1 / 27

An Introduction to Conics

An Introduction to Conics. Let’s try some math aerobics! Stand up and here we go! Establish the x-axis and y-axis. What does x 2 do to the graph?. It creates a CURVE . Hold up your curve! . Think about it: so . What would y 2 do to a graph?. It also creates a

licia
Download Presentation

An Introduction to Conics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. An Introduction to Conics Let’s try some math aerobics! Stand up and here we go! Establish the x-axis and y-axis.

  2. What does x2 do to the graph? It creates a CURVE. Hold up your curve!

  3. Think about it: so What would y2 do to a graph? It also creates a CURVE. Which way does your curve go? Hold up your curve!

  4. PARABOLAS Show me a parabola with your hand. How many curves? ONE So, what do you know about the variables? Only ONE is SQUARED When it opens UP or DOWN, what is squared? The x .

  5. Let’s look at these parabola equations: Explain how we know these equations are parabolas. Do these parabolas open up or down? How do you determine if these parabolas open up or down?

  6. PARABOLAS Show me another parabola with your hand. What makes it open to the LEFT or RIGHT? When the Y is SQUARED .

  7. Let’s look at these parabola equations: Explain how we know these equations are parabolas. Do these parabolas open left or right? How do you determine if these parabolas open left or right?

  8. What happens when you ADD x2 and y2 ? The CURVES go TOGETHER! Hold up your curves!

  9. CIRCLES Show me a circle with your hands. How many curves? TWO So what do you know about x and y? BOTH SQUARED Which way do the curves go? TOGETHER So what do you know about the equation? It has x2 and y2 ADDED.

  10. ELLIPSES Show me an ellipse with your hands. OR How many curves? TWO So what do you know about x and y? BOTH SQUARED Which way do the curves go? TOGETHER So what do you know about the equation? It has x2 and y2 ADDED.

  11. What do you notice about the circle and ellipse equations below? How can we tell circle and ellipse equations apart? CIRCLES: the coefficients of both x2 and y2 are the same ELLIPSES: the coefficients of both x2 and y2 have a different number but the same sign

  12. What happens when you SUBTRACT x2 and y2 ? The CURVES go APART! Hold up your curves!

  13. Hyperbolas Show me an hyperbola with your hands. How many curves? TWO So what do you know about x and y? BOTH SQUARED Which way do the curves go? APART So what do you know about the equation? It has x2 and y2 SUBTRACTED.

  14. NOW, you are ready to work with Conic Sections!CirclesEllipsesHyperbolasParabolas

  15. Which Conic Section is it? Hint: Use your hands to help you out… Horizontal parabola Opens Right Hyperbola Ellipse Vertical parabola Opens downward Circle

  16. Introduction to Conic Sections

  17. In geometry, you learned about cones. In conic sections, we are learning about a “double-napped” cone. Unlike a regular cone, the double-napped cone used to generate conic sections has no bases; each cone is infinite. In a drawing it appears that the cone ends, but imagine that it extends infinitely in both directions.

  18. axis Double-napped cone Nappe Element Apex apex element Regular Cone Lateral Surface Slant Height Vertex

  19. Circles, ellipses, parabolas, and hyperbolas are called conic sections because they are the cross sections formed when a double-napped cone is sliced by a plane. • Match the description of each conic section with its name. • A plane intersects exactly one nappe • and is perpendicular to the axis • A plane intersects both nappes parallel • to the axis • A plane intersects one nappe parallel • to an element • A plane intersects exactly one nappe • and is NOT perpendicular to the axis CIRCLE HYPERBOLA PARABOLA ELLIPSE

  20. Shapes with Double Napped Cone Circle Ellipse Parabola Hyperbola

  21. Other Creations with Double-Napped Cone Line Double Line Point

  22. The equation of any conic section can be written in the General form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A and C are not both 0. What would happen if A and C were both 0? Answer: the equation could represent a point, a line, or a pair of intersecting lines.

  23. Classification of a Conic Section: You can determine the type of conic by looking at the general form. Look at the A and C terms.

  24. Conics Identification Flow Chart yes Parabola Is either x2 or y2 missing? no Hyperbola Is either x2or y2 negative? yes no Do x2 and y2 have the same coefficient? Circle yes no Do x2 and y2 have different coefficients? Ellipse yes

  25. Identify the type of conic by looking at its equation in general form. Circle Ellipse Parabola Ellipse Hyperbola 1. 2. 3. 4. 5.

  26. Identify the type of conic by looking at its equation in general form. Parabola Circle Ellipse Ellipse Hyperbola 6. 7. 8. 9. 10.

  27. Identify the type of conic by looking at its equation in general form. Circle Hyperbola Ellipse Parabola Hyperbola Parabola 11. 12. 13. 14. 15. 16.

More Related