We are investigating the following equation:
To begin with, we will let the domain for our t's range from -2 pi ... 2pi. Our graph looks like this:
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We can derive the rectangular equation from this based on the slope and y-intercept, and get the following:
How do we arrive at that though?
We know that y = 2t -1. We also know that x = t + 1, so t = x - 1. Now we substitute in the y equation and we get y = 2 (x - 1) - 1 = 2x - 3.
So parametrics can be changed to rectangular coordinates by a matter of solving for t and substituting. How does changing a constant a little effect our overall graph? Let's look at the generic equation:
Let's solve. Since x = at + b, we derive:
Substitute this in our y-equation:
After a little simplification, we get the following:
So now let's examine problem # 7 ...
Problem # 7
" Write parametric equations of a line through (7, 5) with slope of 3. Graph the line using your equations."
First, we put that equation in rectangular form:
So for the sake of simplicity, let's make a = 1 and c = 3. Now we have the following:
Where 3b + d has to be 16. Let's make b = 5 and d = 1. This gives us the following final setup:
On the other hand, we can have infinitely many equations. For example, if a = 2, the only thing that has to change is c:
We can see that relationship ... On the other hand, if we were to vary our b, it would effect only our d. Let's make b = 3:
All those are acceptable solutions to our given problem. Let's refresh our setup that we are working with:
Generalizing this process to ANY line written in the form y = mx + b (which we can find given a point and slope), we get the following conditions:
And then it's just a matter of plugging the numbers in.