Let’s start with a small deviation from calculus to introduce (or re-introduce) some ideas of how we might define functions and curves. We have been working with functions and graphs of functions for so long, but that might mean that some of the ideas are so engrained that might not even think about them. Let’s remind ourselves as much as possible.
This is mostly a discussion activity. There’s not a lot to do, other than think and share your thoughts with others. If you’re not in a classroom setting, this can be hard! Find someone else with a relatively similar math background, and talk about this with them!
When you think of a function, what do you think of? What do you imagine in your mind when you hear the word “function” (in a mathematical context)? Is it based on a definition, a visualization, an example, or some other more loosely connected set of phrases?
How do you visualize a function? What does the picture represent, or how would you explain it to others? What are the important aspects of the visualization that we might use to control what these pictures look like, or what do we try to keep consistent when we visualize these functions? Why do we do this?
None of this should be new to you, but it probably has been some time since you’ve thought about it! The reason we’re starting this chapter (and this section) with this little reminder is simple: we’re going to think about how we could slightly alter some of these basic concepts and change the way that we think about functions, curves, and graphs.
We’re going to try to change some of the assumptions about curves and graphs and functions by asking some “What if...?” questions. For this section, we’ll ask a very short one:
What if, instead of the \(y\) variable being an output that depends on the \(x\) variable input, we had both \(x\) and \(y\) as outputs of some third variable, like time?
Instead of thinking of \(y\) as a function output of \(x\text{,}\)\(y=f(x)\text{,}\) we’ll think about the vertical component \(y\) and the horizontal component \(x\) of our curve being defined as functions of \(t\text{.}\) Now, instead of \(x\) being an independent variable and \(y\) being dependent on the value of \(x\text{,}\) we have that \(x\) and \(y\) are independent of each other, both dependent on \(t\text{,}\) which we often call a parameter. Typically, we think of this parameter as time: \(x\) and \(y\) both act independently, but change as time marches forward.
This small change in perspective (we’re changing how we view or think of the relationship between variables) impacts a lot of what we assumed and recalled in Activity 10.1.1.
Fill in the table with the \(x\)-values at different values of \(t\text{.}\) Describe the horizontal motion of the parametric curve using this information.
Fill in the table with the \(y\)-values at different values of \(t\text{.}\) Describe the vertical motion of the parametric curve using this information.
This small interactive graph is a nice one, because it highlights one of the important aspects of a parametric curve: motion. A graph of a parametric curve is really a path that a point travels along, which is pretty different from our original interpretation of a graph—the collection of points \((x,y)\) where \(y=f(x)\text{.}\)
We often try to include some visual clues depicting the orientation or direction of the curve when we visualize it. Often, this can be achieved with a simple animation (where we can see a point move along the curve), but this is also achieved using arrows along a curve.
We could end our section with a list of facts or formulas for different curves that we might recognize, but it will be more fun to explore these and discover them ourselves.
Before you visualize any of this, think about the behavior of the \(x\)-values and \(y\)-values. What do you think they’ll do? What will this curve look like?
It is possible that, in some case, we might want to represent our curve in a more traditional “\(y\) as a function of \(x\)” or “\(x\) as a function of \(y\)” setting. In this case, we need to manipulate our functions for \(x\) and \(y\) in order to eliminate the parameter, \(t\text{,}\) from our function representation.
Typically, this can be done by isolating \(t\) in either the function definition for \(x\) or \(y\text{,}\) and then substituting this new expression for \(t\) into the other variable’s function.
Use the graphing tool below to check to see if these curves are all the same. You’ll need to find appropriate intervals of \(x\)-values or \(y\)-values along the way.
We might notice that this strategy depends on us being able to solve for \(t\) in at least one of these functions. For instance, the following parametric curve will be hard to employ this strategy:
\begin{align*}
x \amp = t - \sin(t)\\
y \amp = t\cos(t)
\end{align*}
This is everything we need for now. We’re going to make one more quick stop to check out another way of representing functions, and then we’ll spend our time thinking about how we might do calculus with these parametric curves.
For each of the following parametric curves, plot the curves in the graphing utility below. Then, find at least 5 different points on the curve, plot them, and describe the curve.
How does the function \(f(t)=\cos(5t)\) compare to the function \(g(t)=\cos(t)\text{?}\) How about \(\ell(t)=\sin(5t)\) compared to the function \(m(t)=\sin(t)\text{?}\)
