THE PARABOLA section


 Lauren Owen
 3 years ago
 Views:
Transcription
1 698 (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. Real numbers. Find two numbers that have a sum of 8 and a product of 0. 0 ft 5. Imaginar numbers. Find two comple numbers whose sum is 8 and whose product is Imaginar numbers. Find two comple numbers whose sum is 6 and whose product is Making a sign. Rico s Sign Shop has a contract to make a sign in the shape of a square with an isosceles triangle on top of it, as shown in the figure. The contract calls for a total height of 0 ft with an area of 72 ft 2. How long should Rico make the side of the square and what should be the height of the triangle? 54. Designing a bo. Angelina is designing a rectangular bo of 20 cubic inches that is to contain new Eaties breakfast cereal. The bo must be 2 inches thick so that it is eas to hold. It must have 84 square inches of surface area to provide enough space for all of the special offers and coupons. What should be the dimensions of the bo? ft FIGURE FOR EXERCISE 53 GRAPHING CALCULATOR EXERCISES 55. Solve each sstem b graphing each pair of equations on a graphing calculator and using the intersect feature to estimate the point of intersection. Find the coordinates of each intersection to the nearest hundredth. a) e 4 b) 3 ln( 3) 2 c) ft 3.2 THE PARABOLA In this section The Geometric Definition Developing the Equation Parabolas in the Form a( h) 2 k Finding the Verte, Focus, and Directri Ais of Smmetr Changing Forms The parabola is one of four different curves that can be obtained b intersecting a cone and a plane as in Fig These curves, called conic sections, are the parabola, circle, ellipse, and hperbola. We graphed parabolas in Sections 0.3 and.2. In this section we learn some new facts about parabolas. Parabola Circle Ellipse FIGURE 3.3 Hperbola
2 The Geometric Definition 3.2 The Parabola (3 ) 699 In Section 0.3 we called the graph of a 2 b c a parabola. This equation is the standard equation of a parabola. In this section ou will see that the following geometric definition describes the same curve as the equation. Parabola Given a line (the directri) and a point not on the line (the focus), the set of all points in the plane that are equidistant from the point and the line is called a parabola. In Section 0.3 we defined the verte as the highest point on a parabola that opens downward or the lowest point on a parabola that opens upward. We learned that b (2a) gives the coordinate of the verte. We can also describe the verte of a parabola as the midpoint of the line segment that joins the focus and directri, perpendicular to the directri. See Fig The focus of a parabola is important in applications. When parallel ras of light travel into a parabolic reflector, the are reflected toward the focus as in Fig This propert is used in telescopes to see the light from distant stars. If the light source is at the focus, as in a searchlight, the light is reflected off the parabola and projected outward in a narrow beam. This reflecting propert is also used in camera lenses, satellite dishes, and eavesdropping devices. Parabola Focus Verte Focus Directri FIGURE 3.4 FIGURE 3.5 p > 0 (0, p) (, ) (0, 0) = p (, p) FIGURE 3.6 Developing the Equation To develop an equation for a parabola, given the focus and directri, choose the point (0, p), where p 0 as the focus and the line p as the directri, as shown in Fig The verte of this parabola is (0, 0). For an arbitrar point (, ) on the parabola the distance to the directri is the distance from (, ) to(, p). The distance to the focus is the distance between (, ) and (0, p). We use the fact that these distances are equal to write the equation of the parabola: ( ) 0 2 ( p) 2 ( ) 2 ( )) ( p 2 To simplif the equation, first remove the parentheses inside the radicals: 2 2 2p p 2 p 2 2 p p p 2 2 2p p 2 2 4p 2 4 p Square each side. Subtract 2 and p 2 from each side.
