# How to compare the similarity between functions?

I'm designing a web service that finds the regression function of a pattern within an image.

I analyzed three images and found the following three regressions:

1) $f(x) = 74.7602 + 0.2005x - 0.00091891x^2$ (domain: 0 ≤ x ≤ 200)

2) $f(x) = 102.337 + 0.349x - 0.002x^2$ (domain: 0 ≤ x ≤ 200)

3) $f(x) = 103.417 + 0.074x + x^2$ (domain: -8 ≤ x ≤ 8)

When these functions are plotted on a graph, functions 1 and 2 are more similar in shape. Function 3 is very narrow and is concave up (whereas functions 1 and 2 are concave down). How can I compare these three functions and mathematically determine that the curves of functions 1 and 2 are more similar? I don't care where on the coordinate plane the curves are, I just want to know if the shape is the same.

I was thinking about the obvious: plugging in points and checking for similarity, however this will not work if the function has a different scale. Do you know how I should begin?

P.S. I'm currently taking precalculus, so if you could describe this on my level (if possible), it would be greatly appreciated!

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Will all the functions you are going to compare be parabolas (of the form $a+bx+cx^2$)? The differences you describe are both due to the sign and value of the coefficient of $x^2$ in your functions. – Alfonso Fernandez Jan 10 '13 at 4:25
@AlfonsoFernandez Yes, they will all be parabolas. – Jack Humphries Jan 10 '13 at 4:27
@AlfonsoFernandez I understand how the coefficient of $x^2$ plays a role in the concave and width of the graph, but what does the coefficient of x do? – Jack Humphries Jan 10 '13 at 4:29
Every parabola can be rewritten in the form $a(x-b)^2 + c$. When in this form, $c$ controls the vertical elevation of the parabola, $b$ controls its horizontal position and $a$ is the $x^2$ coefficient as usual. So if you don't care about translations, you should get your parabolas to that form and compare $a$s. – Alfonso Fernandez Jan 10 '13 at 4:34