For each plane curve, (a) graph the curve, and (b) find a rect...

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.

x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)

Accepted Answer

Welcome back, everyone. Given the parametric equations X equals 4 T + 1 and Y equals tt minus 54 T in the interval between negative infinity and infinity, what is the rectangular equation that represents this curve, graph this curve. So let's consider this problem. As we can see, we're given our X expression, which is fort. Plus 1 and we're given our Y, which is 2 T minus 5. So what we're going to do is just express the parameter T in terms of in terms of X from the first equation. First of all, for T is going to be equal to X minus 1, we are subtracting one from both sides, and now if we divide both sides by 4, we're going to get T equals 4, T equals X minus 1 divided by 4. And now we're going to substitute this expression into the Y equation. Y equals 2 multiplied by T and T equals X minus 1 divided by 4. And then we are subtracting 5. So now let's simplify Y equals 2 divided by 4 gives us 1/2, right? So we get X minus 1 divided by 2 minus 5. And now a less essentially. Simplify it. Y equals x divided by 2 minus 1 divided by 2. We're going to split this fraction, and then we're subtracting 5, which is essentially 10 divided by 2, right? We're using the common denominator. So the equation would be X divided by 2 minus 11 divided by 2. We are combining the like terms, and we have our equation. This is basically a line. Right, because it has a form of Y equals m x plus B. We have our intercept, we have our slope, we have our X. So what we're going to do is first we'll label this as our final equation, and now we're going to plot this line. Notice that we are not restricted by T values, so we can essentially plot this line choosing two points. One of the points that we can take is going to be, let's say 1, right? We can take X equals 1. It doesn't really matter which point we take. Let's assume that the first point that we take is X equals 1, and then we're going to solve for Y. Y at 0.1 is going to be equal to 1/2 minus 11 divided by 2, which gives us -10 divided by 2, and that's basically -5. So the first point that we can plot is 1.5. And now let's consider our 2nd point. For the 2nd point, we can assume that X is equal to -5. We can take any point that we want, right? If X is equal to -5. Then we are going to solve for Y and Y at 0.5 would be equal to -5 divided by 2 minus 11 divided by 2. And this gives us -16 divided by 2, which is -8. So the second point would be located at -5, -8. Now, if we have those two points, we can plot the required line. So the first point that we want to locate is at 1.5. And the second point is at -5, -8. So we are going to label -8. So going down from -5, we get -5, -6, 7, -8. And then let's find the intersection with -5. Right? So we have the 2 points and we can now connect those points in a straight line. To get the desired. Graph Well done. We have our final answer. We have our graph and the required equation. Thank you for watching.

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. See Examples 1 and 2.

x = t + 2 , y = t ―4 , for t in (― ∞ , ∞)

Key Concepts

Parametric Equations

Eliminating the Parameter to Find Rectangular Equations

Graphing Parametric Curves

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