3 700 (3 2) Chapter 3 Nonlinear Sstems and the Conic Sections = p (, p) (0, 0) (, ) (0, p) So the parabola with focus (0, p) and directri p for p 0 has equation 4p 2. This equation has the form a 2 b c, where a 4 p, b 0, and c 0. If the focus is (0, p) with p 0 and the directri is p, then the parabola opens downward as shown in Fig Deriving the equation using the distance formula again ields 4 p 2. p < 0 FIGURE 3.7 Parabolas in the Form a( h) 2 k The simplest parabola, 2, has verte (0, 0).The transformation a( h) 2 k is also a parabola and its verte is (h, k). The focus and directri of the transformation are found as follows: Parabolas in the Form a( h) 2 k The graph of the equation a( h) 2 k (a 0) is a parabola with verte (h, k), focus (h, k p), and directri k p, where a 4 p. If a 0, the parabola opens upward; if a 0, the parabola opens downward. Figure 3.8 shows the location of the focus and directri for parabolas with verte (h, k) and opening either upward or downward. Note that the location of the focus and directri determine the value of a and the shape and opening of the parabola. a > 0 (h, k + p) (h, k) Directri: = k p a 4p = a( h) 2 + k a < 0 Directri: = k p (h, k) = a( h) 2 + k (h, k + p) FIGURE 3.8 CAUTION For a parabola that opens upward, p 0, and the focus (h, k p) is above the verte (h, k). For a parabola that opens downward, p 0, and the focus (h, k p) is below the verte (h, k). In either case the distance from the verte to the focus and the verte to the directri is p. Finding the Verte, Focus, and Directri In Eample we find the verte, focus, and directri from an equation of a parabola. In Eample 2 we find the equation given the focus and directri. E X A M P L E Finding the verte, focus, and directri, given an equation Find the verte, focus, and directri for the parabola 2.
4 ( 3.2 The Parabola (3 3) 70 ( 0, 4 = 2 ( = 4 FIGURE 3.9 Solution Compare 2 to the general formula a( h) 2 k. We see that h 0, k 0, and a. So the verte is (0, 0). Because a, we can use a 4 p to get 4 p, or p 4. Use (h, k p) to get the focus 0, 4. Use the equation k p to get as the equation of the directri. See Fig E X A M P L E 2 Finding an equation, given a focus and directri Find the equation of the parabola with focus (, 4) and directri 3. Solution Because the verte is halfwa between the focus and directri, the verte is, 7 2. See Fig The distance from the verte to the focus is. Because the 2 focus is above the verte, p is positive. So p 2, and a 4 p. The equation is 2 ( 7, 2 (, 4) 2 ( ( )) = 3 Convert to a 2 b c form as follows: 2 ( )2 7 2 FIGURE (2 2 ) = b 2a or = h (h, k) Ais of Smmetr The graph of 2 shown in Fig. 3.9 is smmetric about the ais because the two halves of the parabola would coincide if the paper were folded on the ais. In general, the vertical line through the verte is the ais of smmetr for the parabola. See Fig. 3.. In the form a 2 b c the coordinate of the verte is b (2a) and the equation of the ais of smmetr is b (2a). In the form a( h) 2 k the verte is (h, k) and the equation for the ais of smmetr is h. Ais of smmetr FIGURE 3. Changing Forms Since there are two forms for the equation of a parabola, it is sometimes useful to change from one form to the other. To change from a( h) 2 k to the form a 2 b c, we square the binomial and combine like terms, as in Eample 2. To change from a 2 b c to the form a( h) 2 k, we complete the square, as in the net eample.
5 702 (3 4) Chapter 3 Nonlinear Sstems and the Conic Sections E X A M P L E 3 calculator closeup The graphs of and 2 2( ) 2 3 appear to be identical. This supports the conclusion that the equations are equivalent. 0 Converting a 2 b c to a( h) 2 k Write in the form a( h) 2 k and identif the verte, focus, directri, and ais of smmetr of the parabola. Solution Use completing the square to rewrite the equation: 2( 2 2) 5 2( 2 2 ) 5 2( 2 2 ) 2 5 2( ) 2 3 The verte is (, 3). Because a 4, we have p 4p 2, Complete the square. Move 2( ) outside the parentheses. 5 and p 8. Because the parabola opens upward, the focus is 8 unit above the verte 5, and the directri is the horizontal line 8 unit below the verte, at, 3 8, or, or 2 3. The ais of smmetr is CAUTION Be careful when ou complete a square within parentheses as in Eample 3. For another eample, consider the equivalent equations 3( 2 4), 3( ), and 3( 2) 2 2. E X A M P L E 4 A calculator graph can be used to check the verte and opening of a parabola. calculator closeup Finding the features of a parabola from standard form Find the verte, focus, directri, and ais of smmetr of the parabola , and determine whether the parabola opens upward or downward. Solution The coordinate of the verte is b a 2( 3) 6 2. To find the coordinate of the verte, let 3 2 in : The verte is 3 2, 7 4. Because a 3, the parabola opens downward. To find the focus, use 3 4 p to get p. The focus is 2 of a unit below the verte at 3 2, or 3 2, 5 3. The directri is the horizontal line of a unit above 2 the verte, or 6. The equation of the ais of smmetr is 3 2.
