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Exploring Conic Sections in Algebra 2: Mrs. Spitz's Guide

Understand conic sections in standard form, identify shapes from equations, and learn the properties of parabolas, circles, ellipses, and hyperbolas. Discover how to recognize these curves from their quadratic equations.

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Exploring Conic Sections in Algebra 2: Mrs. Spitz's Guide

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  1. 9.7 Conic Sections Algebra 2 Mrs. Spitz Spring 2007

  2. Objectives • Write the equation of a conic section in standard form, and • Identify a conic section from its equation.

  3. Assignment • Pg. 429 #4-27 all

  4. Graphically speaking

  5. A supersonic jet has a shock wave in the shape of a cone. At the points where the wave hits the ground, a sonic boom is heard. If the jet is traveling parallel to the ground, the sonic boom is heard at the points that form one branch of the hyperbola. What shape is formed when the jet is not traveling parallel to the ground? Application

  6. By Slicing a double cone in different directions, you can form parabolas, circles, ellipses and hyperbolas. For this reason, these curves are called conic sections. The conic sections can all be described by a quadratic equation. Application

  7. Equation of a conic section • The equation of a conic section can be written in the form: Where A, B and C are not all zero.

  8. What else? • You can identify the conic section that is represented by a given equation by writing the equation in one of the standard forms you have learned. Study the table on the next slide.

  9. Table of Standard Forms

  10. Ex. 1: Is the graph of x2 + y2 – 8x + 6y + 24 = 0 a parabola, a circle, an ellipse or a hyperbola?

  11. Ex. 2: Is the graph of x2 +10x+5 = 4y2 + 16y a parabola, a circle, an ellipse or a hyperbola?

  12. Some other useful information . . . • You can easily determine the type of conic section represented by an equation of the form, when B = 0 by looking at A and C. • If A = C, then the equation represents a circle. • If A and C have the same sign and A ≠ C, the equation represents an ellipse. • If A and C have opposite signs, the equation represents a hyperbola. • If A = 0 or C = 0, but not both, the equation represents a parabola.

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