1. Section 15.2 A Brief Catalogue of the Quadratic Surfaces; Projections
Chapter 15
Section 15.2
A Brief Catalogue of the Quadratic Surfaces; Projections

2. Projections (or sometimes referred to as traces) z
Graphs in 3 Dimensions
A function of 2 independent variables 𝑧=𝑓 𝑥,𝑦 has a graph that is called a surface. For each pair of values 𝑥 0 , 𝑦 0 in the xy-plane the value of 𝑧 0 =𝑓 𝑥 0 , 𝑦 0 is the distance the point on the surface is above or below the xy-plane. But, determining the graph by plotting individual points is long and difficult. We will look at 3 methods:
Projections (or sometimes referred to as traces)
Quadratic Surfaces (Paraboloids, Spheres, Ellipsoids, Hyperboloids)
Cylinders (or omitting a variable)
𝑧 0 y
𝑥 0 , 𝑦 0 x
Recognizing the basic shapes of parabolas, ellipses, circles and hyperbolas and in what planes they are oriented is important.
𝑥−ℎ 2 𝑎 2 − 𝑦−𝑘 2 𝑏 2 =1
Hyperbola opening right or left h k
𝑦−𝑘=𝑎 𝑥−ℎ 2
Parabola opening up or down
𝑥−ℎ 2 𝑎 𝑦−𝑘 2 𝑏 2 =1
Ellipse with 𝑏>𝑎 k k h h
𝑥−ℎ 𝑦−𝑘 2 = 𝑟 2
Circle with 𝑎=𝑏=𝑟 h k
𝑥−𝑘=𝑎 𝑦−ℎ 2
Parabola opening right or left
𝑦−𝑘 2 𝑏 2 − 𝑥−ℎ 2 𝑎 2 =1
Hyperbola opening up or down k k h h

3. Graphing With Traces (Projections)
Graphing with traces you graph an entire curve (not a point) at a time for a particular value of x, y or z. Graph enough to give you an outline of the shape in order to tell what the curves “between” the values you picked are doing. To organize this I usually make a table of values of along with the corresponding trace equation.
Example
Graph the surface 𝑧= 𝑥 2 + 𝑦−4 2 and name it. z
𝑦=4
Value
Trace Equation
𝑧=4
𝑥=0
𝑧= 𝑦−4 2
This surface is called a paraboloid.
𝑦=4
𝑧= 𝑥 2
𝑥=0
𝑧=0
0= 𝑥 2 + 𝑦−4 2
𝑧=1
1= 𝑥 2 + 𝑦−4 2
𝑧=1
𝑧=4 y
4= 𝑥 2 + 𝑦−4 2
𝑧=0 4
Example
Graph and name the surface 𝑦−4 2 = 𝑥 2 + 𝑧 2 x z
Value
Trace Equation
𝑦=2
𝑦=6
𝑥=0
𝑦−4=± 𝑧
𝑦=5
𝑧=0
𝑦−4=± 𝑥
This surface is called a double cone.
𝑦=3
𝑥=0
𝑦=4
0= 𝑥 2 + 𝑧 2
𝑧=0 y 4
𝑦=3,5
1= 𝑥 2 + 𝑧 2
𝑦=2,6
4= 𝑥 2 + 𝑧 2 x

4. Both of these surfaces are called cones.
Example
If we solve for a variable we just get part of the surface. Find the graph of each surface below. z
𝑦=4− 𝑥 2 + 𝑧 2 z
𝑦=4+ 𝑥 2 + 𝑧 2
Both of these surfaces are called cones. y y 4 4 x x
Quadratics
Surfaces of the form ± 𝑥−ℎ 2 𝑎 2 ± 𝑦−𝑘 2 𝑏 2 ± 𝑧−𝑙 2 𝑐 2 =1 are called quadratics. The graphs are either a sphere, ellipsoid, hyperboloid of 1 sheet, or hyperboloid of 2 sheets. z
Example
Graph the surface 𝑥 2 + 𝑦 2 + 𝑧 2 4 =1
𝑥=0
Value
Trace Equation
𝑦 2 + 𝑧 2 4 =1
This shape is called an ellipsoid (a football shape) y
𝑥=0
𝑧=0
𝑥 2 + 𝑧 2 4 =1
𝑦=0 x
𝑧=0
𝑥 2 + 𝑦 2 =1
𝑦=0

5. Example
Graph the surface − 𝑥 2 + 𝑦 2 + 𝑧 2 =4
Notice the traces are hyperbolas when y and z are zero and circles of increasing radius when x gets larger in absolute value.
This is called a hyperboliod of 1 sheet.
Example
Graph the surface − 𝑥 2 − 𝑦 2 + 𝑧 2 =4
Notice the traces are hyperbolas when x and y are zero and circles of increasing radius when z gets larger in absolute value than 2.
This is called a hyperboliod of 2 sheets.
Missing Variable
If a variable is omitted from an equation that means all of the traces for that missing variable will be the same shape. Generally we call these shape cylinders. An adjective sometimes describes the shape of a cross section. z
Example
Graph 𝑧=4𝑦− 𝑦 2
For all values of x the traces equation is the parabola 𝑧=4𝑦− 𝑦 2
This shape is called a parabolic cylinder. y x

6. z
Example
Graph the surface 𝑥 𝑧 2 =1
Every trace for all values of y is an ellipse.
This shape is called an elliptical cylinder. x y z
Example
Graph the surface 𝑧= sin 𝑦
For all values of x the trace equation is a sine wave. y x