1. Reflective Property of Parabolas
One more day in Sec. 8.1b…

2. But first, let’s start with a problem:
Prove that the graph of the given equation is a parabola, and
find its vertex, focus, and directrix.
Write in standard form:
V: (1, 7/6), F: (1, 5/3), D: y = 2/3

3. More Practice V: (13/4, 1), F: (9/4, 1), D: x = 17/4
Prove that the graph of the given equation is a parabola, and
find its vertex, focus, and directrix.
Write in standard form:
V: (13/4, 1), F: (9/4, 1), D: x = 17/4

4. More Practice Write an equation for the parabola graphed below.
General equation:
Substitute in vertex:
(11/2,0)
Substitute in point, solve for p:
V(1,–3)
Final Answer:

5. Reflective Property of Parabolas
If a parabola is rotated in 3-D space around its axis, the
parabola “sweeps out” a paraboloid of revolution.
These are important because a signal (light, sound, radio
waves, etc.) emanating from the focus reflects off the
surface of the dish in lines parallel to the axis (and vice
versa – incoming parallel waves are sent directly toward
the focus…).
Check out Figure 8.9 on p.637!!!

6. Practice Problems
During football games, a TV network uses a parabolic reflector
with a microphone at the reflector’s focus to capture the
conversations among players on the field. If the parabolic
reflector is 3 ft across and 1 ft deep, where should the
microphone be placed?
General equation:
A diagram of a cross-section
of the reflector:
Substitute the points:
(–1.5, 1)
(1.5, 1)
F(0, p)
V(0, 0)

7. Practice Problems
During football games, a TV network uses a parabolic reflector
with a microphone at the reflector’s focus to capture the
conversations among players on the field. If the parabolic
reflector is 3 ft across and 1 ft deep, where should the
microphone be placed?
A diagram of a cross-section
of the reflector:
Because p = ft,
or 6.75 inches, the
microphone should be
placed inside the
reflector along its axis
and 6.75 inches from
its vertex.
(–1.5, 1)
(1.5, 1)
F(0, p)
V(0, 0)

8. Practice Problems Because p = 1.125 cm, the filament should be placed
The mirror of a flashlight is a paraboloid of revolution. Its
diameter is 6 cm and its depth is 2 cm. How far from the vertex
should the filament of the light bulb be placed for the flashlight
to have its beam run parallel to the axis of its mirror?
Because p = cm, the filament should be placed
1.125 cm from the vertex along the axis of the mirror.

9. Practice Problems Lengths (in feet): 79.444, 54.444, 35, 21.111,
A main cables of a suspension bridge uniformly distribute the
weight of the bridge when in the form of a parabola. The main
cables of a particular bridge are attached to towers that are
600 ft apart. The cables are attached to the towers at a height
of 110 ft above the roadway and are 10 ft above the roadway
at their lowest points. If vertical support cables are at 50-ft
intervals along the level roadway, what are the lengths of these
vertical cables?
Lengths (in feet): , , 35, ,
12.778, 10, , , 35, ,