In Exercises 21–40, eliminate the parameter t. Then use the re...

Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = t, y = 2t

Accepted Answer

Welcome back everyone. In this problem, we want to write the corresponding rectangular equation for the parametric equation X equals three T and Y equals seven T where T is between negative infinity and positive infinity. We also want to draw a graph of the plane curve using that rectangular equation. And on our graph, the values of X are between negative 2015 and Y is the values of Y are between negative 10 and 10. We also want to indicate the direction of the curve obtained corresponding to increasing values of T using arrows. Now, what are we trying to do here? Well, basically, we're trying to convert this parametric equation that we have into a rectangular equation. And that just means we want to rewrite the equation purely in terms of X and Y eliminating our value of T. So let's do that first. Before we draw our graph, now we know that X equals three T. OK. And we know that this is a linear equation. So all values of X and all values of T would be valid. So if X equals three T and Y equals seven, I can rewrite X in terms of T because T would be equal to X divided by three. And no, that means why instead of being 70 it would have been seven multiplied by X divided by three, which would give me seven X divided by three. So that would be our equation for Y. So now we have our rectangular equation. OK. Now that we have that, let me put rectangular equation right here right beside it. Now that we have that we can go ahead and plot our graph. Now, the interesting thing about this, like we said like and as you notice, it's a, it's a linear equation. So that means it's going to form a line. So our graph is probably going to end up looking something like this. Now, if we're gonna plot that graph, that means we only need two points because we're talking about the line and we can choose any two random points from our grid. No, iii I want to be smart about this. We, we can just use two. So let's say when X equals zero, OK? Now when X equals zero, then Y would be equal to seven multiplied by zero, divided by three, which is zero. So 00 is a point on our graph. And let's also use the value of X being equal to three. So when X equals three, then Y equals seven multiplied by three, all divided by three, the values of three factor out. And that just leaves us with Y being seven. So 37 is also another value on our graph. Now that we have two points, we can go ahead and plot. So one point is at 00 and the next point would be at 77, no sorry 37 when X is three Y is seven. So those would be both our points. And now let's go ahead and plot our line and our line. It's probably going to look something like that. Let me extend it in the opposite direction as well. So now we have our line, OK. Now we have our line and let me write the equation of the line beside it, Y equals seven thirds of X. Finally, we need to indicate the direction of the curve. Now, what's the direction? Well, as you can notice it's going in a positive direction. So our curve would be pointing in a positive direction like that. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Parametric Equations

Eliminating the Parameter

Orientation of Parametric Curves

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