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

Daniel wants to graph the parametric curve defined by the equations π‘₯ = 𝑑 + 1 and 𝑦 = 5𝑑 βˆ’ 1 for βˆ’2 ≀ 𝑑 ≀ 2. Determine the coordinates of the point on the curve where 𝑑 = 1.

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Video Transcript

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

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

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

So, we need to substitute 𝑑 is equal to one into both of these expressions. Let’s start with substituting this into π‘₯ is equal to 𝑑 plus one. Substituting 𝑑 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 𝑦. Substituting 𝑑 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 𝑑 is equal to one, our π‘₯-coordinate is two and our 𝑦-coordinate is four. And we write this as the Cartesian coordinates two, four. Therefore, we were able to show when 𝑑 is equal to one, the parametric curve defined by the equations π‘₯ is equal to 𝑑 plus one and 𝑦 is equal to five 𝑑 minus one will have coordinates two, four.

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