Video: Finding the Coordinates of a Parametric Curve at a Given Value

Daniel wants to graph the parametric curve defined by the equations π₯ = π‘ + 1 and π¦ = 5π‘ β 1 for β2 β€ π‘ β€ 2. Determine the coordinates of the point on the curve where π‘ = 1.

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Video Transcript

Daniel wants the graph the parametric curve defined by the equations π₯ is equal to π‘ plus one and π¦ is equal to five π‘ minus one for π‘ is greater than or equal to negative two and π‘ is less than or equal to two. Determine the coordinates of the point on the curve where π‘ is equal to one.

Weβre told that Daniel is trying to sketch a graph of the parametric curve defined by a pair of parametric equations. Weβre told π₯ is equal to π‘ plus one and π¦ is equal to five π‘ minus one. And weβre told that our values of π‘ range from negative two to two. We need to determine the coordinates that this parametric curve will have when our value of π‘ is equal to one.

The first thing worth pointing out is, this value of π‘ is indeed within our range of values of π‘. So, this is a valid value of π‘. Next, to find the coordinates of our parametric curve, we need to recall what we mean by parametric equations. Weβre given functions for π₯ and π¦ in terms of π‘. Weβre told π₯ is equal to π‘ plus one and π¦ is equal to five π‘ minus one. We can input values of π‘, and these will output the π₯- and π¦-coordinate for this value of π‘.

So, we need to substitute π‘ is equal to one into both of these expressions. Letβs start with substituting this into π₯ is equal to π‘ plus one. Substituting π‘ is equal to one, we get π₯ is equal to one plus one, which we can calculate is equal to two. We can do the same for π¦. Substituting π‘ is equal to one, we get five times one minus one, which simplifies to give us five minus one, which we can calculate is equal to four.

So, when π‘ is equal to one, our π₯-coordinate is two and our π¦-coordinate is four. And we write this as the Cartesian coordinates two, four. Therefore, we were able to show when π‘ is equal to one, the parametric curve defined by the equations π₯ is equal to π‘ plus one and π¦ is equal to five π‘ minus one will have coordinates two, four.