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An Introduction to Conics Let’s try some math aerobics! Stand up and here we go! Establish the x-axis and y-axis. 2 x What does do to the graph? It creates a CURVE. Hold up your curve! 2 y What would do to a graph? It also creates a CURVE. Which way does your curve go? 2 Think about it: y x so y Hold up your curve! x 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 . Let’s look at these parabola equations: y 2x 2 y 4(x 1) 3 2 y 5 4x 2 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? PARABOLAS Show me another parabola with your hand. What makes it open to the LEFT or RIGHT? When the Y is SQUARED . Let’s look at these parabola equations: x 3y 2 x 4(y 1) 3 2 x 4 5y 2 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? What happens when you ADD x2 and y2 ? The CURVES go TOGETHER! Hold up your curves! 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. 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. What do you notice about the circle and ellipse equations below? 2x 2y 32 3x 4y 12 4x 5y 100 5x 5y 100 2 2 2 2 2 2 2 2 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 What happens when you SUBTRACT x2 and y2 ? The CURVES go APART! Hold up your curves! 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. NOW, you are ready to work with Conic Sections! Circles Ellipses Hyperbolas Parabolas Which Conic Section is it? Hint: Use your hands to help you out… x 2y 2 Horizontal parabola 3x 2y 24 2 2 3x 2y 8 2 2 y 3(x 1) 2 2x 2y 32 2 2 Opens Right Hyperbola Ellipse Vertical parabola Opens downward Circle Introduction to Conic Sections In geometry, you learned about cones. In conic sections, we are learning about a “doublenapped” 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. axis element apex DoubleRegular Cone napped cone Lateral Surface Nappe Slant Height Element Vertex Apex 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 CIRCLE - A plane intersects both nappes parallel to the axis HYPERBOLA - A plane intersects one nappe parallel to an element PARABOLA - A plane intersects exactly one nappe and is NOT perpendicular to the axis ELLIPSE Shapes with Double Napped Cone Circle Ellipse Parabola Hyperbola Other Creations with Double-Napped Cone Point Line Double Line 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. 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. If: Then the conic is a(n): A = C, A 0, C 0 Circle AC > 0 (A and C have the same sign and AC) Ellipse AC = 0 (A=0 or B=0, but not both). Parabola AC < 0 (A and C have different signs) Hyperbola Conics Identification Flow Chart Is either x2 or y2 missing? yes Parabola no Is either x2 or y2 negative? yes Hyperbola no Do x2 and y2 have the same coefficient? yes Circle no Do x2 and y2 have different coefficients? yes Ellipse Identify the type of conic by looking at its equation in general form. 1. x y 9 0 Circle 2. 4x y 8x 4 y 4 0 Ellipse 3. y 8x 6 y 1 0 Parabola 4. 2x y 16 0 Ellipse 5. 9x y 18x 6 y 9 0 Hyperbola 2 2 2 2 2 2 2 2 2 Identify the type of conic by looking at its equation in general form. 6. x 6 x 8 y 7 0 Parabola 7. 2x 2 y 4x 0 Circle 2 2 2 Ellipse 9 x 25 y 36 x 150 y 36 0 8. 2 2 9. 4 x y 4 0 Ellipse 10. 16 x 4 y 64 0 Hyperbola 2 2 2 2 Identify the type of conic by looking at its equation in general form. 4 x 4 y 8 x 8 y 7 0 11. Circle x y 4 x 2 y 2 0 12. Hyperbola 13. x 9 y 6x 18 y 9 0 Ellipse 2 y 8x 4 y 4 0 14. Parabola 15. x y 4 0 Hyperbola y 12 x 6 y 3 0 16. Parabola 2 2 2 2 2 2 2 2 2