Question Video: Graphing a Pair of Parametric Equations Mathematics

Video Transcript

Consider the parametric equations 𝑥 of 𝑡 equals 𝑡 minus two and 𝑦 of 𝑡 equals three 𝑡 plus one, where 𝑡 is strictly greater than negative two but strictly less than two. Which of the following is the sketch of the given equations?

Here, we are given a pair of parametric equations and asked to determine which sketch out of the five given correctly represents them when the values of the parameter 𝑡 lie in the open interval from negative two to two. We will do this by finding coordinate pairs that satisfy the given parametric equations over the open interval of 𝑡-values from negative two to two. To find the coordinate pairs, 𝑥, 𝑦, that satisfy the given parametric equations over the open interval of 𝑡-values from negative two to two, we will substitute values of 𝑡 between negative two and two into the parametric equations.

Note that even though the coordinate pairs associated to the values of 𝑡 equal to negative two and two are not to be included in our sketch, we still need to find them as our curve approaches those coordinate pairs. Let’s increase the values of 𝑡 by one each time, starting with negative two. In this question, we simply want to determine which of the given sketches represents the parametric equations. And so, a subinterval of one between values of 𝑡 is okay. However, if we wanted to sketch the parametric equations from the beginning, then we may want to keep a smaller subinterval, say, of 0.5 or 0.25 between values of 𝑡.

Substituting 𝑡 equals negative two into 𝑥 of 𝑡 equals 𝑡 minus two, we obtain negative two minus two, which equals negative four. Substituting 𝑡 equals negative two into 𝑦 of 𝑡 equals three 𝑡 plus one, we obtain three times negative two plus one, which equals negative five. So, the value 𝑡 equals negative two gives us the coordinate pair negative four, negative five. Substituting 𝑡 equals negative one into 𝑥 of 𝑡, we obtain negative one minus two, which equals negative three. Substituting 𝑡 equals negative one into 𝑦 of 𝑡, we obtain three times negative one plus one, which equals negative two. So, the value 𝑡 equals negative one gives us the coordinate pair negative three, negative two.

Substituting 𝑡 equals zero into 𝑥 of 𝑡 and 𝑦 of 𝑡, we obtain zero minus two and three times zero plus one, respectively. So, the value 𝑡 equals zero gives us the coordinate pair negative two, one. Continuing this process and substituting values of 𝑡 equal to one and two into the functions 𝑥 of 𝑡 and 𝑦 of 𝑡, we obtain the coordinate pairs negative one, four and zero, seven, respectively. Now that we have the coordinate pairs associated to each of our chosen values of 𝑡, we will see which out of the five curves given to us contains all of these coordinate pairs.

We could also plot the coordinate pairs on a separate graph if we wanted to. And we would indeed do that if we were sketching the graph of the parametric equations from the beginning. We see that only the curve in option (A) contains the coordinate pair negative four, negative five. The curve in option (A) is also the only curve which contains the coordinate pair negative three, negative two. In fact, we can clearly see that only option (A) contains all the coordinate pairs that we have computed.

Note that when sketching parametric equations, we move in increasing values of 𝑡. And so, the direction arrows on the sketch of our curve should point towards the coordinate pair zero, seven, which they indeed do in option (A). So, the sketch of the parametric equations 𝑥 of 𝑡 equals 𝑡 minus two and 𝑦 of 𝑡 equals three 𝑡 plus one, where 𝑡 lies in the open interval from negative two to two, is option (A).