# Parametric Equations (Part 2)

(This page is Part 2. Click here for Part 1.)

## Constructing a Parametric Curve with Desired Properties

Suppose we want to travel between two fixed points, $\,P_1(x_1,y_1)\,$ and $\,P_2(x_2,y_2)\,,$ in a fixed time. More specifically, we want to be at $\,P_1\,$ at time $\,t_1\,,$ and at $\,P_2\,$ at time $\,t_2\,$ (with $\,t_1\ne t_2\,$).

Of course, there are *many*
different ways to travel between two points!
The simplest solution is a straight line
path between the two.

Using function notation, we want:

$$ \begin{align} \cssId{s6}{x(t_1) = x_1}\cr \cssId{s7}{x(t_2) = x_2}\cr \cssId{s8}{y(t_1) = y_1}\cr \cssId{s9}{y(t_2) = y_2} \end{align} $$Using point-slope form in the $tx$-plane, we get:

$$ \begin{gather} \cssId{s11}{x(t) - x_1 = \frac{x_2-x_1}{t_2-t_1}(t - t_1)}\cr\cr \cssId{s12}{x(t) = x_1 + \frac{x_2-x_1}{t_2-t_1}(t - t_1)}\tag{9a} \end{gather} $$Similarly,

$$ \cssId{s14}{y(t) = y_1 + \frac{y_2-y_1}{t_2-t_1}(t - t_1)}\tag{9b} $$Equations (9a) and (9b) are the desired parametric equations. Together, they give a line through points $\,P_1(x_1,y_1)\,$ and $\,P_2(x_2,y_2)\,$ satisfying:

- at time $\,t_1\,$ you're at $\,P_1\,$
- at time $\,t_2\,$ you're at $\,P_2\,$

### Example

The line above is at:

- $\,P_1(-2,5)\,$ at time $\,t = 1$
- $\,P_2(3,-4)\,$ at time $\,t = 11$

It was generated by substituting the following values into equations (9a) and (9b):

giving

$$ \begin{gather} \cssId{s26}{x(t) = -2 + \frac{1}{2}(t - 1)}\cr\cr \cssId{s27}{y(t) = 5 + \frac{-9}{10}(t - 1)} \end{gather} $$Measuring $\,t\,$ in (say) seconds, each arrow represents the part of the curve traced in two seconds.

## Eliminating the Parameter: Going from a Parametric Curve to $\,y = f(x)\,$ (When Possible)

Suppose a parametric curve passes a vertical line test: like examples 1, 2, 3, 4, 6, 9ab (but not 5, 7, 8). Then, you may be able to get a description of the curve in the form ‘$\,y = f(x)\,$’.

In the
‘$\,y = f(x)\,$’
form, you *lose* information about how the curve is traced out,
but this single-equation form
may be easier to work with for some applications.

When we want to *emphasize*
that an
$x$-value
depends on (say) $\,t\,,$
we can write
‘$\,x = x(t)\,$’.
Read this aloud as:
‘$\,x\,$ equals
$\,x\,$
of $\,t\,$’,
meaning that the
$x$-value
is a function of $\,t\,.$

In going from a pair of parametric equations ‘$\,x = x(t)\,$ and $\,y = y(t)\,$’ to a single equation ‘$\,y = f(x)\,$’, the parameter $\,t\,$ has been eliminated—it is gone! Therefore, this process is typically referred to as ‘eliminating the parameter’.

The process of eliminating
the parameter is often used to help identify
the *graph* of parametric equations.
It may be easier to recognize
a curve in
‘$\,y = f(x)\,$’
form than in parametric equation form.

## Example: Eliminating the Parameter in Examples 1–4

Procedure:

- Drop function notation: that is, write $\,x(t)\,$ as simply $\,x\,$; write $\,y(t)\,$ as simply $\,y\,.$
- Solve for $\,t\,$ in a simplest equation; substitute this expression for $\,t\,$ into the remaining equation.

