Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-1 Chapter 9 Analytic Geometry and the Conic Sections.

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Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-1 Chapter 9 Analytic Geometry and the Conic Sections

9 Chapter Outline 9.1A Brief Introduction to Analytical Geometry 9.2The Circle and the Ellipse 9.3The Hyperbola 9.4The Analytic Parabola 9.5Nonlinear Systems of Equations and Inequalities 9.6Polar Coordinates, Equations, and Graphs 9.7More on the Conic Sections: Rotation of Axes and Polar Form 9.8Parametric Equations and Graphs 1-2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Learning Objectives In Section 9.1 you will learn how to:  A.Verify theorems from basic geometry involving the distance between two points  B.Verify the points (x, y) are an equal distance from a given point and a given line  C.Use the defining characteristics of a conic to find its equation 9.1 A Brief Introduction to Analytical Geometry 1-3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-6 EXAMPLE 1 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-8 EXAMPLE 2 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

1-9 EXAMPLE 2 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-14 EXAMPLE 3 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Learning Objectives In Section 9.2 you will learn how to:  A.Use the characteristics of a circle and its graph to understand the equation of an ellipse  B.Use the equation of an ellipse to graph central and noncentral ellipses  C.Locate the foci of an ellipse and use the foci and other features to write the equation  D.Solve applications involving the foci 9.2 The Circle and the Ellipse 1-15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Learning Objectives In Section 9.3 you will learn how to:  A.Use the equation of a hyperbola to graph central and noncentral hyperbolas  B.Distinguish between the equations of a circle, ellipse, and hyperbola  C.Locate the foci of a hyperbola and use the foci and other features to write its equation  D.Solve applications involving foci 9.3 The Hyperbola 1-32 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-34 EXAMPLE 1 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Learning Objectives In Section 9.4 you will learn how to:  A.Graph parabolas with a horizontal axis of symmetry  B.Identify and use the focus-directrix from of the equation of a parabola  C.Solve applications of the analytic parabola 9.4 The Analytic Parabola 1-55 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Learning Objectives In Section 9.5 you will learn how to:  A.Visualize possible solutions  B.Solve nonlinear systems using substitution  C.Solve nonlinear systems using elimination  D.Solve nonlinear systems of inequalities  E.Solve applications on nonlinear systems 9.5 Nonlinear Systems of Equations and Inequalities 1-70 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-72 EXAMPLE 1 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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1-83 EXAMPLE 6 CONTINUED Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Learning Objectives In Section 9.6 you will learn how to:  A.Plot points given in polar form  B.Convert from rectangular form to polar form  C.Convert from polar form to rectangular form  D.Sketch basic polar graphs using an r-value analysis  E.Use symmetry and families of curves to write a polar equation given a polar graph or information about the graph 9.6 Polar Coordinates, Equations, and Graphs 1-84 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Learning Objectives In Section 9.7 you will learn how to:  A.Graph conics that have nonvertical and nonhorizontal axes (rotated conics)  B.Identify conics using the discriminant of the polynomial form: the invariant B 2 – 4AC  C.Write the equation of a conic section in polar form  D.Solve applications involving the conic sections in polar form 9.7 More on the Conic Sections: Rotation of Axes and Polar Form Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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Learning Objectives In Section 9.8 you will learn how to:  A.Sketch the graph of a parametric equation  B.Write parametric equations in rectangular form  C.Graph curves from the cycloid family  D.Solve applications involving parametric equations 9.8 Parametric Equations and Graphs Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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