6 3.2 The Parabola (3) 703 WARMUPS True or false? Eplain our answer.. There is a parabola with focus (2, 3), directri, and verte (0, 0). 2. The focus for the parabola 4 2 is (0, 2). 3. The graph of 3 5( 4) 2 is a parabola with verte (4, 3). 4. The graph of is a parabola. 5. The graph of is a parabola opening upward. 6. For 2 the verte and intercept are the same point. 7. A parabola with verte (2, 3) and focus (2, 4) has no intercepts. 8. The parabola with focus (0, 2) and directri opens upward. 9. The ais of smmetr for a( 2) 2 k is If a and a, then p (4 p) EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the definition of a parabola given in this section? 2. What is the location of the verte? ( 3) What are the two forms of the equation of a parabola? 2. 4 ( 2) What is the distance from the focus to the verte in an parabola of the form a 2 b c? 3. ( ) ( 4) 2 5. How do we convert an equation of the form a 2 b c into the form a( h) 2 k? 6. How do we convert an equation of the form a( h) 2 k into the form a 2 b c? Find the equation of the parabola with the given focus and directri. See Eample Focus (0, 2), directri 2 6. Focus (0, 3), directri 3 Find the verte, focus, and directri for each parabola. See Eample Focus 0, 2, directri 2 8. Focus 0, 8, directri 8
7 704 (3 6) Chapter 3 Nonlinear Sstems and the Conic Sections 9. Focus (3, 2), directri Focus ( 4, 5), directri 4 2. Focus (, 2), directri Focus (2, 3), directri 23. Focus ( 3,.25), directri Focus 5, 7 8, directri 5 8 Write each equation in the form a( h) 2 k. Identif the verte, focus, directri, and ais of smmetr of each parabola. See Eample Solve each problem. 43. World s largest telescope. The largest reflecting telescope in the world is the 6meter (m) reflector on Mount Pastukhov in Russia. The accompaning figure shows a cross section of a parabolic mirror 6 m in diameter with the verte at the origin and the focus at (0, 5). Find the equation of the parabola (0, 5) Find the verte, focus, directri, and ais of smmetr of each parabola (without completing the square), and determine whether the parabola opens upward or downward. See Eample m FIGURE FOR EXERCISE Arecibo Observator. The largest radio telescope in the world uses a 000ft parabolic dish, suspended in a valle in Arecibo, Puerto Rico. The antenna hangs above
8 3.2 The Parabola (3 7) 705 the verte of the dish on cables stretching from two towers. The accompaning figure shows a cross section of the parabolic dish and the towers. Assuming the verte is at (0, 0), find the equation for the parabola. Find the distance from the verte to the antenna located at the focus Antenna at focus 200 ft 200 ft ft FIGURE FOR EXERCISE 44 Graph both equations of each sstem on the same coordinate aes. Use elimination of variables to find all points of intersection Solve each problem. 53. Find all points of intersection of the parabola and the ais. 54. Find all points of intersection of the parabola and the ais. 55. Find all points of intersection of the parabola and the line Find all points of intersection of the parabola and the line. 57. Find all points of intersection of the parabolas 2 and 2.
