. 8.7 The Parabola MHR 66

10 B 8. Parabolic reflector TV technicians use parabolic reflectors to pick up the sounds from the plaing field at sporting events. The reflector focuses the incoming sound waves on a microphone, which is located at the focus of the reflector. Suppose the microphone is located 5 cm from the verte of the reflector. a) Write an equation in standard form for the parabolic reflector. Assume that the verte is at the origin and that the microphone is to the left of the verte on the -ais. b) Find the width of the reflector, to the nearest centimetre, at a horizontal distance of 30 cm from the verte. 9. Application The stream of water from some water fountains follows a path in the form of a parabolic arch. For one fountain, the maimum height of the water is 8 cm, and the horizontal distance of the water flow is 2 cm. (0, 0) a) Find an equation in standard form that models the continuous flow of water. Assume that the water spout is located at the origin. b) At a horizontal distance of 0 cm from the origin, what is the height of the water, to the nearest tenth of a centimetre? 0. Skateboard ramp For a skateboarding competition, the organizers would like to use a parabolic ramp with a depth of 5 m, and a width of 5 m. Assume that the starting point of a skateboarder at the top of the ramp is (0, 0), as shown. a) Find an equation in standard form that models the parabolic ramp. b) Find the depth of the ramp, to the nearest tenth of a metre, at a horizontal distance of 0 m from the origin. (0, 0). Parabolic antenna A parabolic antenna is 320 cm wide at a distance of 50 cm from its verte. Determine the distance of the focus from the verte. 8 cm 2 cm 5 m 5 m 662 MHR Chapter 8

11 2. Motion in space A spacecraft is in a circular orbit 50 km above Earth. When it reaches the velocit needed to escape the Earth s gravit, the spacecraft will follow a parabolic path with the focus at the centre of the Earth, as shown. Suppose the spacecraft reaches its escape velocit above the North Pole. Assume the radius of the Earth is 6400 km. Write an equation in standard form for the parabolic path of the spacecraft. Circular orbit 0 North Pole Parabolic orbit Earth 3. Hubble Space Telescope The primar reflector of the Hubble Space Telescope is parabolic and has a diameter of 4.27 m and a depth of 0.75 m. a) If a camera is recording pictures at the focus of the reflector, how far is the camera from the verte, to the nearest hundredth of a metre? b) Write an equation in standard form that models the parabolic reflector. Sketch the location of the reflector on the coordinate aes for this equation. 4. Technolog Use a graphing calculator to graph each parabola. a) + 3 = 4 ( 2) 2 b) + = ( 5) 2 2 c) 3 = 8 ( + 2) 2 5. Communication a) Use a graphing calculator to graph the famil of parabolas of the form = 2 for p =, 2, 3. b) Now graph for p = 2, 3, 4. c) How are the graphs alike? How are the different? d) What happens to the parabola as p gets closer to 0? e) What happens to the foci as p gets closer to 0? C 6. Inquir/Problem Solving Use the locus definition of the parabola to derive the equation in standard form for a parabola with verte (0, 0), focus (0, p) and directri = p. 8.7 The Parabola MHR 663

12 7. Standard form Use the locus definition of the parabola to derive the equation in standard form for a parabola with verte (0, 0), focus (p, 0) and directri = p. A CHIEVEMENT Check Knowledge/Understanding Thinking/Inquir/Problem Solving Communication Application A parabolic bridge is 40 m across and 25 m high. What is the length of a stabilizing beam across the bridge at a height of 6 m? CAREER CONNECTION Communications The need for people to communicate with each other has been an important aspect of human histor. Modern communication between people can take man forms, including travelling to see each other, mailing a letter, or making a phone call. For a countr as large as Canada to compete economicall, a highl developed communications industr is essential. Sending information b electronic means, including radio, television, and the Internet, is an increasingl important aspect of the communications industr.. TV satellite dish A satellite dish picks up TV signals from a satellite. The signals travel in parallel paths. When the signals reach the dish, the are reflected to the focus, where the detector is located. Suppose that the focus is located 20 cm from the verte. a) Find an equation in standard form that models the shape of the satellite dish. Sketch the location of the dish on the coordinate aes for this equation. b) Find the width of the dish 20 cm from the verte. 2. Research Use our research skills to investigate each of the following. a) a career that interests ou in the communications industr, including the education and training required and the tpe of work involved b) the work of Marshall McLuhan (9 980), a Canadian who was world famous for his work on communications and the media 664 MHR Chapter 8

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

7.3 Parabolas. 7.3 Parabolas 505

7.3 Parabolas. 7.3 Parabolas 505 7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of

More 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

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

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

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

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

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

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

Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.

Warm-Up y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. CST/CAHSEE: Warm-Up Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and

More information

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)

In this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1) Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and

More information

REVIEW OF CONIC SECTIONS

REVIEW OF CONIC SECTIONS REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result

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

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.

Colegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t. REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()

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

LINEAR FUNCTIONS OF 2 VARIABLES

LINEAR FUNCTIONS OF 2 VARIABLES CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for

More information

Graphing Quadratic Equations

Graphing Quadratic Equations .4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph first-degree equations. Similar methods will allow ou to graph quadratic equations

More information

A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION

A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION A CLASSROOM NOTE ON PARABOLAS USING THE MIRAGE ILLUSION Abstract. The present work is intended as a classroom note on the topic of parabolas. We present several real world applications of parabolas, outline

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

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X

Ax 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus

More information

Answers (Anticipation Guide and Lesson 10-1)

Answers (Anticipation Guide and Lesson 10-1) Answers (Anticipation Guide and Lesson 0-) Lesson 0- Copright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0- NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get

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

Quadratic Equations and Functions

Quadratic Equations and Functions Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In

More information

8.9 Intersection of Lines and Conics

8.9 Intersection of Lines and Conics 8.9 Intersection of Lines and Conics The centre circle of a hockey rink has a radius of 4.5 m. A diameter of the centre circle lies on the centre red line. centre (red) line centre circle INVESTIGATE &

More information

9.5 CALCULUS AND POLAR COORDINATES

9.5 CALCULUS AND POLAR COORDINATES smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet

More information

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin.

135 Final Review. Determine whether the graph is symmetric with respect to the x-axis, the y-axis, and/or the origin. 13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, -6); P2 = (7, -2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the -ais, the -ais, and/or the

More information

5.3 Graphing Cubic Functions

5.3 Graphing Cubic Functions Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a ( - h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b ( - h) 3 ) + k Resource Locker Eplore 1

More 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

Slope-Intercept Form and Point-Slope Form

Slope-Intercept Form and Point-Slope Form Slope-Intercept Form and Point-Slope Form In this section we will be discussing Slope-Intercept Form and the Point-Slope Form of a line. We will also discuss how to graph using the Slope-Intercept Form.

More information

What is a parabola? It is geometrically defined by a set of points or locus of points that are

What is a parabola? It is geometrically defined by a set of points or locus of points that are Section 6-1 A Parable about Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directrix).

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

2.3 Quadratic Functions

2.3 Quadratic Functions 88 Linear and Quadratic Functions. Quadratic Functions You ma recall studing quadratic equations in Intermediate Algebra. In this section, we review those equations in the contet of our net famil of functions:

More information

Teacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image.

Teacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image. Teacher Page Geometr / Da # 10 oordinate Geometr (5 min.) 9-.G.3.1 9-.G.3.2 9-.G.3.3 9-.G.3. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric

More information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

Higher. Polynomials and Quadratics 64

Higher. Polynomials and Quadratics 64 hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining

More information

Polynomial Degree and Finite Differences

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

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 24, 2012 9:15 a.m. to 12:15 p.m. INTEGRATED ALGEBRA The Universit of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesda, Januar 4, 01 9:15 a.m. to 1:15 p.m., onl Student Name: School Name: Print our name and

More information

Linear Equations in Two Variables

Linear Equations in Two Variables Section. Sets of Numbers and Interval Notation 0 Linear Equations in Two Variables. The Rectangular Coordinate Sstem and Midpoint Formula. Linear Equations in Two Variables. Slope of a Line. Equations

More information

SAMPLE. Polynomial functions

SAMPLE. Polynomial functions Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through

More information

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:

6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions: Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(-, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions - 4, use the graph at the right.. Eplain wh the graph

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

Precalculus. What s My Locus? ID: 8255

Precalculus. What s My Locus? ID: 8255 What s My Locus? ID: 855 By Lewis Lum Time required 45 minutes Activity Overview In this activity, students will eplore the focus/directri and reflection properties of parabolas. They are led to conjecture

More information

I think that starting

I think that starting . Graphs of Functions 69. GRAPHS OF FUNCTIONS One can envisage that mathematical theor will go on being elaborated and etended indefinitel. How strange that the results of just the first few centuries

More information

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x

Downloaded from www.heinemann.co.uk/ib. equations. 2.4 The reciprocal function x 1 x Functions and equations Assessment statements. Concept of function f : f (); domain, range, image (value). Composite functions (f g); identit function. Inverse function f.. The graph of a function; its

More information

Section 2-3 Quadratic Functions

Section 2-3 Quadratic Functions 118 2 LINEAR AND QUADRATIC FUNCTIONS 71. Celsius/Fahrenheit. A formula for converting Celsius degrees to Fahrenheit degrees is given by the linear function 9 F 32 C Determine to the nearest degree the

More information

Unit 9: Conic Sections Name Per. Test Part 1

Unit 9: Conic Sections Name Per. Test Part 1 Unit 9: Conic Sections Name Per 1/6 HOLIDAY 1/7 General Vocab Intro to Conics Circles 1/8-9 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics

