CONIC SECTIONS
If someone were
to slice a cone, the shape of the slice might produce a parabola, a circle, an
ellipse, or a hyperbola. This section
describes these shapes.
PARABOLAS
Parabolas look a
little like cups. The simplest
parabolas are of the form y=
and the sign of the 
term determines
whether we have an "up cup" or a "down cup."
The equation y =
has the graph
And the equation
y=-
term has the graph
The standard
form of a parabola is
y = a
+ bx + c
where the line
of symmetry, or the line that cuts the x-axis in half, is the line
The roots of a
parabola can be found by solving the quadratic equation
y = a
+ bx + c = 0
The roots is
where the parabola crosses the x-axis.
You probably
won't need to memorize this or use it, but the roots can also be found by using
the quadratic formula:
Example:
y = 
- 5x + 6
Setting the
value of y equal to 0 and factoring yields
-5x+6 = 0
(x-2)(x-3) = 0
Therefore,
either x-2=0 or x-3=0.
If x-2=0 then
x=+2, and if x-3=0, then x=+3.
The values of x
that satisfy the equation are 2 or 3
The roots of the
equation can also be found using the quadratic formula:
a=1, =-5, and
c=6
Therefore, x =
2, 3.
The line for the
axis of symmetry is
The graph of y =
-5x+6 is shown below.
Note that the axis of symmetry is the line x=2.5.
Example:
For the equation y = 
+4x-5 sketch the parabola and find the axis of symmetry.
We can factor to
find the roots:
y=
+4x-5=(x-1)(x+5)
Then if y=
+4x-5=0, x-1=0 or x+5=0, so x=-1 or 5.
Using the
quadratic formula to find the roots we have
Therefore,
So that x=2, -5
And the axis of
symmetry is the line
CIRCLES
The standard
form of a circle is 
where the center is
(h, k) and the radius is r.
A circle with
radius 1 and center at (0, 0) is 
+
= 1.
Example: the circle with center (4, -2) and radius 5
has the equation
.
Problem: What is the equation of the circle below?
Note that the
center is (-6,1) and the radius is 6 so that the equation of the circle is
ELLIPSES
Ellipses are a
little like stretched circles.
The standard
form of an ellipse is
where the
x-intercepts are (-a, 0), (a, 0) and the y-intercepts are (b, 0), (-b, 0).
The x-intercept, for example, is found by
setting y = 0, so that
and 
so x = 
a.
For example, the
equation of the ellipse
has x-intercepts
4 and y-intercepts 
3.
HYPERBOLAS
Hyperbolas are a
little like pairs of parabolas but with different shapes.
A hyperbola with
the center at the origin and that opens to the side has the form
where the x
intercepts are 
a.
For example:
has x intercepts
-3, 3.
A hyperbola with
the center at the origin and that opens to the top and bottom has the form
where the
y-intercepts are 
b.
For example:
has y-intercepts
at -4, 4.
Shortcut: If the 
term is positive, then
the hyperbola crosses the x axis.
If the 
term is positive,
then the hyperbola crosses the y
axis.
We might also
encounter a hyperbola of the form
xy = 
,
where the
hyperbola intercepts the line y=x at (c, c) and(-c, -c)
Example: xy = 1
Example: xy = 4
Lastly, if the 
term is negative, the
line of symmetry will be the line y=-x.
Example: xy = -4
SUMMARY
|
Type of Curve |
Description |
Examples |
|
Line |
No terms
squared |
y = mx + b ax + by = c |
|
Parabola Form: a |
One term
squared |
y = a Roots: solution of a or
Line of
symmetry:
|
|
Parabola Form: y = |
One term squared |
y = Vertex = (h,
k) |
|
Circle |
Both terms
squared, x, y have same
coefficient |
Radius: r |
|
Ellipse |
Both terms
squared, x, y have
different coefficient |
|
|
Hyperbola Form:
|
Both terms
squared, coefficients have different signs |
x intercepts
are ± a.
y-intercepts
are ± b. |
|
Hyperbola Form: xy= |
Neither term
squared |
x y = Hyperbola
intercepts the line y=x at (c, c) and(-c, -c) |
+ bx + c = 0
+ bx + c
+ bx + c = 0
+ k 
+ 
Example: Which of the following equations describes
the graph below?
A. y = (x+3)
+ 1
B. y = (x-3)
- 1
C. y = (x-3)
+ 1
D. y
- 1 = x - 3
E. y
+ 1 = x - 3
Answer: C
The vertex of
the equation y = (x-h)
+ k is (h, k). In
this case, the vertex of the parabola is (3, 1) so that y = (x-3)
+ 1 is the equation of the graph.
Example: Find an equation which describes a circle
with radius 5 and center at (5,5).
A. (x + 5)2 + (y + 5)2 =
1
B. (x + 5)2 + (y + 5)2 =
25
C. (x-5)2 + (y-5)2 = 25
D. (x + 5)2 + (y + 5)2 =
50
E. (x - 5)2 + (y - 5)2 =
50
Answer: B
The formula for
a circle is (x-h)2+(y-k)2=r2
where r is the
radius and (h, k) is the center.
Therefore, h=5,
k=5 and r2=25.
Looking
back: Draw a picture, plug in.
Choose an easy
point like (5,10). Remember, do not do
all the computations, just enough to eliminate answers!
A. (5 + 5)2 + (10 + 5)2 =
100 + 225 ¹ 1
B. (5 + 5)2 + (10 + 5)2 =
100 + 225 ¹ 25
C. (5-5)2 + (10-5)2 = 0
+ 25 Correct answer!
D. (5 + 5)2 + (10 + 5)2 =
25 + 225 ¹ 50
E. (5 - 5)2 + (10 - 5)2 =
0 + 25 ¹ 50
Example: For how many values of x will y = 0?
y =
x(x-5)(x+2)(x+4)
A. 0
B. 1
C. 2
D. 3
E. 4
Answer: E
Solution: If y = 0 then x(x-5)(x+2)(x+4)=0. Solving for x we will have four
solutions:
x=0,
x-5=0Þx=0,
x+2=0Þx=-2, and
x+4=0Þx=-4.
Example: Which of the following equations describes
the graph below?
Answer: C
The standard
form of an ellipse is
where the
x-intercepts are (-a, 0), (a, 0) and the y-intercepts are (b, 0), (-b, 0).
Therefore, if
the x-intercepts are 
2 and the y-intercepts are 
5,
represents the
equation of the ellipse.
Example: Which of the following graphs is an example
of the curve xy = c?
A. 
B. 
C. 
D. 
E. 
Answer: D
The line xy=c
describes a hyperbola. Note that xy = c
is an inverse relationship. As x
increases, y decreases; as y decreases, x increases.
A. (x+2)2+(x-3)2=36
B. y2-x2=4
C. 
D. xy = 9
E. y=x2+4 and -y=x2-4