Question

Can someone help find a quadratic function that best fits this data.
x y
19.5 12965
29.5 32131
39.5 41636
49.5 45693
59.5 41477
69.5 23500

Answer 1

okay? still a bit confuse

Answer 2

1) Define "best fits".

Answer 3

okay but its asking for the points a, b, and c ..not sure how to get the value from the equations

Answer 4

best fit is a line that fits the graph. doesn't have to hit all points but near the points.

Answer 5

You can just pick three of the values and set up three equations, as Hero has demonstrated. Then you get to pull from your pocket all your best algebra to solve the system of three equatios in three unknowns. Note: I wouldn't use the first three. Spread the choices out a bit. Unfortunately, this used the information from only three points. You are ignoring the ones you don't use.

"best fits" still not a good definition. There are infinitely many ways to "fit the graph". You need a plan. How shall we decide what makes one curve better than another?

Answer 6

okay?? i understand best fit but not sure how I would get it from the three points. how would you describe best fit?

Answer 7

I would make a row of \(\Delta x\) and then a row of \(\Delta \Delta x\)

Answer 8

\(2a = \Delta \Delta x\)

Answer 9

39.5 41636
49.5 45693
59.5 41477

Notice how here \(\Delta x\) changes from positive to negative numbers. This means we have a vertex near (49.5,45693)

Answer 10

A parabola (quadratic equation) is given by \[
y-k=a(x-h)^2
\]Where \((h,k)\) is the vertex.

Answer 11

okay i understand that.

Answer 12

\(a\) measures steepness. Basically \(2a\) = \(\Delta \Delta x\)

Answer 13

Again, the second differences (\(\Delta\Delta x\)) are a great way to proceed. This method can provide yet another approximation. However, the second differences are not constant, so you'll have to pick something that might be representative of all of them.

Answer 14

If you wanted to try "Least Squares", and use all the data, minimizing the squared differences, \(y = -42.964x^{2} + 4066x - 50426\) might prove useful. However, you'll have to learn something about Normal Equations to get there.

Answer 15

okay let me just think about this for a hour or so...im quite a slow learner.

Answer 16

No you're not. Whoever told you that intended to provide an excuse to fail. Please reject this theory. It can be quite appropriate to take some time to ponder. It has nothing to do with being "slow". If you're anywhere near this material, you can't possibly be "slow". Never believe that again.

Answer 17

HAHA THANK YOU! boosted up my energy!

Answer 18

okay i got the answer!
a=-42.964
b=4066.005
c=-50426.042

Answer 19

Sweet. How did you do that?

Answer 20

calculator lol

Answer 21

there is a quadreg button

Answer 22

Very good. Now you know how your calculator does it. That is a "Least Squares" solution. The interested student can look that up and read all about it.

Would you like to talk some more about how "slow" you are?! Seriously, excellent work and good, creative thinking.

Answer 23

haha thanks! im not slow just a person who likes to take her time!!