How Do Cones Relate to Graphing?

Question

Seriously, how?

Answer 1

Take right circular cones and put them on top of each other with only their vertices intercepting (double napped cone). Then, use planes to slice the structure in different ways. The cross sections will form different graphs.



horizontal parallel slice through one cone (not through tip) = circle



slanted slice through one cone (not through tip)(must go from side to side not side to base) = ellipse



vertical slice (not through center) = hyperbola



slanted slice through one cone (not through tip)(must go from side to base) = parabola



horizontal slice through point of intersection (the tips) only = point



plane tangent to "opposite sides" of both cones = line



plane vertically through point of intersection but instead of looking at the cross section, focusing on where the plane intersects the sides of the cones = pair of lines with opposite slopes





Search some pictures if you need some help visualizing all this. The "Khan's Academy" website provides some awesome videos for this.

Answer 2

If you put a cut plane through a cone you get the so called conic sections:



includes circle, parabola, hyperbola and elipse

Answer 3

Parabolas can be seen as sections through a cone.