More information

To Be or Not To Be a Linear Equation: That Is the Question

To Be or Not To Be a Linear Equation: That Is the Question To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not

More information

When I was 3.1 POLYNOMIAL FUNCTIONS

When I was 3.1 POLYNOMIAL FUNCTIONS 146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we

More information

STRAND: ALGEBRA Unit 3 Solving Equations

STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

More information

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude

ACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height

More information

4Unit 2 Quadratic, Polynomial, and Radical Functions

4Unit 2 Quadratic, Polynomial, and Radical Functions CHAPTER 4Unit 2 Quadratic, Polnomial, and Radical Functions Comple Numbers, p. 28 f(z) 5 z 2 c Quadratic Functions and Factoring Prerequisite Skills... 234 4. Graph Quadratic Functions in Standard Form...

More information

Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012

Rotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012 Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to

More information

Lesson 9.1 Solving Quadratic Equations

Lesson 9.1 Solving Quadratic Equations Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte

More information

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

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

Use the following information to deduce that the gravitational field strength at the surface of the Earth is approximately 10 N kg 1.

Use the following information to deduce that the gravitational field strength at the surface of the Earth is approximately 10 N kg 1. IB PHYSICS: Gravitational Forces Review 1. This question is about gravitation and ocean tides. (b) State Newton s law of universal gravitation. Use the following information to deduce that the gravitational

More information

Polynomial and Rational Functions

Polynomial and Rational Functions Polnomial and Rational Functions 3 A LOOK BACK In Chapter, we began our discussion of functions. We defined domain and range and independent and dependent variables; we found the value of a function and

More information

Physics 30 Worksheet #10 : Magnetism From Electricity

Physics 30 Worksheet #10 : Magnetism From Electricity Physics 30 Worksheet #10 : Magnetism From Electricity 1. Draw the magnetic field surrounding the wire showing electron current below. x 2. Draw the magnetic field surrounding the wire showing electron

More information

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship

Direct Variation. 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship 6.5 Direct Variation 6.5 OBJECTIVES 1. Write an equation for a direct variation relationship 2. Graph the equation of a direct variation relationship Pedro makes $25 an hour as an electrician. If he works

More information

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM

EQUATIONS OF LINES IN SLOPE- INTERCEPT AND STANDARD FORM . Equations of Lines in Slope-Intercept and Standard Form ( ) 8 In this Slope-Intercept Form Standard Form section Using Slope-Intercept Form for Graphing Writing the Equation for a Line Applications (0,

More information

Applications of the Pythagorean Theorem

Applications of the Pythagorean Theorem 9.5 Applications of the Pythagorean Theorem 9.5 OBJECTIVE 1. Apply the Pythagorean theorem in solving problems Perhaps the most famous theorem in all of mathematics is the Pythagorean theorem. The theorem

More information

Shake, Rattle and Roll

Shake, Rattle and Roll 00 College Board. All rights reserved. 00 College Board. All rights reserved. SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Tet, Visualization, Interactive Word Wall Roller coasters are scar

More information

Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 314 318) Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

More information

Solving Absolute Value Equations and Inequalities Graphically

Solving Absolute Value Equations and Inequalities Graphically 4.5 Solving Absolute Value Equations and Inequalities Graphicall 4.5 OBJECTIVES 1. Draw the graph of an absolute value function 2. Solve an absolute value equation graphicall 3. Solve an absolute value

More information

Core Maths C2. Revision Notes

Core Maths C2. Revision Notes Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...

More information

For each Circle C, find the value of x. Assume that segments that appear to be tangent are tangent. 1. x = 2. x =

For each Circle C, find the value of x. Assume that segments that appear to be tangent are tangent. 1. x = 2. x = Name: ate: Period: Homework - Tangents For each ircle, find the value of. ssume that segments that appear to be tangent are tangent. 1. =. = ( 5) 1 30 0 0 3. =. = (Leave as simplified radical!) 3 8 In

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

SECTION 2-2 Straight Lines

SECTION 2-2 Straight Lines - Straight Lines 11 94. Engineering. The cross section of a rivet has a top that is an arc of a circle (see the figure). If the ends of the arc are 1 millimeters apart and the top is 4 millimeters above

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle - in particular

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Tuesday, August 16, 2005 8:30 to 11:30 a.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Tuesday, August 16, 2005 8:30 to 11:30 a.m. MATHEMATICS B The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Tuesday, August 16, 2005 8:30 to 11:30 a.m., only Print Your Name: Print Your School's Name: Print your

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, January 24, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

Introduction to Quadratic Functions

Introduction to Quadratic Functions Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

More information

2.5 Library of Functions; Piecewise-defined Functions

2.5 Library of Functions; Piecewise-defined Functions SECTION.5 Librar of Functions; Piecewise-defined Functions 07.5 Librar of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: Intercepts (Section.,

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name: GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, June 17, 2010 1:15 to 4:15 p.m., only Student Name: School Name: Print your name and the name of your

More information

Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6. Instructions: 1. In the formula F = qvxb:

Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6. Instructions: 1. In the formula F = qvxb: Physics 121 Sample Common Exam 3 NOTE: ANSWERS ARE ON PAGE 6 Signature Name (Print): 4 Digit ID: Section: Instructions: Answer all questions 24 multiple choice questions. You may need to do some calculation.

