Chapter 10 Conic Sections.

Slides:



Advertisements
Similar presentations
Conics D.Wetzel 2009.
Advertisements

Lesson 10-1: Distance and Midpoint
Section 11.6 – Conic Sections
6.5 Graphing Linear Inequalities in Two Variables Wow, graphing really is fun!
Chapter 9 Analytic Geometry.
Colleen Beaudoin February,  Review: The geometric definition relies on a cone and a plane intersecting it  Algebraic definition: a set of points.
Hyperbolas and Rotation of Conics
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 8 Systems of Equations and Inequalities.
Copyright © Cengage Learning. All rights reserved. Conic Sections.
Section 5 Chapter Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives 2 Second-Degree Inequalities and Systems of Inequalities Graph.
10.3 Hyperbolas. Circle Ellipse Parabola Hyperbola Conic Sections See video!
Rev.S08 MAC 1140 Module 11 Conic Sections. 2 Rev.S08 Learning Objectives Upon completing this module, you should be able to find equations of parabolas.
THE HYPERBOLA. A hyperbola is the collection of all points in the plane the difference of whose distances from two fixed points, called the foci, is a.
Review Day! Hyperbolas, Parabolas, and Conics. What conic is represented by this definition: The set of all points in a plane such that the difference.
Slide Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
10.6 – Translating Conic Sections. Translating Conics means that we move them from the initial position with an origin at (0, 0) (the parent graph) to.
Copyright © 2011 Pearson Education, Inc. The Parabola Section 7.1 The Conic Sections.
Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Section 9.1 Quadratic Functions and Their Graphs.
Copyright © Cengage Learning. All rights reserved. 10 Parametric Equations and Polar Coordinates.
Hyberbola Conic Sections. Hyperbola The plane can intersect two nappes of the cone resulting in a hyperbola.
CHAPTER TWO: LINEAR EQUATIONS AND FUNCTIONS ALGEBRA TWO Section Linear Inequalities in Two Variables.
Exponential and Logarithmic Functions
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Slide 1- 1 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Distance The distance between any two points P and Q is written PQ. Find PQ if P is (9, 1) and Q is (2, -1)
Copyright © 2011 Pearson Education, Inc. Conic Sections CHAPTER 13.1Parabolas and Circles 13.2Ellipses and Hyperbolas 13.3Nonlinear Systems of Equations.
Chapter 10 Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
Warm-up: Solve for x. 6x + 12 = x 2x + 3(x – 5) = 25
Chapter 10 Conic Sections
Analyzing Conic Sections
Chapter 6 Analytic Geometry. Chapter 6 Analytic Geometry.
Copyright © Cengage Learning. All rights reserved.
Quadratic and Other Nonlinear Inequalities
Translating Conic Sections
6-3 Conic Sections: Ellipses
Conic Sections College Algebra
Section 13.2 The Ellipse.
Graphing Quadratic Functions Rational Functions Conic Sections
Topics in Analytic Geometry
Chapter 3 Graphs and Functions
Introduction to Graphing
Introduction to Conic Sections
9.6A Graphing Conics Algebra II.
THE HYPERBOLA.
Hyperbolas.
6-3 Conic Sections: Ellipses
Chapter 9 Conic Sections.
Review Circles: 1. Find the center and radius of the circle.
Chapter 10 Conic Sections
Introduction to Conic Sections
Chapter 1 Graphs, Functions, and Models.
Test Dates Thursday, January 4 Chapter 6 Team Test
Day 138 – Equation of ellipse
10-4 Hyperbolas Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2.
Introduction to Graphing
Analyzing Conic Sections
Hyperbolas.
Chapter 10 Conic Sections.
THE HYPERBOLA.
Section 11.6 – Conic Sections
Intro to Conic Sections
STANDARD FORM OF EQUATIONS OF TRANSLATED CONICS
Chapter 10 Conic Sections.
10.6 – Translating Conic Sections
Jeopardy Solving for y Q $100 Q $100 Q $100 Q $100 Q $100 Q $200
Hyperbolas 12-4 Warm Up Lesson Presentation Lesson Quiz
Chapter 7 Analyzing Conic Sections
Presentation transcript:

