In Exercises 57–58, the parametric equations of four plane curves...

Question

In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other.

x = t and y = t² − 4

Accepted Answer

Welcome back. I am so glad you're here. We are asked to plot the plain curve defined by the given set of parametric equations. Our equations are X equals T plus one and Y equals T cubed minus nine. And then we're given a graph, we have a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. The domain for what's shown for our X axis is from negative eight to positive eight. And the range for what's shown for our Y axis is from negative 40 to positive 40. All right. So we're calling the steps in the book. The first step for graphing a plane curve with parametric equations is that we are going to select T values within our given interval. Now, we're not given an interval for T here. So we're just going to pick some T values that are going to work with what's shown on our graph. So I'm drawing a table. The first column is going to be for our T values. The second column is for the equation X equals T plus one. The third column is for the equation Y equals T cubed minus nine. And the fourth column is for our XY values. So for the T values here, we can see that the T values are very closely correlated with our X values. So we want to choose T values, there are within the domain of what's shown for our X values. But when you start to pick, if you were to pick a T value of negative eight, and if you plug that in for solving for Y, you're going to get a very big number that is outside of the range of what's shown. So I've selected some T values that are going to stay within what's shown for our graph for the T values, I'm choosing negative three, negative two, negative 1012 and three. So now let's solve for our X values. So X equals not T but negative three plus one and negative three plus one is negative two. And let's just keep going with the X values. So A T value of negative two, X equals not T but negative two plus one. Negative two plus one is negative one. For nat value of negative one, X equals negative one plus one. Negative one plus one is zero. For A T value of zero, we have X equals zero plus one which is a positive one. For A T value of one, we have X equals one plus one that equals a positive two for A T value of two, two plus one equals positive three, X equals three. And for A T value of three, X equals three plus one and X equals four. So we've got our X values now let's solve four Y. All right. Our first T value is negative three. So Y equals not T cubed but negative three cubed minus nine, negative three cubed is negative 27. So negative 27 minus nine, that's going to give us a negative T value of negative two Y equals not T but negative two cubed minus nine, negative two. Cute is negative eight, negative eight minus nine is negative 17. 0, that's not a seven negative 17. All right. For A T value of negative one, that's negative one cubed minus nine, negative one cubed is negative one negative one minus nine is negative T value of 00 cubed minus cubed is 00 minus nine is negative nine T value of one. That's one cubed minus 91 cubed is 11 minus nine is negative eight T value of 22 cubed minus nine two cubed is 88 minus nine is negative one and A T value of 33 cubed minus 93 cubed is 27 27 minus nine is positive 18. So now let's plot these points first one, negative two, negative 36 from the origin, we go to the left two and down 36. Next point negative one negative 17 from the origin we go to the left one and down 17, next 0.0 negative 10 from the origin. We go down 10, next 0.1 negative nine from the origin. We go to the right one and then we go down nine, next 0.2 negative eight from the origin. We go to the right two down eight, next 0.3 negative one from the origin. We go to the right three down one next 0.4 18 from the origin, we go to the right four, we go up 18. And now we connect these as smoothly as possible and we have our plane curve and we can indicate its direction by seeing what direction our T values are going. The T values are increasing as we move from our points from left to right. So we can draw our arrows pointing from left to right along our plane curve. And there is our graph well done. We'll catch you on the next one.

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In Exercises 57–58, the parametric equations of four plane curves are given. Graph each plane curve and determine how they differ from each other. x = t and y = t² − 4