Note:

- In examples 1–4, the equation for $\,x(t)\,$ is simplest to solve for $\,t\,.$
- Upon eliminating the parameter, all four curves give $\,y = x^2\,.$

### Example (1)

$$ \begin{gather} \cssId{s54}{x(t) = t\,,\ \ y(t) = t^2}\cr \cssId{s55}{\text{Original parametric equations}}\cr\cr \cssId{s56}{x = t\,,\ \ y = t^2}\cr \cssId{s57}{\text{Drop function notation}}\cr\cr \cssId{s58}{t = x}\cr \cssId{s59}{\text{Solve for $\,t\,$ in simplest equation}}\cr\cr \cssId{s60}{y = t^2 = x^2}\cr \cssId{s61}{\text{Substitute into remaining equation}}\cr\cr \cssId{s62}{y = x^2}\cr \cssId{s63}{\text{($\,y = f(x)\,$ form)}} \end{gather} $$### Example (2)

$$ \begin{gather} \cssId{s65}{x(t) = 2t\,,\ \ y(t) = 4t^2}\cr \cssId{s66}{\text{Original parametric equations}}\cr\cr \cssId{s67}{x = 2t\,,\ \ y = 4t^2}\cr \cssId{s68}{\text{Drop function notation}}\cr\cr \cssId{s69}{t = \frac x2}\cr \cssId{s70}{\text{Solve for $\,t\,$ in simplest equation}}\cr\cr \cssId{s71}{y = 4t^2 = 4\bigl(\frac x2\bigl)^2 = \frac{4x^2}{4} = x^2}\cr \cssId{s72}{\text{Substitute into remaining equation}}\cr\cr \cssId{s73}{y = x^2}\cr \cssId{s74}{\text{($\,y = f(x)\,$ form)}} \end{gather} $$### Example (3)

$$ \begin{gather} \cssId{s76}{x(t) = \frac 12t\,,\ \ y(t) = \frac{t^2}4}\cr \cssId{s77}{\text{Original parametric equations}}\cr\cr \cssId{s78}{x = \frac 12t\,,\ \ y = \frac{t^2}4}\cr \cssId{s79}{\text{Drop function notation}}\cr\cr \cssId{s80}{t = 2x}\cr \cssId{s81}{\text{Solve for $\,t\,$ in simplest equation}}\cr\cr \cssId{s82}{y = \frac{t^2}4 = \frac{(2x)^2}4 = \frac{4x^2}4 = x^2}\cr \cssId{s83}{\text{Substitute into remaining equation}}\cr\cr \cssId{s84}{y = x^2}\cr \cssId{s85}{\text{($\,y = f(x)\,$ form)}} \end{gather} $$### Example (4)

$$ \begin{gather} \cssId{s87}{x(t) = -t\,,\ \ y(t) = t^2}\cr \cssId{s88}{\text{Original parametric equations}}\cr\cr \cssId{s89}{x = -t\,,\ \ y = t^2}\cr \cssId{s90}{\text{Drop function notation}}\cr\cr \cssId{s91}{t = -x}\cr \cssId{s92}{\text{Solve for $\,t\,$ in simplest equation}}\cr\cr \cssId{s93}{y = t^2 = (-x)^2 = x^2}\cr \cssId{s94}{\text{Substitute into remaining equation}}\cr\cr \cssId{s95}{y = x^2}\cr \cssId{s96}{\text{($\,y = f(x)\,$ form)}} \end{gather} $$## Example: Eliminating the Parameter in 9ab

To eliminate the parameter,
you don't *have* to solve for (just) $\,t\,.$

As illustrated in this example, it's sometimes easier to solve for a different expression (involving $\,t\,$) that appears in both $\,x(t)\,$ and $\,y(t)\,.$ Here, we solve for $\,t - t_1\,$:

$$ \begin{gather} \cssId{s106}{t-t_1 = \frac{(x-x_1)(t_2-t_1)}{x_2-x_1}}\cr\cr \cssId{s107}{\text{Solve for $\,t-t_1\,$ in the equation for $\,x\,$}} \end{gather} $$

$$ \begin{gather} \cssId{s108}{y = y_1 + \frac{y_2-y_1}{t_2-t_1}\cdot\frac{(x-x_1)(t_2-t_1)}{x_2-x_1}}\cr\cr \cssId{s109}{\text{Substitute into remaining equation}} \end{gather} $$

$$ \begin{gather} \cssId{s110}{y = y_1 + \frac{y_2-y_1}{x_2-x_1}(x - x_1)}\cr\cr \cssId{s111}{\Large\substack{\text{Cancel; re-arrange factors;}\\ \text{this is $\,y = f(x)\,$ form}}} \end{gather} $$

$$ \begin{gather} \cssId{s112}{y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)}\cr\cr \cssId{s113}{\Large\substack{\text{Point-slope form for line}\\ \text{through $\,(x_1,y_1)\,$ and $\,(x_2,y_2)\,$}}} \end{gather} $$

## Parametric Equations at WolframAlpha

Hop up to WolframAlpha and type in (say):

parametric plot (0.5 + cos(t) , 0.5*tan(t) + sin(t)), t=0 to 2pi

(You can cut-and-paste.) Voila! How easy is that?

Congratulations on finishing Precalculus!! Next up—Calculus!