Parametric Equations

by Brant Chesser


Today we want to explore different parametric equations using Graphing Calculator with respect to COS and SIN involved in today's equation. We want to see in a graph as for t between 0 and two pi or approximately 6.28. We will change the values in A and B to investigate what is going on in today's assignment. Our first graph below is when A=5 and B=1.





We can see that with this equation with A=5 it stretches the graph out along the x-axis. It becomes an oblong shape with B=1 also because it seems like from our first observation that B will coincide with the distance along the y-axis. Lets explore another graph to get a better grasp on it.


Lets see what it will look like when we keep A=5 and set B=3.




As we might have expected while making B larger it becomes more of an egg shape because when B is equal to three then it will grow taller on the y-axis.


So we can keep the same graph and see what we get when we set A and B equal to each other. What should we get? A circle would be a good assumption now that we see what is going on in the equation when we change variables A and B.






As we can see when A and B are set equal to one another in this parametric equation then we will get a circle as our graph.


We find out that as B gets larger we will move more along the y-axis than the x-axis. So lets make B=7 and keep A=5 which will give us the following graph:




Now lets see the shapes as A is smaller than B which we would imagine to be similar to the equations before but now the objects should be stretched along the y-axis instead of the x-axis. So set A=2 and B=4 and we get this:



As we had thought the graph will go 4 down and 4 up along the y-axis corresponding to B and 2 positive and 2 negative along the x-axis corresponding to A.


We can stretch it out even further and let A=2 and B=7. We will get the following graph:



We see that the graph expands vertically with B being larger than before. So through our investigations we can now conclude that with t between 0 and 6.28 and being able to change A and B that with changes to A we can reduce and expand it along the x-axis. Changes made to B will allow us to reduce and stretch the shape long the y-axis. So we should now be able to tell exactly what figure we will end up with on the graph.


Whatever B is equal to the shape will correspond to the positive and negative of that number along the y-axis(vertical). Also, whatever A is set equal to, the shape will correspond to the positive and negative of that number along the x-axis(horizonal). So from now one when we are given this parametric equation we have a good idea of what the shape is going to look like on the graph.



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