By Na Young Kwon
A parametric curve in the plane is a pair of functions
x = f(x)
y = g(x)
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.
2. For various a and b, investigate
x = cos (at)
y = sin (bt) for 0<= t <= 2¹
At first, let a and b be the same number, say a=b=1.(0<t<2¹)
We can check this graph is a circle. Investigate the graph changing value.
When a and b have the same value the graph is always a circle regardless
of the value. (Only 0 < t < 2¹). We can check this truth algebraically.
For X=cosat , Y= sinat for any value of a .
Then by the rule of
Next, change the b value. To compare graphs we fix a value of a
is 1. (0 < t < 2¹).
When the value of b is changing, a number of curves increase.
I investigate graphs in the case that b is even and odd. Especially,
when b is odd, the graph passes a point (1,0) and (-1,0) and
the shape is similar to the graph of coskx and -coskx.
In this case we can find that each graph has concave curves
above the x-axis by b numbers. When b is even, the graph
passes an original point and the shape is similar to sinkx.
In this case each graph has also concave curves above
the x-axis by b numbers. And all the graph have the symmetry
graph by x-axis and y-axis.
Next, change the value of a. Fix b is 1(0 < t < 10).
We can find an interesting point in this graph.
For example, when a value of a is equal to 3 and b=1, the graph
is congruent to the graph which is rotated a graph of b=3 and a=1
by 90 degrees. I found the result that a graph which is x=cos(at)
and y=sin(bt) is rotated a graph which is x=cos(bt) and sin(at)
by 90 degrees when a and b is odd. Strangely, when b=1 and
a value of a is changing, any graph doesnÕt pass an original point.
In this case all graphs passes (0,1).
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