Answers to “Why Do We Need Parametric Equations



Answers to “Why Do We Need Parametric Equations?”

I. Parametric equations are 2 or more equations (usually for x and y – or x, y and z) written in terms of a common variable or variables (usually t – but (, r, s and p are also commonly used).

II. Substitute in the different values of t.

At t = 0, the position is (2, 1), at t = 1 the position is (1, 1.5) and at t = 2, the position is (0, 2). However, as soon as we graph these three points, we lose all information about t.

The mode must be changed from FUNCTION to PARAMETRIC

The y= command looks different, because now there are two equations to type instead of one. Note that the x,t,θ,n key now provides us with a T symbol.

The TABLE gives us values of x, y and t all at the same time. This is a nice feature to have.

Notice that the ( command gives you options for values of t now, as well as values for x and y. Although in science t values are positive, we accept negative t values in math, because we aren’t always limited to a physical setting. In fact, we are often interested in what t values are possible – it’s like a “domain” for our parametric equations. Values of (10 < t < 10, (4.7 < x < 4.7 and (3.1 < y < 3.1 provide a nice graph. If you change your value of Tmin to 0, you only get part of your graph. The values of t are important to your graph.

The Tstep changes what you “count by”. It has the same effect as the ÃTbl did in your TABLE SETUP. The smaller your Tstep is, the more points your calculator will plot. This will take more time, but might provide a more accurate graph.

The TRACE command works in increasing values of t. This may drive you crazy at first, but you get used to it.

There are several ways to eliminate the parameter algebraically. I’d multiply the equation for y by 2 to get 2y = t + 2. Now, just add the equations to get x + 2y = 4. Note that the intercepts of this line are exactly where they should be on your graph.

III. Your t values should allow you to draw a good graph of a sine or cosine function. Therefore if you are in degrees, use 0 < t < 360, step 5, but if you are in radians, use 0 < t < 6.29 (or 2() step .1 .

It graphs an ellipse, and a far better one that you got when you solved for y in the conics chapter. To eliminate the parameter, solve for x and y and then square both sides.

[pic] (And it’s even in standard form!)

Translating these equations is intuitive: x = 4⋅cos t + 1 and y = 2⋅sin t - 2

IV. Think simple. Let x = t. Then y = .5t ( 2.

However, if x = t + 3, y = .5(x + 3) + 3.

In general, you can write x (the independent variable) as almost any function of t (although using x = t is the easiest) and then write y by replacing all the x values with the same function of t that you chose for x.

This means that there are ( many ways to write a Cartesian equation in parametric

form. This can be a good thing or a confusing thing…

V. Let x = .5t2 and let y = t. It’s best to let t be whatever your independent variable is. The graph of this is wonderful, but you’ll need to look at your t values and tstep carefully to get the entire parabola. You want a large enough set of t values to get an entire picture of the graph and you want the t-step to be small enough to get the details of the graph.

VI. Let x1 = t and y1 = t2 ( 3. Then let x2 = y1 and y2 = x1. (You really do switch your x and your y.)

VII. In Cartesian form, the equation would be: y = (x + 2)2 + 1. In other words, we would subtract the (2 because it changed the x, but add the 1, because that changed the y.

However, parametric equations are honest. The original parametric equations are

X = t and y = t2, and the translated ones are: x = t (2 and y = t2 + 1.

In order to eliminate the parameter, we’d need to solve the first equation for t and substitute that into the equation for y. So x + 2 = t and y = (x + 2)2 + 1. In solving the x equation for t, we “undid” all that had been done to x. That’s why all transformations involving x act the exact opposite way from the way we expect them to do.

Y = sin x becomes x = 2t and y = 3sin t + 2 and eliminating the parameter yields y = 3sin (x/2) + 2.

VIII. You have the answers EXCEPT – “How close do Tom and Jerry get?” You know that the distance between any two points is d = [pic], and you have values for x1, y1, x2¸ and y2, so substitute in and you have d as a function of t. Graph that in FUNCTION mode and use the MINIMUM feature of the calculator to find how close they get. You don’t want to use parametric equations to graph this function because if you’ve looked at the CALC feature, you’ll see that you can’t calculate minimum values of parametric equations. However, take that equation back to function form and you’ll find that at t = 4.353, Tom and Jerry distance is only 1.213, and that’s as close as they get.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download