Linear vs non linear question?

Question

I had a question in regards to linear and non linear graphs;

If for example; one is comparing time (x axis) and velocity (y axis)

& the table is as follows:

X;Y
0;0
1;1.6
2;4.2
3;5.7
4;8.4
5;11

I observe that the velocity increase hovers roughly around 2.3 for each second increase; but because this increase is not exact can it still be linear? (I graphed it and the dots to me seemed close enough to create a 'best fit' line; am I then correct that this is a 'best fit' linear line? Or is it a nonlinear parabolic line?

Thanks much

Answer 1

It's a rising linear function. Parabolic functions make a little "U" shapes in different directions.

Answer 2

Hovers?

The least squares fit has a slope of 2.197 call it 2.2. Not sure what is hovering around 2.3 - perhaps a UFO?

In the absence of a physical model of the motion, a "best fit" line is (for a single independent variable) generally a polynomial - (such as a straight line or a quadratic (what you call parabolic). It is a fact that for n points a nth order polynomial will fit the points EXACTLY.

A rough comparison of the linear r² vs the quadratic r² shows that the quadratic is a better fit than the linear. But since there are more degrees of freedom (adjustable parameters ( coefficients of the powers of x )) we would expect a better fit as we increase the degree of the polynomial. ( although depending on the shape of the curve, often a lower order even/odd polynomial will fit better than the next higher power (odd/even).

One indication that the quadratic might be better [besides the r²=0.996 vs 0.992] is that at both ends the linear best fit line is below the end points - suggesting that the "real" line may be curved.

Choosing a best fit line is a judgment call. There are some pseudo-mathematical formulas to help decide, but none that I'm aware of that are solidly based.

With just 5 points, and with a difference in r² only in the third decimal place, my opinion is that parsimony should rule. you should choose the simplest polynomial that "adequately" describes the curve. My vote is for the linear fit.

BTW "nonlinear parabolic" is that compared to a "linear parabolic" curve?



y=2.2x - 0.34

Finally, graphing will often indicate the shape of a curve, but it is a poor way to determine the "best fit". You should use least squares analysis since a graph can be constructed so that the fit may seem quite bad but the regression coefficient r² is quite high (quite high is over 0.95, say - true in Physics but 0.6 is "quite high" for Psychology! ) or constructed so that the fit seems good but r² is low.

[[FYI - The best fit is obtained with a valid physical model, even if fitting the parameters of that model results in a lower r² than a polynomial.]]

Answer 3

Hello Linear refers to a function with an equation of y = mx+b. M being the slope and b equaling the y-intercept. So an example would be y = 5x -4 or y = 7x. Non linear is any equation that is not in this form. Thus the equation of a parabola would work: x^2 - 10x + 5. Hope this helps

Answer 4

Well this is definitely not a parabolic line because I graphed it on my TI-84 and it does seem like a best-fit line.

Answer 5

Just go with the best fit line.