Exploring Parametric Equations

by

Laura Kimbel

A parametric curve in the plane is a pair of functions x = f(t) and y = g(t) 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 to the range of t .

Graph several sets of curves for x = a + t and y = b + kt for selected values of a, b, and k in an appropriate range for t.

Let's fix a and b and investigate how k affects the graph of (x, y).

x = 1 + t

y = 1 + kt

In this video, our value of k varies from -10 to 10. You can see that this is the graph of a line, and as we change k, the slope of the line is affected. However, we can also see that 1 point on this graph does not change no matter what value of k we have. That is the point that goes through (x, y) = (1, 1). This is when t = 0 since x = 1 + t and y = 1 + kt in this particular situation.

Let's now set b and k and let the value of a vary.

x = a + t

y = 1 + t

Here, b = 1, k = 1, and a is changing from -5 to 5. We observed earlier that k affects the slope. We can see here that the slope of this line remains the same as a changes and the line is being translated. If we pause the video, we can find out what that slope is. The slope of this line is 1.

Let's see what happens to a if b and k are fixed but unequal.

x = a + t

y = 1 + 2t

Even though b and k are fixed but unequal, this is consistent with our previous videos. k has affected the slope of our line and a performs a shift as it varies from -5 to 5.

x = 2 + t

y = b + 2t

Let's now fix a = 2 and k = 2 and have the value of b vary from -5 to 10.

By letting our b value change, we can see that we have a translation of our line, with our slope remaining the same throughout. However, it is interesting to note that in this case, I found that I needed to extend my domain of b to [-5, 10] to produce the same picture we had for our analysis of a. This is because the value of k is 2.

Let's look at 1 more example as b changes where a and k are fixed but not equal to each other.

x = -1 + t

y = b + (1/2)t

We have a new slope yet the other characteristics of the graph are consistent with the previous one. However, b now varies from -5 to 2.5. This is because the value of k is 1/2.

Parametric equations are quite interesting. Since they can become very complicated, the use of technology is extremely important in exploring the characteristics or these equations.