Chapter 10 Conic Sections

The Parabola and the Circle § 10.1 The Parabola and the Circle

Conic Sections Conic sections derive their name because each conic section is the intersection of a right circular cone and a plane. Circle Ellipse Parabola Hyperbola

The Parabola Just as y = a(x – h)2 + k is the equation of a parabola that opens upward or downward, x = a(y – k)2 + h is the equation of a parabola that opens to the right or to the left. y = a(x – h)2 + k x = a(y – k)2 + h x y x y x y x y a > 0 (h, k) (h, k) (h, k) y = k a < 0 y = k (h, k) a < 0 a > 0 x = h x = h

The Parabola Example: Graph the parabola x = (y – 4)2 + 1. a > 0, so the parabola opens to the right. The vertex of the parabola is (1, 4). The axis of symmetry is y = 4.

The Parabola Example continued: x y 2 The table shows ordered pairs of the solutions of x = (y – 4)2 + 1. x y 1 4 y = 4 2 3 2 5 17 17 8

The Parabola Example: Graph the parabola y = x2 + 12x + 25. Complete the square on x to write the equation in standard form. y – 25 = x2 + 12x Subtract 25 from both sides. The coefficient of x is 12. The square of half of 12 is 62 = 36. y – 25 + 36 = x2 + 12x + 36 Add 36 to both sides.

The Parabola Example continued: y + 11 = (x + 6)2 Simplify the left side and factor the right side. y = (x + 6)2 – 11 Subtract 11 from both sides. a > 0, so the parabola opens upward. The vertex of the parabola is (– 6, – 11). The axis of symmetry is x = – 6.

The Parabola Example continued: x y 3 y = x2 + 12x + 25

The Distance Formula Distance Formula The distance d between any two points (x1, y1) and (x2, y2) is given by y x (x1, y1) (x2, y2) d a = x2 – x1 b = y2 – y1

The Distance Formula Example: Find the distance between (– 6, – 6) and (– 5, – 2).

The Midpoint The midpoint of a line segment is the point located exactly halfway between the two endpoints of the line segment. Midpoint Formula The midpoint of the line segment whose endpoints are (x1, y1) and (x2, y2) is the point with the coordinates

The Midpoint Example: Find the midpoint of the line segment that joins points P(0, 8) and Q(4, – 6).

The Cirlce A circle is the set of all points in a plane that are the same distance from a fixed point called the center. The distance is called the radius. Circle The graph of (x – h)2 + (y – k)2 = r2 is a circle with center (h, k) and radius r. y x (h, k) r

The Circle Example: Graph (x – 3)2 + y2 = 9. The equation can be written as (x – 3)2 + (y – 0)2 = 32. h = 3, k = 0, and r = 3. x y r = 3 (3, 0)

The Circle Example: Find the equation of the circle with center (– 7, 6) and radius 2. h = – 7, k = 6, and r = 2. (x – h)2 + (y – k)2 = r2. The equation can be written as [x – (– 7)2] + (y – 6)2 = 22. (x + 7)2 + (y – 6)2 = 4. Simplify.

The Ellipse and the Hyperbola § 10.2 The Ellipse and the Hyperbola

The Ellipse An ellipse can be thought of as the set of points in a plane such that the sum of the distances of those points from two fixed points is constant. Each of the two fixed points is called a focus. (The plural of focus is foci.) The point midway between the foci is called the center.

The Ellipse Ellipse with Center (0, 0) The graph of an equation of the form is an ellipse with center (0, 0). The x-intercepts are (a, 0) and (– a, 0), and the y-intercepts are (0, b) and (0, – b). y x a b – b – a

The Ellipse Example: The equation is of the form a = 2 and b = 3. y x – 2 – 3 3

The Hyperbola Hyperbola with Center (0, 0) y x y a a The graph of an equation of the form is a hyperbola with center (0, 0) and x-intercepts (a, 0) and (– a, 0). x y b b The graph of an equation of the form is a hyperbola with center (0, 0) and y-intercepts (0, b) and (0, – b).