9 706 (3 8) Chapter 3 Nonlinear Sstems and the Conic Sections 58. Find all points of intersection of the parabolas 2 and ( 3) 2. c) Sketch the graphs 2( 3) 2 and ( ) 2 2. GETTING MORE INVOLVED 59. Eploration. Consider the parabola with focus ( p, 0) and directri p for p 0. Let (, ) be an arbitrar point on the parabola. Write an equation epressing the fact that the distance from (, ) to the focus is equal to the distance from (, ) to the directri. Rewrite the equation in the form a 2, where a. 4 p 60. Eploration. In general, the graph of a( h) 2 k for a 0 is a parabola opening left or right with verte at (k, h). a) For which values of a does the parabola open to the right, and for which values of a does it open to the left? b) What is the equation of its ais of smmetr? GRAPHING CALCULATOR EXERCISES 6. Graph 2 using the viewing window with and 0. Net graph 2 2 using the viewing window 2 2 and 7. Eplain what ou see. 62. Graph 2 and 6 9 in the viewing window 5 5 and Does the line appear to be tangent to the parabola? Solve the sstem 2 and 6 9 to find all points of intersection for the parabola and the line. 3.3 THE CIRCLE In this section Developing the Equation Equations Not in Standard Form Sstems of Equations r (h, k) (, ) In this section we continue the stud of the conic sections with a discussion of the circle. Developing the Equation A circle is obtained b cutting a cone, as was shown in Fig We can also define a circle using points and distance, as we did for the parabola. Circle A circle is the set of all points in a plane that lie a fied distance from a given point in the plane. The fied distance is called the radius, and the given point is called the center. We can use the distance formula of Section 9.5 to write an equation for the circle with center (h, k) and radius r, shown in Fig If (, ) is a point on the circle, its distance from the center is r. So FIGURE 3.2 ( ) h 2 ( k) 2 r. We square both sides of this equation to get the standard form for the equation of a circle. Standard Equation for a Circle The graph of the equation ( h) 2 ( k) 2 r 2 with r 0, is a circle with center (h, k) and radius r.
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 informationWarmUp 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: WarmUp 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 informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationREVIEW 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 informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information7.3 Parabolas. 7.3 Parabolas 505
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree 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 informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationNonlinear Systems and the Conic Sections
C H A P T E R 11 Nonlinear Systems and the Conic Sections x y 0 40 Width of boom carpet Most intense sonic boom is between these lines t a cruising speed of 1,40 miles per hour, the Concorde can fly from
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationAnswers (Anticipation Guide and Lesson 101)
Answers (Anticipation Guide and Lesson 0) Lesson 0 Copright Glencoe/McGrawHill, a division of The McGrawHill Companies, Inc. 0 NAME DATE PERID Lesson Reading Guide Midpoint and Distance Formulas Get
More informationLesson 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 yvalues. b. The verte in the third quadrant and no intercepts. c. The verte
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, 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 informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More information2014 2015 Geometry B Exam Review
Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists
More informationAlgebra II. Administered May 2013 RELEASED
STAAR State of Teas Assessments of Academic Readiness Algebra II Administered Ma 0 RELEASED Copright 0, Teas Education Agenc. All rights reserved. Reproduction of all or portions of this work is prohibited
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
More informationGraphing Quadratic Equations
.4 Graphing Quadratic Equations.4 OBJECTIVE. Graph a quadratic equation b plotting points In Section 6.3 ou learned to graph firstdegree equations. Similar methods will allow ou to graph quadratic equations
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationA 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 informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationThe Distance Formula and the Circle
10.2 The Distance Formula and the Circle 10.2 OBJECTIVES 1. Given a center and radius, find the equation of a circle 2. Given an equation for a circle, find the center and radius 3. Given an equation,
More informationQuadratic Equations and Functions
Quadratic Equations and Functions. Square Root Propert and Completing the Square. Quadratic Formula. Equations in Quadratic Form. Graphs of Quadratic Functions. Verte of a Parabola and Applications In
More informationSection 23 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 informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree 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 informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More informationSLOPE OF A LINE 3.2. section. helpful. hint. Slope Using Coordinates to Find 6% GRADE 6 100 SLOW VEHICLES KEEP RIGHT
. Slope of a Line () 67. 600 68. 00. SLOPE OF A LINE In this section In Section. we saw some equations whose graphs were straight lines. In this section we look at graphs of straight lines in more detail
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More informationSECTION 22 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 informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
More informationComplex Numbers. (x 1) (4x 8) n 2 4 x 1 2 23 No realnumber solutions. From the definition, it follows that i 2 1.
