Conics: a crash course MathScience Innovation Center Betsey Davis

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

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

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

Conics B. Davis MathScience Innovation Center Slicing a cone Let’s visit 1http://id.mind.net/~zona/mmts/miscellaneousMat h/conicSections/conicSections.htmhttp://id.mind.net/~zona/mmts/miscellaneousMat h/conicSections/conicSections.htm 2http://ccins.camosun.bc.ca/~jbritton/jbconics.htmhttp://ccins.camosun.bc.ca/~jbritton/jbconics.htm 3http:// onics.htmlhttp:// onics.html 4http:// st.cfm?categoryID=1http:// st.cfm?categoryID=1 Take notes on first site! You will be responsible for knowing some real-world applications of each of the conics.

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

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

Conics B. Davis MathScience Innovation Center 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 Parabola Circle Ellipse Hyperbola

Conics B. Davis MathScience Innovation Center 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 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 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 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 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 = Center at (0,0) Major axis 10, minor 4

Conics B. Davis MathScience Innovation Center 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 = Center at (1,-3) Major axis 20, minor 8

Conics B. Davis MathScience Innovation Center 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 = Center at (1,-3) Transverse axis length is 8

Conics B. Davis MathScience Innovation Center 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 = Center at (0,0) Transverse axis length is 8