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April 5, 2021

Parametric Equations

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Textbook Reference: Section 8.7 Parametric Equations: Graphs

Conic sections are great. They have cool graphs from relatively simple equations (all it takes is squaring one or two variables), and they have all sorts of real-world applications.

There's just one problem. I wonder if you've noticed it.

In general, they're not functions.

With the exception of certain parabolas, they don't pass the Vertical Line Test.

So what the heck are they doing in a class called MAT 130 Advanced Functions??

Well, to be entirely honest, saying that conic sections "aren't functions" is a bit too vague. What we really mean to say is that \(y\) isn't a function of \(x\) — given the value of \(x\), we can't always give a single value of \(y\). For example, if we're looking at the unit circle (which, remember, is a conic section!), and we want a point on it where \(x=0\), we could have \(y=1\) or \(y=-1\).

But, there's another way in which we can think of a circle as a function. And you've seen it already.

Parametric Equations

Instead of thinking of \(y\) as a function of \(x\) — input a value of \(x\), get a value of \(y\) — we can think of both \(x\) and \(y\) as functions of the angle \(\theta\): \[ \begin{cases} x = \cos(\theta)\\ y = \sin(\theta) \end{cases} \]

We call these parametric equations, and we call \(\theta\) a parameter, which both \(x\) and \(y\) depend on separately. The curve we get is called a parametric curve, and the above equations are a way to parametrize the circle.

However, our parameter doesn't necessarily have to be an angle, so instead of \(\theta\), we usually use \(t\) for our parameter. \[ \begin{cases} x = \cos(t)\\ y = \sin(t) \end{cases} \] We often think of \(t\) as representing time, and imagine that the curve is being "drawn" as \(t\) increases, say from \(t=0\) to \(t=2\pi\):

When drawing a parametric curve, we can draw arrows that show the direction the curve gets drawn as our parameter \(t\) increases. (This is called the orientation of the curve.) We might also label where the "pen" is at various values of \(t\).

It might not seem it right now, but allowing \(x\) and \(y\) to separately depend on \(t\) gives us a ton of flexibility, blowing the possibilities for what new curves we can draw wide open.

Preview Activity 23

Answer these questions and submit your answers as a document on Moodle. (Please submit as .docx or .pdf if possible.)

  1. Consider the following parametric equations: \[ \begin{cases} x = 1+2t\\ y = 3-t \end{cases} \]

    1. Create a table showing the values of \(x\) and \(y\) when \(t=-3,-2,-1,0,1,2,3\).
    2. Sketch the \((x,y)\) points. At each point, put the value of \(t\).
    3. What shape appears to be drawn as \(t\) increases? In which direction is the shape drawn? Mark the orientation using some arrows.
  2. There may be more than one way to parametrize a given curve like the unit circle.

    For example, consider the parametric equations: \[ \begin{cases} x = \sin(t)\\ y = \cos(t) \end{cases} \]

    1. Create a table showing the values of \(x\) and \(y\) when \(t=0,\frac\pi2,\pi,\frac{3\pi}2,2\pi\).
    2. Sketch the \((x,y)\) points. At each point, put the value of \(t\).
    3. Compare and contrast this parametrization with the one given earlier.
  3. The fact that \(t\) usually represents time makes parametric equations perfect for representing objects in motion.

    Suppose a soccer ball kicked into the air follows a path given by the following parametric equations: \[ \begin{cases} x = 6t\\ y = 32t-16t^2 \end{cases} \]

    1. Create a table showing the values of \(x\) and \(y\) when \(t=0,\frac14,\frac12,\cdots,2\).
    2. Sketch the \((x,y)\) points. At each point, put the value of \(t\).
    3. What appears to be the path of the projectile?
    4. What real-world significance do the points at \(t=1\) and \(t=2\) have?
  4. Answer AT LEAST one of the following questions:
    1. What was something you found interesting about this activity?
    2. What was an "a-ha" moment you had while doing this activity?
    3. What was the muddiest point of this activity for you?
    4. What question(s) do you have about anything you've done?

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