Asymptotes x y ( a, b) b a (a,  b) ( a,  b) (a, b)  a  b The asymptotes of the hyperbola are dashed lines used to sketch the graph of the hyperbola. To sketch the asymptotes, draw a rectangle with vertices (a, b), (– a, b), (a, – b), and (– a, – b).

The Hyperbola Example: The equation is of the form so its graph is a hyperbola that opens to the left and right. It has center (0, 0) and x-intercepts (3, 0) and (– 3, 0). The asymptotes of the hyperbola are the extended diagonals of the rectangle with corners (3, 6), (– 3, 6), (3, – 6), and (– 3, – 6 ).

The Hyperbola Example continued: y x ( 3, 6) (3,  6) ( 3,  6) 2 – 2 ( 3, 6) (3,  6) ( 3,  6) (3, 6)

Solving Nonlinear Systems of Equations § 10.3 Solving Nonlinear Systems of Equations

Nonlinear Systems of Equations A nonlinear system of equations is a system of equations where at least one of the equations is not linear. The substitution method or the elimination method may be used to solve the system.

Nonlinear Systems of Equations Example: Solve the system. The substitution method will work best to solve this system. y + 2 = x Second equation x2 + y = 4 First equation (y + 2)2 + y = 4 Replace x with y + 2. y2 + 4y + 4 + y = 4

Nonlinear Systems of Equations Example continued: y2 + 4y + 4 + y = 4 y2 + 5y = 0 y(y + 5) = 0 y = 0 or y = – 5 Let y = 0 and then let y = – 5 to find the corresponding x values. Let y = 0 Let y = – 5 y + 2 = x y + 2 = x 0 + 2 = x – 5 + 2 = x x = 2 x = – 3

Nonlinear Systems of Equations Example continued: The solutions are (2, 0) and (– 3, – 5). Check both solutions in both equations. x2 + y = 4 y + 2 = x x2 + y = 4 y + 2 = x 22 + 0 = 4 0 + 2 = 2 (– 3)2+ (– 5) = 4 – 5 + 2 = – 3 4 = 4 2 = 2 9 – 5 = 4 – 3 = – 3 x y y + 2 = x x2 + y = 4 A graph of the system also verifies the solution. (2, 0) (– 3, – 5).

Nonlinear Systems of Equations Example: Solve the system. The elimination method will work best to solve this system. (– 1) x2 + (– 1) 2y2 = (– 1) 4 Multiply the first equation by – 1. – x2 + – 2y2 = – 4 x2 – y2 = 4 – 3y2 = 0 Add the two equations.

Nonlinear Systems of Equations Example continued: – 3y2 = 0 y = 0 Let y = 0 to find the corresponding x values. x2 + 2y2 = 4 x2 + 2(0)2 = 4 x2 = 4 x = 2 or x = – 2 The solutions are (– 2, 0) and (2, 0). Check both solutions in both equations.

Nonlinear Inequalities and Systems of Inequalities § 10.4 Nonlinear Inequalities and Systems of Inequalities

Nonlinear Inequalities Nonlinear inequalities in two variables are graphed in a similar way to linear inequalities in two variables.

Nonlinear Inequalities Example: x y 4 – 4 – 3 3 Graph the equation Sketch a dashed curve since the graph of does not include the graph of

Nonlinear Inequalities Example continued: x y 4 – 4 – 3 3 Select a test point to determine which region contains the solutions. Select the test point (0, 0) if it is not on the boundary line. (0, 0) This is a true statement, so the test point is part of the solution.

Nonlinear Inequalities Example: x y – 2 2 Graph the equation Sketch a solid curve since the graph of does include the graph of

Nonlinear Inequalities Region A Example continued: x y – 2 2 (0, 4) The hyperbola divides the plane into three regions. Select a test point in each region to determine the solutions. Region B (0, 0) Region A Region B Region C (0, – 4 ) Region C False True False

Nonlinear Inequalities Example continued: The graph of the solution set includes the shaded region B only. It also includes the boundary lines. x y – 2 2

Nonlinear Inequalities Example: Graph the system. Graph each inequality on the same set of axes. x y – 2 2

Nonlinear Inequalities Example continued: x y – 2 2 Shade the solution set for each inequality. The solution to the system is the dark green area where both regions are shaded.