In Exercises 15–16, eliminate the parameter and graph the plane c...

Question

In Exercises 15–16, eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve.        _x = √t , y = t + 1; −∞ < t < ∞

Accepted Answer

Hello, today we're going to be finding the rectangular version of the given parametric equations. Then we're going to be graphing the parametric equations. So let's go ahead and start with the first part and find the rectangular version of the parametric equations. Now, the parametric equations given to us is X is equal to five square root of T and Y is equal to T plus 13. Now, in order to get the rectangular version of the parametric equations, we're first going to need to solve for T in the first equation of X. In order to solve for T, we can start by dividing both sides of the equation by five. That will leave us with the square root of T is equal to X over five. Next, in order to remove T from the square root, we're going to have to square both sides of the equation. And that will leave us with T is equal to X over five squared, which will simplify to X squared over 25. Now, that means that T is equal to X squared over 25. And we can use this value of T as a substitution for the Y value of the parametric equations. So if we substitute T uh with X squared over that will give us Y is equal to X squared over 25 plus 13, this will be the rectangular version of the given parametric equations. Now we're going to move on to the second part where we're going to graph the parametric equations. Now, in order to graph the parametric equations, we want to go ahead and pick values of T to plug into the original parametric equations which was given to us as X is equal to five square root of T and Y is equal to T plus 13. Now, one thing to note about the first value X is that we have T inside of the square root. That means that when we are picking values of T to plug into X and Y, we only, we want to follow the domain of the square root. And that means that any values of T must be greater than, or equal to zero. So let's go ahead and start when T is equal to zero. When T is equal to zero, we get X is equal to five multiplied by the square of zero, which will simplify to give us zero and we'll have Y is equal to zero plus 13, which will give us the value of 13. That means that the position when T is equal to zero is going to be at the coordinate zero comma 13 next, since we are dealing with the square root, let's go ahead and pick a value of T that is a perfect square. The next value of T we can test is when T is equal to one, when T is equal to one, we get X is equal to five, multiplied by the square root of one, which will give us a final product of five. Then we will have Y is equal to one plus 13, which will give us the final value of 14. That means the second coordinate that we can graph is going to be five comma 14. Next, let's pick T is equal to four. When T is equal to four, we get X is equal to five multiplied by the square of four, which will give us a final product of 10. And we'll have Y is equal to four plus 13, which will give us the final value of 17. The next coordinate we can plot is 10 comma 17. Next, let's pick a final point and we'll pick when T is equal to nine, when T is equal to nine, we get X is equal to five multiplied by the square root of nine, which will give us a final product of 15. We will also get Y is equal to nine plus 13, which will give us a final sum of 22. That means the final point that we can plot will be 15 comma 22. So we have four points that we can plot on the given axis. We have zero comma 13, we have five comma 14, 10 comma 17 and 15 comma 22. Let's start by plotting zero comma 13. Now zero comma 13 will be roughly located about here. Next, we're going to plot five comma 14 and five comma 14 will be located slightly above this point. Next, we're going to plot 10 comma 17 and 10 comma 17 will be located slightly above that point. And we're going to plot 15 comma 22 and 15 comma 22 will be located just below the 30 mark. Now, we can go ahead and connect these dots with a single line and noticed that when we were plotting the four given points, we were moving from left to right. The convenient thing about parametric equations is that it tells us what direction the plot is moving in. And since we were moving left to right, the direction of the parametric equations are going to be moving from left to right. And this is going to be the rough graph of the parametric equation. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 15–16, eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. _x = √t , y = t + 1; −∞ < t < ∞

Key Concepts

Parametric Equations

Elimination of the Parameter

Graphing Plane Curves

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