Investigation 2: Solving Parametric Equations

New Concept

When two variables are expressed in terms of a third variable, the equations used are called parametric equations.

What’s next

Next, you'll use this concept to model real-world scenarios, like painting a house and hitting a golf ball, by plotting their paths over time.

When two variables, like $x$ and $y$, are both expressed in terms of a third variable, such as $t$, the equations used are called parametric equations. For example, the area a painter covers and her earnings can be described by the equations $x = 200t$ and $y = 30t$.

Example 1: A painter's progress can be tracked with area painted $x = 150t$ and earnings $y = 25t$ dollars.
Example 2: A golf ball's flight path is modeled by its horizontal position $x = 100t$ and vertical position $y = -16t^2 + 80t$.
Example 3: A puppy's growth is shown by its height $x = 1.5t + 12$ and its weight $y = 9t$.

Think of parametric equations as a dynamic duo! Instead of relating $x$ and $y$ directly, they both rely on a third 'manager' variable, usually time. This manager tells both $x$ and $y$ what to do at every moment, creating a path or a story that unfolds over time. It's the ultimate behind-the-scenes coordinator for motion and change.

In a set of parametric equations, the third variable that both main variables depend on is called the parameter. In the equations $x = 200t$ and $y = 30t$, the variable $t$ is the parameter. Often, time is the parameter in real-world models.

Example 1: In the painting equations $x = 200t$ and $y = 30t$, the parameter $t$ represents the number of hours worked.
Example 2: For the golf ball's flight described by $x = 120t$ and $y = -16t^2 + 96t$, the parameter $t$ is the time in seconds after being hit.
Example 3: In a bus fuel problem where distance is $x = 40t$, the parameter $t$ represents travel time in hours.

The parameter is the secret controller, often 't' for time, that drives the action from behind the scenes. As the parameter changes, it dictates the values for both $x$ and $y$ simultaneously. It’s like the clock in a video game, where every tick of time moves your character to a new $(x, y)$ coordinate on the screen.

To graph parametric equations, you first use the parameter, $t$, to generate a set of $(x, y)$ coordinate pairs. Then, you plot these coordinate pairs on a standard Cartesian plane and connect them in order. This process reveals the path or relationship between the $x$ and $y$ variables, independent of time.

Example 1: For $x = 3t$ and $y=t+2$: if $t=1$, the point is $(3, 3)$; if $t=2$, the point is $(6, 4)$. Plot these points to see the line.
Example 2: For $x = t^2$ and $y=t-2$: at $t=0$, you get $(0, -2)$; at $t=1$, you get $(1, -1)$; at $t=2$, you get $(4, 0)$. Connect them to form a curve.
Example 3: Using a table for $x=120t, y=96t-16t^2$: calculate each $(x,y)$ for $t = 0, 1, 2...$ and then plot the points $(0,0), (120,80), (240,128)$ to see the parabola.

Time to play matchmaker! For each value of your parameter, $t$, calculate the corresponding $x$ and $y$. This gives you an $(x, y)$ point. Once you have a few of these points, plot them on a graph and connect the dots in order of increasing $t$. Voila! You have just revealed the secret path that $x$ and $y$ follow.