Hyperboloid of TWO Sheets a.k.a “Two Opposing Facing Bowls” 𝑥2 𝑦2 𝑧2 − 2− 2+ 2=1 𝑎 𝑏 𝑐 The Shape Looks like two opposite facing bowls, giving it the nickname, “two opposite facing bowls” Example: 𝑥 2 (𝑦 − 2)2 (𝑧 + 1)2 − − =1 4 25 9 Some Features that tell you it’s a Hyperboloid of Two Sheets: This Equation represents a Hyperboloid of Two Sheets with a center of (0,2,-1) and the “bowls” will be flaring out in the X direction. In the Equation: o The variables all equal 1 o All three Variables are squared o 2 of the variables are negative Horizontal Traces are ellipses when |z|>c are ellipses Vertical traces are Hyperbolas The XY Plane is an Axis of Symmetry for the graphs of Hyperboloids of TWO sheets Hollow The Graph looks like two opposing bowls Trace for x=0 − (𝑦−2)2 25 − (𝑧+1)2 9 =1 This trace is impossible because 2 negative numbers can not add up to a positive 1 Trace for y=0 𝑥2 4 − (𝑧+1)2 9 =1 This trace results in hyperbolas Trace for z=0 𝑥2 4 − (𝑦−2)2 25 =1 This trace results in hyperbolas as well Underview: Overview: