how do you find the rectangular equation for the plane curve defined by the parametric equations: x=t-3 and y=t^2+5?
finding rectangular equations from parametric equations
Converting parametric equation to a cartesian equation or rectangular form involves solving for t in terms of x and then plugging this into the y equation. Therefore,
y = (x+3)^2 + 5
y = x^2+6x + 9 + 5
y= x^2 + 6x + 14
Hope that helps. Jim
Yes, Jim is entirely correct.
I would only augment his answer by pointing out that in general you need to manipulate the parametric equations to eliminate the parametric variable.
It made the most sense to eliminate t by solving the first equation for t, in this particular problem, because that equation was linear so each value of x only corresponded to one value of t. Things would have gotten a bit messy in this case if you had tried to solve the second equation for t, in this problem, but in general you eliminate the non rectangular-grid parameter by whatever means possible if you want the expression converted to its rectilinear coordinates.
Which equation should be solved for the parametric variable depends on the problem -- whichever equation can be most easily solved for that parametric variable is typically the best choice. For instance had the problem been y = t -3, and x = t^2 + 5, I hope you see that solving for t in terms of y would make more sense, for exactly the same reasons already discussed.
So you solve for t in terms of x if that's easier, or solve for t in terms of y if that's easier, then put the result into the other equation. In this problem, the former was easier.