Identifying Parametric Equations in the Plane

Question

Exercises 1–6 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation and indicate the direction of motion and the portion traced by the particle.

x = √t, y = 1 − √t, t ≥ 0

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Identify the particle's path by finding a Cartesian equation for it. Graph this equation indicating both the direction of motion and the proportion of the graph tracraced by the particle for the following parametric equations x is equal to t2. Y is equal to 2 minus t 2, where t is greater than or equal to 0. OK, so it appears for this particular problem, we are asked to solve for a few different answers here. So essentially, the main answer that we're ultimately trying to solve for is we're trying to create a graph for this equation, and we're solving for the Cartesian equation for this particular particle's motion. And we're also asked when we draw our graph to show the direction of motion of this particle. So, is it going to the left, is it going to the right? Is it going from top to bottom, bottom to right? We will need to indicate that. So, with that in mind, noting that we are ultimately trying to solve for the Cartesian equation first, and this Cartesian equation will be what we're graphing, and then when we graph our Cartesian equation, we need to show the direction of motion. So now that we know what we're ultimately trying to solve for. Let us first come up with a plan for how we're going to approach solving this problem. And based on our prior knowledge, we need to note that we can solve for this specific problem by substituting the parametric expression for X into the equation for Y in order to eliminate the parameter T, which will result in a single linear relationship between the values X and Y. And we will then need to determine the start point at T equals 0, and this will help us calculate additional points to help us identify the specific portion of the line that is being traced and the direction of the particle's movement. So now that we have a basic outline for how we're going to solve this problem, our first step that we need to take is we need to eliminate the parameter. So in order to find the Cartesian equation, we will need to eliminate the value t. So we are told that x is equal to t 2, and we're told that y is equal to 2 minus t 2. And we can see that in both expressions, T2 is common, so that will mean that from the first equation we will have that T 2 is equal to x, and when we substitute x for T 2 in the second equation, it will give us that Y is equal to 2 minus x. Fantastic. And this represents a straight line that has a slope of -1 and a Y intercept of 2. For our 2nd step, we need to identify the portion of the graph that is to be traced. And since the parameter is restricted to T is greater than or equal to 0, as a result, we will need to determine the possible values for X and Y. Considering x is equal to t to the 2, since any real number squared is non-negative, that means we can go ahead and write that x will be greater than or equal to 0. And this will mean, as a result, the particle does not trace the entire line of Y is equal to 2 minus x. However, it only traces the part where X starts at zero and increases. So keeping that in mind, for our 3rd step, we now need to determine the direction of motion. So in order to find the direction of motion, we need to consider the following. So we need to find the direction by testing the values of T as the time increases. So T represents time. So we need to find the direction once again by testing the values of T as these time values increase. So considering, we will consider T equals 0, T equals 1, and T equals 2. So when we plug in a value of zero in for T, we will find that X is equal to 0, and Y will be equal to 2, so that means the particle starts at 0.2. And plugging in a value of 1 in for T, we will find that X is equal to 1 and Y is equal to 1. So that means our particle will move to 1.1. And finally, when we plug in a value of 2 in fort, we will find that x is equal to 4 and y is equal to -2, so that will mean that our particle will move to 42. And we need to note that as T increases, the particle will move further to the right as X increases, and it will also move further down as Y decreases. So, considering this, or considering this key piece of information, or I should say key pieces of information, our step number 4, which is our final step, is we can now begin to graph our equation. So essentially we will need to plot our ray starting at the y intercepts 0.2, and then we will need to draw a line passing through the points 1.1 and 2.0, and then we will need to add arrowheads to indicate direction along the line that are pointing downward to the right to once again indicate our direction of motion. So with that in mind, going back to look at the blank graph that was provided to us by the prom itself. We can safely take all the key pieces of information that we've gathered to produce the following graph. So as we can see, we have our ray, which starts at 0.2, and then we connect our, so once again, starting at 0.2, we will draw our ray to 0.1.1 and then keep going to 2.0, and then keep going down. And as we can see, we have arrowheads pointing down to the right direction, and this is for Y is equal to 2 minus X, and once again, that's our Cartesian equation, and we need to note that this is for X is greater than or equal to 0. And that's it. We've officially solved for this problem. So to recap, once again, the Cartesian equation for this particular problem is Y is equal to 2 minus x, and this is for the portion x is greater than or equal to 0, and the motion once again is downward and to the right, starting from the point 0.2 in parentheses, and that's it. We've officially solved for this problem. Thank you so much for watching, hopefully that helped, and I can't wait to see you in the next video, bye.

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