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

More information

Area of Parallelograms (pages 546 549)

Area of Parallelograms (pages 546 549) A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

More information

Complex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No real-number solutions. From the definition, it follows that i 2 1.

Complex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No real-number solutions. From the definition, it follows that i 2 1. 7_Ch09_online 7// 0:7 AM Page 9-9. Comple Numbers 9- SECTION 9. OBJECTIVES Epress square roots of negative numbers in terms of i. Write comple numbers in a bi form. Add and subtract comple numbers. Multipl

More information

MATH 185 CHAPTER 2 REVIEW

MATH 185 CHAPTER 2 REVIEW NAME MATH 18 CHAPTER REVIEW Use the slope and -intercept to graph the linear function. 1. F() = 4 - - Objective: (.1) Graph a Linear Function Determine whether the given function is linear or nonlinear..

More information

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

FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. FINAL EXAM REVIEW MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine whether or not the relationship shown in the table is a function. 1) -

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

Algebra II. Administered May 2013 RELEASED

Algebra II. Administered May 2013 RELEASED STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited

More information

State Newton's second law of motion for a particle, defining carefully each term used.

State Newton's second law of motion for a particle, defining carefully each term used. 5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

More information

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

More information

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145: MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

More information

MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS

MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS MECHANICS OF SOLIDS - BEAMS TUTORIAL TUTORIAL 4 - COMPLEMENTARY SHEAR STRESS This the fourth and final tutorial on bending of beams. You should judge our progress b completing the self assessment exercises.

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

More information

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes

2.6. The Circle. Introduction. Prerequisites. Learning Outcomes The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures. In this brief Section we discuss the basic coordinate geometr of a circle - in particular the basic equation representing

More information

2.3 TRANSFORMATIONS OF GRAPHS

2.3 TRANSFORMATIONS OF GRAPHS 78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and time-and-ahalf for the number of hours he works above 0. Let denote the number of hours he

More information

11.1 Parabolas Name: 1

11.1 Parabolas Name: 1 Algebra 2 Write your questions and thoughts here! 11.1 Parabolas Name: 1 Distance Formula The distance between two points, and, is Midpoint Formula The midpoint between two points, and, is,, RECALL: Standard

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Volume of a Cylinder

Volume of a Cylinder Volume of a Cylinder Focus on After this lesson, you will be able to φ determine the volume of a cylinder How much water do you use? You might be surprised. The water storage tank shown has a height of

More information

3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7?

3 e) x f) 2. Precalculus Worksheet P.1. 1. Complete the following questions from your textbook: p11: #5 10. 2. Why would you never write 5 < x > 7? Precalculus Worksheet P.1 1. Complete the following questions from your tetbook: p11: #5 10. Why would you never write 5 < > 7? 3. Why would you never write 3 > > 8? 4. Describe the graphs below using

More information

All I Ever Wanted to Know About Circles

All I Ever Wanted to Know About Circles Parts of the Circle: All I Ever Wanted to Know About Circles 1. 2. 3. Important Circle Vocabulary: CIRCLE- the set off all points that are the distance from a given point called the CENTER- the given from

More information

Satellites and Space Stations

Satellites and Space Stations Satellites and Space Stations A satellite is an object or a body that revolves around another object, which is usually much larger in mass. Natural satellites include the planets, which revolve around

More information

Not for distribution

Not for distribution SHPE, SPE ND MESURES Volume Volume of a cuboid Volume is the amount of space inside a -D shape. he common units for volume are: mm, cm or m. Volume = length x width x height height V = l x w x h V = lwh

More information

Wednesday 15 January 2014 Morning Time: 2 hours

Wednesday 15 January 2014 Morning Time: 2 hours Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Wednesday 15 January 2014 Morning Time: 2 hours Candidate Number

More information

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

More information

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal

Physics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3-D We have defined the velocit and acceleration of a particle as the first and second

More information

Fluid Pressure and Fluid Force

Fluid Pressure and Fluid Force 0_0707.q //0 : PM Page 07 SECTION 7.7 Section 7.7 Flui Pressure an Flui Force 07 Flui Pressure an Flui Force Fin flui pressure an flui force. Flui Pressure an Flui Force Swimmers know that the eeper an

More information