Parametric Curves

Graph x = cos (t) and y = cos (t) for 0 < t < 2 Pi.

How would you change the equations to explore other graphs?

A parametric curve in the plane is a pair of functions x = f(t) and y = f(t) where the two continuous functions define ordered pair (x, y). The two equations are usually called the parametric equations of a curve.

Let's first look at the original graph:

We can see that the equations form a circle with the center at (0, 0) and the radius of the circle is 1. It goes from -1 to 1 on the x-axis and on the y-axis from -1 to 1.

Now lets look at what happens when we put a constant infront of both equations. How is the graph going to change?

When we put the same number infront of both equations, the graph remains a circle where the radius is the constant by which we multiply the equation.

What about when the two constants we are multiplying the equations by are diffferent? Let's look.

We notice when the constants are different the graphs form ellipses, instead of circles. However, the center still remains at (0, 0). The constant which we multiply the x parametric equation by is how far the ellipse reaches on the x-axis. The constant which we multiply the y parametric equation by is how far the ellipse reaches on the y-axis. Notice that the negative sign does not have much effect on the equations. The ellipse looks no different whether the constants are negative or positive.

Now lets look at what happens when you use a scalar in the parametric equations. In one set, a is held constant at 1 and b is changed, while in the other set, b is changed and a is held constant at 1.

Let's look at just one now to see if we see anything that it is more difficult to see when plotting multiple equations.

First we notice that the graph is contained in a box ranging from -1 to 1 on both axes. When a is odd, the graph does not end. It continually goes around and around. When a is even, there are two tails that are disjointed. Let's say that the scalar for the x parametric equation is 3, then there are three curves which touch the edge of the box along the line x = 1 and x = -1.

When b is odd and a is constant at 1, they all connect and there are no tails. Let's say that the scalar for the y parametric equation is 4, then there are three curves which touch the edge of the box along the line y = 1 and y = -1. All the curves are symmetrical across both axes.

What happens when you square the parametric equations?

When the equation is squared and there is no scalar in the equation, but there is a constant, the graph forms a straight line running from a to b, whether positive or negative. The y-intercept point is b and the x-intercept point is a. When there is a scalar, the curve remains in the first quadrant because the quantity is squared, so the value always remains positive.

If you want to know more, feel free to investigate yourself in graphing calculator.

By Carolyn Amos