Exploring Conics with Graphing Technology
Richard Parr
Rice University School Mathematics Project
There are several ways to explore conic sections on the TI-84+ graphing calculator. In
this presentation we will look at not only the CONICS Application on the TI-84+
calculator but also how to use function, parametric and polar graphing modes to draw
more mathematics out of the study of conic section.
First let lets look at conic sections using function mode. For conic sections expressed in
standard form, solving the equation for y in terms of x allows. All conic sections except
for parabolas with vertical axes of symmetry can be represented by two functions. For
example to graph the ellipse:
1
9
)2(
16
)3(
22
=
+
+
− yx
We can solve this equation for y resulting in:
16
)3(9
92
2
−
−±−=
x
y
Graphing these equations in a standard window results in the following graph:
A couple of comments need to be made about this graph in the standard window. First, it
is obvious that ellipse does not look like a closed figure. Second it is not in the correct
aspect since the units on the x-axis are longer than units on the y-axis. A much better
graph can be viewed by using the viewing window [-9.4, 9.4] x [-6.2, 6.2]:
In Pre-Calculus, we can study conic sections that have axes that are not parallel and
perpendicular to the coordinate axes. All conic sections have an equation in general form
given by:
0
22
=+++++ FEyDxCyBxyAx
This equation can be re-expresses as a quadratic in y:
0)(
22
=+++++ FDxAxyEBxCy
Using the quadratic formula this can be solved for y:
C
FDxAxCEBxEBx
y
2
)(4)()(
22
++−+±+−
=
This is a cumbersome expression to enter in the calculator. A nice calculator program can
be written that will automatically graph the two functions. The program can be modified
to take care of special cases and to give the angle of rotation.
For example graphing the conic section:
020886
22
=−−+++ yxyxyx
Results in the following graph:
For rotated conic sections, the discriminant ACB 4
2
− determines what type of conic
section is present. If 04
2
>− ACB the resulting figure is an hyperbola, if 04
2
<− ACB
the conic is an ellipse and if
04
2
=− ACB
then the figure is a parabola.
Now let us turn our attention to conics represented in parametric form.
The Pythagorean trigonometric identities allow for easy parametric representation of
ellipses and hyperbolas. Parabolas are most easily represented without the use of
trigonometry.
Ellipses
A comparison of the Pythagorean identity: cos si
n
22
1tt
+
=
and a standard form for the
equation of an ellipse :
()
xh
a
yk
b
−
+
−
=
2
2
2
2
1
()
allows for two simple substitutions :
cos
()
2
2
2
t
xh
a
=
−
and sin
()
2
2
2
t
yk
b
=
−
Solving these two equations for x and y yields a pair of parametric equations:
x
ath=+cos
yb tk=+sin
A specific example; to graph
()()xy−
+
+
=
3
9
2
4
1
22
on the TI-83, one would put the
calculator in parametric mode. The choice of degree or radian mode is one of personal
choice. For the first examples I would do in my class I would use degree mode. Then
with a window of [0, 360]
5
x [-15, 15]
1
x [ -10,10]
1
, the appropriate equations and
graphs are shown:
A few personal comments are important at this point:
I chose substitutions I did to reinforce the use of x and y coordinates of a unit circle to
represent sine and cosine respectively. In using this method I am de-emphasizing the idea
that “a” corresponds to the major axis, etc. I focus on the idea that “a” is a stretch in the x
equation and therefore a horizontal stretch. Likewise, “b” is a vertical stretch. I’d just as
soon not use the letters “a” and “b” at all, but focus on the major axis being the axis with
the “largest” stretch.
Some students see a contradiction in the transformation in parametric representation
when compared to the Cartesian representation, by re-writing
x
t
=
+33cos in the
form
x
t−=3 3cos , I try to show that there is really no contradiction.
Hyperbolas
By using the Pythagorean identity: sec ta
n
22
1tt
−
=
and a standard form for a hyperbola
:
1
)()(
2
2
2
2
=
−
−
−
b
ky
a
hx
One can derive the following pairs of parametric equations to represent hyperbolas:
x
ath=+sec
x
bth
=
+
ta
n
yb tk=+tan ya tk
=
+
sec
(horizontal transverse axis) (vertical transverse axis)
In a hyperbola, unlike an ellipse, it makes a difference which trigonometric function
corresponds with which variable. Therefore using the sane window settings as before to
graph the hyperbola
()()yx−
−
+
=
2
25
1
36
1
22
on the TI-83 we get as a result:
Parabolas
Parabolas are most easily graphed parametrically without the use of trigonometric
functions. All non-rotated parabolas can either be written in the form yfx= ()
or xfy= ( ). Parametrically, parabolas that can be written yfx
=
( )can be graphed using
x
t= and yft= ( ) , likewise parabolas that can be represented as xfy
=
( ) can be
graphed parametrically using xft= () and
yt
=
. In this case the T step of the window
must be adjusted to include negative values for t or the entire parabola will not appear.
Extensions
Some possible ideas for extension include trying to develop a trigonometric method of
graphing parabolas parametrically, and extending the use of parametric graphing to graph
rotated conic equations.
Polar graphing of conic sections:
In polar mode, conic sections with one focus at the pole (origin) can be graphed. The
general equation of a polar conic can be expressed by the equation:
θ
sin1 e
ep
r
±
= (horizontal directrix)
or
θ
cos1 e
ep
r
±
=
(vertical directrix)
Where e is the eccentricity of the conic and p is the distance from the focus to the
directrix. Polar equations of conics can be used to emphasize the definition that
eccentricity is the ratio of the distance from a point on the graph to the focus to the
distance of the same point to the directrix. If the eccentricity is greater than 1, the graph is
a hyperbola, if the eccentricity is between zero and one the graph is an ellipse, and if the
eccentricity is 1 the graph is a parabola.
By drawing the directrix in polar form you can illustrate this relationship of the polar
conic.
For example:
Finally let’s look at the conics application. The conics application on the TI-84+ allows
us to easily graph conic sections in any of the three ways that we have looked at
previously. We will take a tour of the screens that appear when choosing the various
modes on the application:
Function mode:
Parametric mode:
Polar Mode:
The conic application does allow you to easily graph conic sections with ease; however
the problems with the applications need to be noted:
•
Conic sections with axes that are not parallel or perpendicular to the coordinate
axes can not be graphed.
•
Automatic window can cause make all ellipses look similar
•
Difficult to add additional graphs to the graph of the conic sections
Overall there are many ways to study conic sections using the TI-84+ , that can make the
once difficult study of 2
nd
degree relations easier for students to visualize.
