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# Quadratic Regression: Using a calculator to perform a quadratic regression

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- 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.

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