conics: a crash course mathscience innovation center betsey davis
TRANSCRIPT
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/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
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 = 1
25 4
•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 = 1
16 100
•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 = 1
16 100
•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 = 1
16 100
•Center at (0,0)
•Transverse axis length is 8