Video Transcript
Consider the parametric equations 𝑥 of 𝑡 is equal to two minus 𝑡 and 𝑦 of 𝑡 is equal to two minus three 𝑡, where 𝑡 is greater than 0.5 and 𝑡 is less than three. Which of the following is the sketch of the given equations? Option (A), option (B), option (C), option (D), or option (E).
We’re given a pair of parametric equations. And we’re given five possible sketches of this pair of parametric equations. We need to determine which sketch represents our parametric equations. Remember, our values of 𝑡 must be bigger than 0.5 and less than three.
To do this, we need to recall what parametric equations are. We input a value of 𝑡. And then this gives us the 𝑥-coordinate and 𝑦-coordinate of our curve. In other words, by substituting different values of 𝑡 into our functions 𝑥 of 𝑡 and 𝑦 of 𝑡, we can find coordinates that our curve must pass through. But the question is, which values of 𝑡 should we substitute in?
Well, we know that our values of 𝑡 must be bigger than 0.5 and they must be less than three. But remember, here, our values of 𝑡 can be very, very close to 0.5, and they can be very close to three. This means the endpoints of our parametric curve will be close to the values where 𝑡 is 0.5 and 𝑡 is three. So we’d include the endpoints of this interval even though our parametric curve technically does not pass through these points. It will help us find out where the endpoints of our curve lie. We’ll then increase our values of 𝑡 by 0.5 each time.
Let’s now start finding the coordinates for these values of 𝑡. Remember, 𝑥 of 𝑡 is two minus 𝑡 and 𝑦 of 𝑡 is two minus three 𝑡. So when 𝑡 is equal to 0.5, we get 𝑥 of 0.5 is two minus 0.5. And of course we can calculate this is equal to 1.5. We can do the same with 𝑦 of 𝑡. When 𝑡 is 0.5, we get 𝑦 of 0.5 is two minus three times 0.5, which we can calculate is equal to 0.5.
So we now have our first point. But we can keep going. When 𝑡 is equal to one, we get 𝑥 of one is equal to two minus one. And 𝑦 of one is equal to two minus three times one. And if we calculate both of these, we get one and negative one, respectively.
We can do exactly the same when 𝑡 is 1.5. We get 𝑥 of 1.5 is two minus 1.5, and 𝑦 of 1.5 is two minus three times 1.5. And once again, calculating these, we get 0.5 and negative 2.5, respectively. And we can do the same for the rest of our values of 𝑡. This gives us the following entries in our table.
We can now start plotting these onto our graphs. Let’s start with our first point. That will be when 𝑥 is 1.5 and 𝑦 is 0.5. If we plot this point on all four of our graphs, we can see that clearly options (C), (B), and (E) do not pass through this point. We can see in option (A), this is one of the endpoints of our graph, which is what we expected. And in option (D), our curve gets very close to this value. However, because of the scaling, we can’t be sure.
So for due diligence, let’s try plotting another point. This time, we’ll plot the point where 𝑥 is equal to one and 𝑦 is equal to negative one. And once again, we get a similar story. We can see that options (B), (C), (D), and (E) do not pass through both of these points. But option (A) does pass through both of these points. In fact, if we were to plot every single point we found in our table onto our sketch in curve (A), we would get the following points. And we can see that yes, in fact, this curve does pass through all of these points.
There is one more thing worth pointing out. When we sketch a parametric curve, we need a direction for this curve. And to find this direction, we see what happens when we increase our values of 𝑡. In this case, as we were increasing our values of 𝑡, our curve was going from right to left and top to bottom. And we can represent this on our curve with arrows pointing in this direction. And in fact, we can see that our sketch already has this.
Therefore, given the parametric equations 𝑥 of 𝑡 is equal to two minus 𝑡 and 𝑦 of 𝑡 is equal to two minus three 𝑡. Where 𝑡 is greater than 0.5 and 𝑡 is less than three. We were able to show that option (A) was the correct sketch of this parametric curve.