I have some data here.
It says a golf ball is
hit down a fairway.
The following table shows the
height of the ball with
respect to time.
The ball is hit at an angle of
70 degrees with the horizontal
with a speed of 40 meters
per second.
And then they give us a bunch
of, essentially data samples.
At time 0, we're at 0 meters.
At time 0.5 seconds, we're
at 17.2 meters.
So they give us a bunch
of data points.
And what I want to do is use
these data, or use these data
points, to essentially find a
quadratic function that fits
it best. Or I guess to view it
the other way, I want to model
this data or I want to model
this phenomenon, I want to
model the golf ball being hit
down the fairway, or the
height of the golf ball using
a quadratic equation, or
quadratic function.
And we're going to do it using
a graphing calculator.
And actually, you know, if you
do a little bit of physics,
you can figure out what
the actual equation
would be in a vacuum.
But this is actual data taken
from a golf ball hit in air.
And that changes things a bunch,
it makes the physics
really complicated.
So we'll actually look at the
measurements and then try to
figure out what the actual
function is.
And actually, the calculator
is going to do most of the
work for us.
I won't go into the details
of what the
calculators algorithm is.
I just want to show
you how to do it.
It's safe to say that this is
going to be well approximated
by some type of quadratic
function or a parabola.
So let's try to fit
a function to it.
So what we want to do is we
want to hit stat on our
calculator.
You hit stat, then
you hit edit.
That's what we use to input
our information.
We can click enter twice.
This just says these are the two
variable names for where
we input our data.
I'm just hitting enter twice.
And now I can start
entering the data.
And I'll use my keyboard
to do this.
So the first data point is 0.
Enter.
The x is 0, y is 0.
Press enter again.
The next data point,
0.5 and 17.2.
I'm just looking at this right
here, 0.5 and 17.2.
Next data point.
We have x is 1.5.
When x is 1.5, y is 42.9.
When x is 2, y is 51.6.
When x is 2.5, y is 57.7.
When x is 3, y is 61.2.
I just keep pressing
enter every time.
When x is 3.5, y is 62.3
meters in the air.
When x is 4, y is 61
meters in the air.
And when x is 4.5, y is 57.2.
So I've entered all
of the data.
And then the next thing I
want to do, I'm going
to plot all of it.
But before I plot all of this
data, I want to make sure that
I have the proper range
on my display.
So I'm going to go to
graph, hit range.
And see.
My x minimum, I want
it to be 0.
That's the lowest value here.
And my x maximum, I could
set it actually,
a little bit lower.
Well, I set my x maximum,
that's fine.
I'll go up to 7 seconds.
I'll make the scale, that's
the measurements on the
x-axis, as 1.
y minimum is 0.
y maximum, 70 looks
pretty good.
It seems like it'll cover all of
these numbers, and some of
these get pretty close.
They're in the 60's, so
70's pretty good.
And I'll make the y scale 5.
I don't have to make any changes
here, but you could
make changes if your calculator
isn't set to this already.
And so we can go back to stat.
And then what we want to do is
have the calculator calculate
a quadratic function that
matches these points as well
as possible.
So we'll go to calc.
I'm assuming that means
for calculate.
This is saying, where
is the data?
It's the same variables we used
to input the data, so
I'll just click enter.
And enter twice.
And now here are the
different types of
regressions we can do.
We can do a linear regression.
We can do an exponential
regression, power regression.
You click more and you'll see
a power 2 regression.
This means a second
degree regression.
So here, we can literally
click-- we
just select this option.
That means a quadratic or this
would be a third degree
regression, a fourth
degree regression.
So we're just going to do
a quadratic regression.
And it figured out the
coefficients of the quadratic
that best matches this data.
So I'm going to write it down.
So it's negative 5.2.
It's negative 5.2.
Scroll over a bit.
35.99999.
And then 0.292514.
So let me write these
down over here.
So let me take it off
the screen so I can
keep looking at it.
And so we have-- it's telling
us, the calculator has told
us, that if the quadratic is
equal to ax squared plus bx
plus c, it's telling us that the
first coefficient, a, is
negative 5.20128.
I'll probably round that
off a little bit.
It's telling us b is equal to
35.9-- let me scroll over my
calculator a bit-- 993.
I'll probably round these.
9934.
And it's telling my c values
that it calculates for the
best fit quadratic is 0.2925.
Which is interesting.
Before I did this problem, I
actually figured out what this
quadratic would be
in a vacuum.
I just used kind of some of
the physics principles.
And then you have this
number in a vacuum.
If you didn't have air
resistance, this number would
be negative 4.9.
This number I think turned out
to be something like 37.
And this number turned out to--
well, this was 0, because
you start at the ground.
But this is interesting.
This is actual data and actual
data is always going to be a
little bit different, or
sometimes a lot different than
the theoretical ideal, if you
had no air resistance.
So that's what we're
getting right here.
And so we can now use this.
This is telling me, my graphing
calculator's telling
me, that the quadratic I should
use-- you know, y is a
function of x-- is negative
5.2x squared
plus 35.99x plus 0.29.
I rounded them off
a little bit.
So now we can use this
information, or maybe I
should say of t.
I could have made this-- well,
x is equal to time.
I could have made it
t instead, but I
think you get the idea.
And now we can use this
information to figure out what
the ball was probably doing
at other points in time.
So, for example, if we wanted
to know what was
happening at time 5.
What's y of, let's say
5.2 seconds into it?
So what happens a little
bit further off?
We're extrapolating, we're going
beyond what the data
that we have right here.
So y of 5.2.
We can do it two ways.
We can look at it graphically.
So now that we've calculated
this regression, here are the
coefficients again.
As you see here, I'm just
scrolling to the left.
It gave me the coefficients.
Negative 5.2, 35.99, and then
it gave me the 0.2925.
So now that it calculated, we
could actually plot all the
points and draw the
regression.
I want to pick this draw
option up here.
So I can either exit and
then I pick draw.
And first I'll just
draw the data.
So that's scattered.
It'll make a scattered plot
of all of the data.
I select it there.
And notice, it drew the data
points as they were given.
That's the data points
right there.
5.2 is going to be out
here some place.
And now I can draw the
actual regression.
This is draw the regression
using these
coefficients right here.
So draw regression.
And you see it actually
fits the data very,
very, very, very well.
So let's use this function that
seems to fit the data
very, very, very well to
actually come up with how high
was the ball at 5.2 seconds?
So let's exit from here.
And let's just calculate it.
It is negative 5.2 times
5.2 squared.
Just by coincidence, I picked a
time that equals that first
coefficient.
Therefore, it's the positive
version of it.
So 5.2 squared plus 35.99 times
5.2-- remember, we're
dealing with when time is equal
to 5.2, or where x is
equal to 5.2-- plus 0.29.
So we're just going to evaluate
this function at time
or at x is equal to 5.2.
And we get 46.83 feet.
So it's equal to 46.83, or
I should say 0.83 meters.
So I hope you enjoyed that.
I just wanted to show you a
quick thing of how a graphing
calculator is useful.
Maybe in the future, I'll make
a more advanced video on the
actual algorithm or the actual
process that the calculator
used to figure out
this best curve.
