In Exercises 21–40, eliminate the parameter t. Then use the recta...

Question

In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞.
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x = √t,  y = t − 1

Accepted Answer

Welcome back everyone. In this problem, we want to write the rectangular equation for the parametric equation X equals the square root of two ty equals T minus three and T is between negative infinity and positive infinity. After we do that, we want to draw a graph of the plane curve using the rectangular equation where our X axis is bounded by eight and negative eight and our Y axis is bounded by negative five and 40. And afterwards, we want to indicate the direction of the curve obtained corresponding to increasing values of T using arrows. Now, what do we want to know it? It sounds like a lot. But basically, we want to take our parametric equation and change it to a rectangular equation. That is an equation expressed in terms of X and Y. The, our parametric equation is related by a common parameter T. So we just need to link both equations. Find some way to remove T from it, substitute T out. And then we'll have our rectangular equation. After we do that, then we will worry about joining the graph of the curve. So what do we know? Well, we know that X equals the square root of two T. And what that tells us as well is that our, our, our values for T that we're going to use would therefore have to be greater than, or equal to zero. Because if T is less than, or equal to zero, then our value under our square root would be negative. And the square root of a negative number is not a real number. OK. So let's see if we can solve four T in our equation for X. So if X equals A squared of two T by squaring both sides, that means X squared equals two, multiplied by T. Therefore, T equals X squared divided by two. Now that I have a value for A T, I can substitute that value so I can substitute T equals X squared divided by two into my equation for Y. So no, that tells me that Y is equal to X squared divided by two minus three. And we now have a rectangular equation from our parametric. We also know that four or rectangular equation, the value of X is also going to be greater than, or equal to zero because our value of T was greater than or equal to zero as well. Now, if we're going to draw this graph, OK, we're going to go ahead and draw this graph, then we can use different points just by looking at the format of our equation here. We know that this is going to graph a parabola. So we could use any points really to draw our graph. But let's be smart about it. We have a denominator of two for our variable. So if we choose even numbers, then we'll get some nice numbers, some friendly numbers to plot to draw our curve. So I'm going to create a table of values for X and Y. OK. And notice that our table goes up from our X values, our positive X values go from 0 to 8. So I'm going to use the X values for 024 and six. By then, we should be able to establish a pattern for what our para will look like. So let's go ahead and figure out what those points would be. Now when X is equals zero, Y equals zero squared divided by two, which would just be zero minus three, which would be equal negative three when X equals two, Y equals two squared, which is four divided by two, which would be equal to two. And then with subject three, which would be equal to negative one when X equals four, Y equals four squared divided by two. And that's 16, divided by two, which is eight and eight minus three equals five. Finally, when X equals six, Y equals six squared divided by two, which is 36 divided by two, which is 18 and 18 minus three equals 15. Therefore, to plot our graph, we're going to use the points, zero, negative three to negative one 45 and 6 15. So let's plot those points on our curve on our graph. So we have zero, negative three, two, negative one 45. Hm. And 6 15. 0, that's a, that's a big jump, right? Know that we have those four points, we can go ahead and plot our curve and our curve would look something like this. That's what our looks like, right? And finally, now that we have our curve, we just need to state the direction of our increasing values of T. We'll notice that as X gets larger, T would also have gotten larger. So our increasing values of T would be going upwards and that would be our graph for the parametric equation that we changed to the rectangular equation. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Parametric Equations

Elimination of the Parameter

Graphing Rectangular Equations

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