7_Ch09_online 7// 0:7 AM Page 99. 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 informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationPrecalculus. 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 informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
More informationPOLYNOMIAL 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 informationCharacteristics of the Four Main Geometrical Figures
Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.
More information3 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 informationUnit 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 1/89 More Circles Ellipses 1/10 Hyperbolas (*)Pre AP Only 1/13 Parabolas HW: Part 4 HW: Part 1 1/14 Identifying conics
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More information9.3 OPERATIONS WITH RADICALS
9. Operations with Radicals (9 1) 87 9. OPERATIONS WITH RADICALS In this section Adding and Subtracting Radicals Multiplying Radicals Conjugates In this section we will use the ideas of Section 9.1 in
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationMEMORANDUM. 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 informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationof surface, 569571, 576577, 578581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433
Absolute Value and arithmetic, 730733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property
More information1. A plane passes through the apex (top point) of a cone and then through its base. What geometric figure will be formed from this intersection?
Student Name: Teacher: Date: District: Description: MiamiDade County Public Schools Geometry Topic 7: 3Dimensional Shapes 1. A plane passes through the apex (top point) of a cone and then through its
More informationSURFACE AREA AND VOLUME
SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has
More information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
More informationRotated 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 informationArea 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 information4Unit 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 information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationFluid 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 informationTeacher Page. 1. Reflect a figure with vertices across the xaxis. 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 informationTeacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.
Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 91.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles
More information6. 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 informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More information39 Symmetry of Plane Figures
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationGEOMETRY COMMON CORE STANDARDS
1st Nine Weeks Experiment with transformations in the plane GCO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,
More informationPerimeter, Area, and Volume
Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive 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 informationCore 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 informationSandia High School Geometry Second Semester FINAL EXAM. Mark the letter to the single, correct (or most accurate) answer to each problem.
Sandia High School Geometry Second Semester FINL EXM Name: Mark the letter to the single, correct (or most accurate) answer to each problem.. What is the value of in the triangle on the right?.. 6. D.
More informationWhy should we learn this? One realworld connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY
Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the intercept. One realworld connection is to find the rate
More informationSolids. Objective A: Volume of a Solids
Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationHigher. 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 informationContents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles...
Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................
More informationFACTORING QUADRATICS 8.1.1 through 8.1.4
Chapter 8 FACTORING QUADRATICS 8.. through 8..4 Chapter 8 introduces students to rewriting quadratic epressions and solving quadratic equations. Quadratic functions are any function which can be rewritten
More informationSummer Math Exercises. For students who are entering. PreCalculus
Summer Math Eercises For students who are entering PreCalculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn
More informationPolynomial 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 informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationG r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e  C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam
G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d P r e  C a l c u l u s M a t h e m a t i c s ( 2 0 S ) Final Practice Exam G r a d e 1 0 I n t r o d u c t i o n t o A p p l i e d a n d
More informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationCHAPTER 8, GEOMETRY. 4. A circular cylinder has a circumference of 33 in. Use 22 as the approximate value of π and find the radius of this cylinder.
TEST A CHAPTER 8, GEOMETRY 1. A rectangular plot of ground is to be enclosed with 180 yd of fencing. If the plot is twice as long as it is wide, what are its dimensions? 2. A 4 cm by 6 cm rectangle has
More informationArea 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 informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationGeometry Notes VOLUME AND SURFACE AREA
Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationMATH 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 informationTo 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 informationExponential 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 informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More informationFor 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 informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationCircle Name: Radius: Diameter: Chord: Secant:
12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane
More information