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

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 = 2ᵗ, y = 2⁻ᵗ; t ≥ 0

Accepted Answer

Welcome back everyone. In this problem, we want to write the corresponding rectangular equation for the parametric equation X equals three, raise to the power of ty equals three race to the power of negative and is greater than or equal to zero. By eliminating T afterwards, we want to draw a graph of the plane curve using that rectangular equation and indicate the direction of the curve obtained corresponding to increasing values of T. By using arrows for our graph, we have our X axis that is bounded by negative one and seven and our Y axis that's bounded by negative one and five. No, that sounds like a lot. But let's see if we can make sense of it. Let's try to break it down. What do we know? Well, here we have a parametric equation and that means it's some equation that relates the coordinates of X and Y by a common parameter. T basically what we're trying to do is eliminate that common parameter. So we want to look at both equations, see if we can get that common parameter out and then go ahead and plot our graph on screen. So let's see if we can figure that out. What do we know about our parametric equations? Well, we know that X equals three race to the power of T and Y equals three race to the power of negative T. But if you think about it, those two are pretty similar and we can use what we know about in the sea to help recall that in the laws of indices, it tells us that if we have a base raise to a negative power is the same thing as the reciprocal of the base risk to the power. OK. So a to the negative end is the same thing as one divided by areas to the power of negative N. So that tells us then that and if Y equals three to the power of negative T is the same thing as saying Y is equal to one divided by three races to the power of T. But remember we know that X equals two to the power of T. So therefore, we can substitute in our equation for Y and that tells us that Y equals one divided by X. So now we have eliminated T and we have our rectangular equation since we have that rectangular equation. The next thing we want to do is to draw a graph of the plane curve that it represents no Y equals one divided by X, one divided by X is an X A synoptic equation. And that means it's a curve that runs as close as it can as possible to one of the axis without touching it. OK? No, the only thing we need to keep in mind for this curve is that there's one limitation, it works only if X is not equal to zero. And that's because if X equals zero, then our denominator will be zero and one divided by zero is undefined. So if we're going to draw it, we know that it's gonna look something like this, but it's gonna look that way once X is not equal to zero. So let's start at that point. We know over here that's gonna be invalid. What about when X equals one? Let's take at least one point. So we know where to start. Well, when X equals one, obviously, Y equals one divided by one, which is one. So our first point for a curve would be here afterwards, we have the general idea of what it would look like. It would be as asymptotic. So that means we can take it from there and continue it like a curve that's going all the way to the X axis but never touching it. And the general direction of our curve would be in the direction of the positive X axis. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Parametric Equations

Eliminating the Parameter

Graphing Rectangular Equations

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