A parametric curve in the plane is a pair of functions
where the two continuous functions
define ordered pairs (x,y). The two equations are usually called the parametric
equations of a curve. The extent of the curve will depend on the range of t and your work with parametric
equations should pay close attention the range of t . In many applications, we think
of x and y "varying with time t " or the angle of rotation
that some line makes from an initial location.
Various graphing technology, such as the TI-81, TI-82, TI-83, TI-85, TI-86, TI-89, TI- 92, Ohio State Grapher, xFunction, Theorist, Graphing Calculator 3.2, and Derive, can be readily used with parametric equations. Try Graphing Calculator 3.2 or xFunction for what is probably the friendliest software.
Note: Graphing technologies compute values of (x,y) for
increments of t and then construct a
line segment connecting them. When the increment of t is small then these are very short segments and the curve
is simulated. The TI instruments include a 'step' setting for the increments of
't' and it is possible for consecutive (x,y) to be rather far apart. This can produce
interesting drawings but misrepresent the parametric curve given by the set of
points. In other words, the 'step' setting is a way of drawing segments between
regularly spaced but not adjacent points on the parametric curve.
for different values of a and b. What is the curve when a < b? a = b? a > b?
LetŐs start with Case 1: a<b
LetŐs first try a=1 and b=4
The graph is an ellipse and its major axis is vertical.
You can see that the vertices of the ellipse is (0, +4) and (0, -4).
Remember that the standard equation for an ellipse whose major axis is vertical can be found by:
The greater the eccentricity of an ellipse, the more elongated the ellipse. Eccentricity (e) can be found by
LetŐs look at several different graphs with values of a and b such that a<b.
Can you tell which equations go with which of the graphs? While youŐre at it, note the eccentricity of each ellipse.
This graph shows another important property of ellipses. The maximum and minimum x values can easily be seen along the x-axis and is the same as the b values, given by (0, b) or (0, -b). So as b gets larger, the eccentricity increases.
One last observation regarding a<b. What if a and b are negative? Does the ellipse have the same characteristics?
LetŐs look at some more graphs to see what happens.
The eccentricity of the ellipses are much the same as before. However, the major axis is horizontal when the values of a and b are negative.
CASE 2: a=b
LetŐs start with some graphs such that the values a=b.
a, b = -5
a, b = 10
a, b = 5
a, b = 1
a, b = -3
Notice that each graph is a circle with the origin at (0,0) and with a radius equal to the absolute value of a and b. So why did I graph five equations, but there are only four graphs showing?
The answer is that a,b =5 and a,b =-5 graph to be the same.
Case 3: a>b
LetŐs take a look using Graphing Calculator 3.2
When a>b it is easy to see that the ellipsesŐ major axis is horizontal. The value for a determines the y maximum and minimum values.
What do you think will happen if a and b are negative values?
What is changed if the equations are
where h is any real number?
with graphs for small h (e.g. -3 < h < 3).
To figure out what is going on, letŐs look at some
graphs using the equations above.
LetŐs first try h=-3, let a=b
a, b = 10
a, b = 4
a, b = 5
a, b = 3
The equations graphed here are:
As a=b gets larger so does the graph.
So, what if we try h=-2 with the same a=b values above.
As h increases, the eccentricity decreases. Below are the graphs for h=0.
Graphing h= 1, 2, 3 the eccentricity of the ellipse corresponds with the eccentricity of h=-1, -2, and Đ3 respectively. However, it is a reflection of the negative values.
Playing around with the values of a, b, and h yields different ellipses some with more or less eccentricity then above. Click here to see a dynamic graph with different h values. You can change the a and b values as well.
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