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Conics: a crash course

Conics: a crash course. MathScience Innovation Center Betsey Davis. Why “conics”?. The 4 basic shapes are formed by slicing a right circular cone. What is a right circular cone?. A cone, with a circular base, whose axis is perpendicular to that base. Not right circular cone:.

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Conics: a crash course

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  1. Conics: a crash course MathScience Innovation Center Betsey Davis

  2. Why “conics”? • The 4 basic shapes are formed by slicing a right circular cone What is a right circular cone? • A cone, with a circular base, whose axis is perpendicular to that base. Conics B. Davis MathScience Innovation Center

  3. Not right circular cone: Conics B. Davis MathScience Innovation Center

  4. What are the 4 basic conics? • Parabola • Circle • Ellipse • Hyperbola Conics B. Davis MathScience Innovation Center

  5. What is the relationship between the cone and the 4 shapes? • It’s how you slice ! Conics B. Davis MathScience Innovation Center

  6. Take notes on first site! You will be responsible for knowing some real-world applications of each of the conics. Slicing a cone Let’s visit 1http://id.mind.net/~zona/mmts/miscellaneousMath/conicSections/conicSections.htm 2http://ccins.camosun.bc.ca/~jbritton/jbconics.htm 3http://www.keypress.com/sketchpad/java_gsp/conics.html 4http://www.exploremath.com/activities/activity_list.cfm?categoryID=1 Conics B. Davis MathScience Innovation Center

  7. General Equation: • Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 • For us, • B = 0 always • (this rotates the conic between 0 and 90 degrees) Conics B. Davis MathScience Innovation Center

  8. General Equation: • Ax^2 + By^2 + Cx + Dy + E = 0 • What is the value of A or B if it is a parabola? • B=0 or A =0 but not both Conics B. Davis MathScience Innovation Center

  9. General Equation: • Ax^2 + By^2 + Cx + Dy + E = 0 • If circle • B=A Conics B. Davis MathScience Innovation Center

  10. General Equation: • Ax^2 + By^2 + Cx + Dy + E = 0 • If ellipse • B is not equal to A, but they have the same sign Conics B. Davis MathScience Innovation Center

  11. General Equation: • Ax^2 + By^2 + Cx + Dy + E = 0 • If hyperbola • B and A have opposite signs Conics B. Davis MathScience Innovation Center

  12. Parabola • Circle • Ellipse • Hyperbola General Equation: • 3x^2 + 3y^2 + 2x + y + 8 = 0 • 3x^2 - 3y^2 + 2x + y + 8 = 0 • 3x^2 + 9y^2 + 2x + y + 8 = 0 • 3x^2 + 2x + y + 8 = 0 Conics B. Davis MathScience Innovation Center

  13. Parabola Reminders • Parabolas opening up and down are the only conics that are functions • Y = (x-3)^2 +4 • Vertex? • Axis of symmetry? • Opening which way? (3,4) X = 3 up Conics B. Davis MathScience Innovation Center

  14. Parabola Reminders • Y^2 –4Y + 3 –x = 0 • Vertex? • Axis of symmetry? • Opening which way? (-1,2) Y=2 right Conics B. Davis MathScience Innovation Center

  15. Circles • Ax^2 + Ay^2 +Cx + Dy + E= 0 • (x-h)^2 + (y-K)^2 = r^2 • Where (h,k) is the center and r is the radius • X^2 + y^2 = 36 • Centered at origin • Radius is 6 Conics B. Davis MathScience Innovation Center

  16. Circles • Ax^2 + Ay^2 +Cx + Dy + E= 0 • (x-h)^2 + (y-K)^2 = r^2 • Where (h,k) is the center and r is the radius • (X-1)^2 +( y-3)^2 = 49 • Center at (1,3) • Radius is 7 Conics B. Davis MathScience Innovation Center

  17. Ellipses • Ax^2 + By^2 +Cx + Dy + E= 0 • (x-h)^2 + (y-K)^2 = 1 a^2 b^2 • Where (h,k) is the center and a is the long radius and b is the short radius • (X)^2 +( y)^2 = 1 • 25 4 • Center at (0,0) • Major axis 10, minor 4 Conics B. Davis MathScience Innovation Center

  18. Ellipses • Ax^2 + By^2 +Cx + Dy + E= 0 • (x-h)^2 + (y-K)^2 = 1 a^2 b^2 • Where (h,k) is the center and a is the long radius and b is the short radius • (X-1)^2 +( y+3)^2 = 1 • 16 100 • Center at (1,-3) • Major axis 20, minor 8 Conics B. Davis MathScience Innovation Center

  19. Hyperbolas • Ax^2 - By^2 +Cx + Dy + E= 0 • (x-h)^2 - (y-K)^2 = 1 a^2 b^2 • Where (h,k) is the center and 2a is the transverse axis • (X-1)^2 -( y+3)^2 = 1 • 16 100 • Center at (1,-3) • Transverse axis length is 8 Conics B. Davis MathScience Innovation Center

  20. Hyperbolas • Ax^2 - By^2 +Cx + Dy + E= 0 • (x-h)^2 - (y-K)^2 = 1 a^2 b^2 • Where (h,k) is the center and 2a is the transverse axis • (y)^2 - ( x)^2 = 1 • 16 100 • Center at (0,0) • Transverse axis length is 8 Conics B. Davis MathScience Innovation Center

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