In Exercises 59–62, sketch the plane curve represented by the giv...

Question

In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range.

x = t² + t + 1, y = 2t

Accepted Answer

Welcome back everyone. In this problem, we want to plot the plain curve defined by defined by the set of parametric equations. X equals T squared plus two T plus 11 and Y equals T. Also, we want to provide the domain and range of our curve on our graph. And screen, we have the X axis that's bounded by negative 10 and 20 the Y axis bounded by eight and a negative five. Now, what do we already know? Well, we know that these equations are parametric. And what that means is that the X and Y values are related by some common parameter. In this case, that parameter is T. So in order to plot the graph of the curve, we will need to figure out what the X and Y values are for this parameter. And then after we plot the graph, we'll be able to better see how we can figure out the domain and the range, especially because this is a plane curve. So let's start by doing that. So first we need to choose a set of values for T, you could find those values by plotting around any and playing around to see which ones best fit the graph. But in this case, we're going to let T be between negative two and positive tool for our graph. OK. So that's, that's the value for the parameter we will be using. So that means I'm going to set up a table that will have the varying values of T, what whatever the X values will be. Remember X equals T squared plus two T plus 11 and Y equals T. So let me draw that table pretty quickly. OK. So now to find the values of X and Y, all we need to do is to substitute for our values of T. So for example, when T is negative two and T equals negative two, then X would be equal to negative two squared plus two multiplied by negative two plus 11. And when we work that out, negative two squared equals +42 multiplied by negative two is negative four and four minus 400 plus 11 equals 11. So when T equals negative two, X equals 11, likewise, since Y equals T, that means Y would also be equal to negative two when T equals negative one, X equals negative one squared, which is one plus two multiplied by negative one. So that's minus two plus 11 and one minus two plus 11 equals when T is zero? Oh And when T is negative one, why is also negative one as well? Remind me not to forget that when T equals zero, X equals zero squared plus two, multiplied by zero plus 11, which would just be 11. And why would be zero when T equals one, X would eventually work it down to one squared plus two plus 11, which equals 14 and Y would be one. And finally, when T equals two, X equals two squared, which is four plus two, multiplied by two, which is also four plus 11 and four plus four plus equals 19. And why would be two? So now we have our values of X are coordinates of X and Y. OK. So now we can plant. So for our first point would be negative two, which would be I plotted on my grid. Next, we'd have 10 negative one to that plot. Next, we'll have 11 0. OK? And then we'll have 14 1, the next card in it and then 2. OK. So now that we have these coordinates, we can plot, I'm gonna try to draw the best curve that I can ever draw, please. I hope it works out. Oh, that's not too bad. OK. I think that can work. I know you'll do a much better job than me, but I think that one's OK. So this is what our curve would look like. And I can draw some arrows to indicate where are the general direction of our plane curve. And now we have drawn our curve on the graph. We've plotted finally, we just need to provide the domain and the range of our curve. Now, the reason why I said leave the domain and the range until we've seen the plot because by looking at the plot, you can kind of get an idea of what those would be. Firstly, our domain represents our X values, the lowest and highest X value. If you look at our curve, it comes at 10 and then it goes back in the other direction. So the minimum value for a plain curve in this case, a Parabola is 10. OK. So for our domain, our domains minute, let me put that in red actually. OK. So our domain it starts at 10 and it includes 10. And then if you look in the direction it goes off into the negative infinity for ever and ever and ever. So our upper boundary would be negative infinity and our lower boundary would be negative infinity and it never will reach infinity. So we can leave that as an own boundary. Now, when we're talking about our range, what do we know about why our range are our Y values? And when you look at the Y values Y equals T, well, T could be anything right? And if t could be anything that means Y could be anything. So since Y is just any real number, its range will be from positive infinity all the way to negative infinity. So that would be our plan curve along with its domain and its range. Thanks for watching everyone. I hope this video helped.

In Exercises 59–62, sketch the plane curve represented by the given parametric equations. Then use interval notation to give each relation's domain and range. x = t² + t + 1, y = 2t

Key Concepts

Parametric Equations

Domain and Range

Sketching Curves

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