By: Keith Schulte
Parametric Equations are another way to write functions to be graphed. When we have the continuous functions x = f(t) and y = g(t), they define ordered pairs (x,y). With parametric equations, the range of the graph that we can see, is determined by the range of ‘t’ that we are using. If we remember our unit circle from trigonometry, we must have a range from 0 to 2 pi, to go completely around the unit circle one time.
The equation for the unit circle, written as Cartesian coordinates would be:
One is the radius of the circle. In Trigonometry, we learn that cosine is the ‘x’ value and sine is the’y’ value. We use this idea to construct our parametric equation for a circle. The unit circle in parametric form would be:
Where a and b are the radial measures of x and y. When they are both equal to one and the range of t is from 0 to 2 pi, we would have the following unit circle, with the center at the origin, of course!
As long as a and b are the same we will have a circle. If a=b=1, we get the unit circle. If a=b=2, then we have a circle with radius of 2. A quick reminder, to see the whole circle, the range of t must be from 0 to 2 pi! If we only had a range of 0 to 1pi, what would we see? Look below!
What would happen if a and b were not the same? What curve would we get if a < b? What if a > b? Since we know that a is the radial measure of ‘x’ and b is the radial measure for y, if a=1, then our radial measure along the x-axis will be one and if b is equal to 2, then we would have a radial measure of 2 along the y axis. So we should see an ellipse centered at the origin, with the bottom at –2 and top at +2 and the sides at –1 and 1. In the graph below, we have blue when a = 1 and b = 2; and purple when a = 2 and b = 1.
So, when we have the parametric equation:
If a=b, we have the equation of a circle and if a and b are different, we have an ellipse, given an appropriate range of t.
How would we change our equation to move the center of the circle or ellipse from the origin, to (1,1) or (-1,1) or (-1,-1) or (1,-1)? Let’s look at the circles first. Look at the graph below before you look at the equations below the graph. How did I change the parametric equation to get the blue circle, the purple one, the red one, the black one?
Did you figure out how I moved them? If you look at the parametric equation above, we said that a and b are the radial measures for x and y, but what in the equation tells us where the graph is to be centered? If we add zero to acos(t) and zero to bsin(t), we don’t change the equations, but now we see that the graph will be centered at the (x,y) coordinates of (0,0). Now do you see how to change the equations? The equation to move the center of the circle from (0,0) to (1,1) – the blue circle above is:
Try the others for yourself! You can move the ellipses the same way!
What kind of curve would we have from the following equation?
With a=b=1 we have a line segment connecting (0,1) and (1,0). Remember that our parametric equation is centered at (0,0), unless we change it. Here, we haven’t changed it, so a is the distance from the origin along the x-axis and b is the distance from the origin along the y-axis. When we have a=1 and b=2, we would have a line segment from (0,2) to (1,1). So our graph of this equation when a=b=1 (in blue) and when a=1 and b=2 (in red) look like this:
By changing a and b from positive numbers to negative numbers, such as a=b=-1, the line segment would go from (-1,0) to (-1,-1). Can you change a and b so that you would get the following diagram? What did you need to do?
The red segment is when a=b=1, the blue segment is when a=b=-1, the green segment is when a=-1 and b=1, the black segment is when a=1 and b=-1. The purple circle is from the previous section with the equation below, with a=b=1:
We could also inscribe line segments on ellipses from the previous section, like below. What are a and b in the graph below? What is the equation for the ellipse?
What would the graphs look like if the equations were like this?
If we have these equations, what happens when we change a and b? We can look at these equations when a=b=1, when a=2 and b=1 and when a=4 and b=1. These graphs would look like this:
We see that these graphs are similar in that a and b are still the radial measures along the x and y axes.
For more fun, explore equations of higher degree. When the equations were to the first power, we had circles if a=b and ellipses if a was not equal to b. For second degree equations, we had line segments. For third degree equations, we had the graphs above. What would you expect if you had higher degree equations with even exponents and with odd exponents?