10-5 Precalculus A 2016: Parabolas
Standard Form
of the Equation of
a Parabola
Orientation when p 0
Description
Standard Form
of the Equation of
a Parabola
y
(x h)2 4p(y k)
x
O
F
V
vertex: (h, k)
focus: (h, k p)
axis of symmetry: x h
directrix: y k p
opening: upward if p 0
downward if p 0
Orientation when p 0
y
V
(y k)2 4p(x h)
F
x
O
Description
vertex: (h, k)
focus: (h p, k)
axis of symmetry: y k
directrix: x h p
opening: right if p 0
left if p 0
For the equation of each parabola, find the coordinates of the
vertex and focus, and the equations of the directrix and axis of
symmetry. Then graph the equation.
1. x 2 2x 8y 17 0
2. y 2 6y 9 12 12x
Write the equation of the parabola that meets each set of
conditions. Then graph the equation.
3. The vertex is at (2, 4) and
the focus is at (2, 3).
4. The focus is at (2, 1), and the
equation of the directrix is x 2.
General Form
for the
Equation of
a Parabola
The general form of the equation of a parabola is
y 2 Dx Ey F 0, when the directrix is parallel to the y-axis, or
x 2 Dx Ey F 0, when the directrix is parallel to the x-axis.
parabola
ellipse
hyperbola
e1
e 1, e 0
e1
P
M
F
P
e PF
PM
e1
F
e PF
PM
M
F
e1
M
e PF
PM
e1
F
P
F
For those conics having more than one focus and directrix,
F and represent alternates that define the same conic.
5. Satellite Dish Suppose the receiver in a parabolic dish antenna
is 2 feet from the vertex and is located at the focus. Assume that
the vertex is at the origin and that the dish is pointed upward.
Find an equation that models a cross section
of the dish.
6.
7.
Business
An airline has been charging $140 per seat for a one-way flight.
This flight has been averaging 110 passengers but can transport up to
180 passengers. The airline is considering a decrease in the price for a
one-way ticket during the winter months. The airline estimates that for each
$10 decrease in the ticket price, they will gain approximately 20 passengers
per flight.
a. Based on these estimates, what ticket price should the airline charge to
achieve the greatest income on an average flight?
b. New estimates reveal that the increase in passengers per flight is closer to
10 for each $10 decrease in the original ticket price. To maximize income,
what should the new ticket price be?
Construction The Golden Gate Bridge in San
Francisco, California, is a catenary suspension
10 ft
500 ft
bridge, which is very similar in appearance to a
parabola. The main span cables are suspended
between two towers that are 4200 feet apart
4200 ft
and 500 feet above the roadway. The cable
extends 10 feet above the roadway midway
between the two towers.
a. Find an equation that models the shape of the cable.
b. How far from the roadway is the cable 720 feet from the bridge’s center?