Relating these representations of functions
It is important to understand the relationship between both ways of
Let's begin with a famous example. Pretend that you drop a ball
and that you would like to know how far it falls after a certain time.
A convenient way to understand this would be to use a strobe light to
take a picture of where the ball is at regular intervals after being
released. Click in the box below and you will see what happens when
we flash the strobe every five seconds.
As you can see, this gives us a collection of data points which
tell us where the ball is every five seconds.
However, we might not feel very confident about where the ball is
after one second (maybe it visits Moose Jaw in between our
So we could do the experiment again and check where
the ball is every second.
Still, we don't have a good feel for where
the ball would be after one one-tenth of a second. Again, we could
increase the frequency with which we check. You can imagine how this
goes: to get a better understanding of where the ball is, you can
simply make more measurements. But no matter how many measurements
you make, you will not know what happens at every time.
This is where something very interesting happens. If we put the
data points on a graph as in this picture, we can see that the
points form a pattern. They all seem to lie on a parabola with the
property that the vertical distance fallen, measured in meters, seems
to always be given by 4.9 times the square of the time since release,
measured in seconds. At this stage, this is just an observation which
we have not explained (we will do that later). However, once we make
this observation, we can make predictions about where we
think the ball should be. For instance, after 0.001 seconds, we might
expect that the ball would only have fallen 0.0000049 meters. This is
something we could verify with our strobe light.
From this, we are lead to write something like y = 4.9t2
to denote the relationship between t, the time since the ball was
released, and y, the vertical distance the ball has fallen.
For your consideration:
- What kinds of measurements can we make to figure out if the trajectory
of the ball is really smooth or just a lot of jumps at very short intervals?
- What is our visual system doing to make a film or video clip (which
is really discrete) seem to be smooth?
- Where do you think the ball would be after one year?
When answering this last question, you need to think about the process
that the function is describing. What are the assumptions you
made to arrive at the answer that you gave? This is an important part
of scientific thinking---when making a prediction, you must always
consider whether there will be changes in the process which
invalidate your assumptions. I would imagine that, in one year, the
ball will be sitting neglected in a closet somewhere (sniff!).
This example illustrates something important about functions. We
can try to understand functions that describe scientific processes by
making some measurements. In general, this cannot tell us the whole
story but rather only what is going on when we happen to be looking
(that is, making a measurement). However, if we make enough
measurements, we often develop confidence about what is happening
between the measurements (or maybe after the measurements end) so that
we feel like we understand the entire process and can hence make
In this class, we will encounter functions in both of these forms.
Sometimes, we will assume that the data about our function is
obtained from just making some measurements. Other times, we will assume
that we can understand what happens at every time and write things
like y=4.9t2. Eventually, we will see why it is useful to
be able to think in both ways.