Algebra III
Lesson 54
Parabolas
Parabolas
Common formula Î y = x2
All have an axis of symmetry.
(y) = (x)2
This is the parabola centered
(bottom of ‘bowl’) at the origin
symmetric about the y-axis.
Getting the formula for a moved
parabola is the same as has been
done for moving any graph.
This is:
(y – 1) = -(x - -2)2
y = -(x + 2)2 + 1
General Formula of a Parabola
y = ax2 + bx + c
In this form it is hard to graph the parabola.
So, completing the square is used to convert this to
the common form.
Sample:
y = x2 + 4x – 3
y = (x2 + 4x
)–3
y = (x2 + 4x + 4) – 3 – 4
y = (x + 2)2 – 7
Example 54.1
Complete the square to graph y = x2 – 4x + 2.
y = (x2 – 4x
)+2
y = (x2 – 4x + 4) + 2 – 4
y = (x – 2)2 – 2
For these all that is needed is:
1) center at correct coordinates
2) Roughly correct (‘U’) shape
in proper orientation
Example 54.2
Complete the square to graph y = -x2 – 6x – 8.
y = -(x2 + 6x
)–8
y = -(x2 + 6x + 9) – 8 + 9
y = -(x + 3)2 + 1
Practice
a) Complete the square to graph y = -x2 – 4x + 6.
y = -(x2 + 4x
)+6
y = -(x2 + 4x + 4) + 6 + 4
y = -(x + 2)2 + 10
b) An automobile whose wheels are 70 centimeters in diameter is
traveling at 30 kilometers per hour. What is the angular velocity of the
wheels in rad/min?
v=rω
ω = v/r
=
30kph
35cm
6 km
=
1 hr
6km
7cm ⋅ hr
1000 m
7 cm·hr 60 min 1 km
= 1428.57 rad/min
100 cm
1m
c) Convert 40 miles per hour to centimeters per second.
40 mi
hr
1 hr
1 min
60 min 60 sec
5280 ft
12 in
2.54 cm
1 mi
1 ft
1 in
40 × 5280 ×12 × 2.54 m
=
60 × 60
s