If given a graph that was either a quadratic function, or a quartic function, how would you tell which it was?Only given a graph of a parabolic shape, nothing else.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Assuming that you know that the graph is of a quadratic or a quartic (not some higher even degree, a part of an odd degree, or something else), the easiest way to confirm which it is seems to be using 2nd order differences.

Choose 4 points that are equally spaced in the x-coordinates. (Pick a starting x-coordinate, and then go up by the same amount e.g. x=1,2,3,4). List the function outputs (the y values). Then subtract pairwise. Repeat:

Ex: `y=4x^2` ; you would find the points (1,4),(2,16),(3,36),and (4,64)

   4     16     36     64
     \   /   \   /   \   /
       12     20    28
        \    /    \    /
          8         8

The row 12,20,28 is called the first-order differences.

The row 8,8 is called the second-order differences.

** See notes on Newton's method of finite differences**

Since the second order differences are the same, the function is quadratic.

Ex: Let `y=x^4` ; you will need at least 6 points. You will get (1,1),(2,16),(3,81),(4,256),(5,625),(6,1296)

   1     16     81     256     625     1296
     \   /   \   /   \   /    \   /     \    /
       15    65    175    369     671
        \    /  \    /   \    /    \    /
          50    110    194     302
          \     /    \     /   \    /
            60        84      108
             \   /       \     /
              24           24

Since the 4th order differences are the same, it is a quartic.

In practice, unless you have rational coefficients, you will not find lattice points to work with (a lattice point being a point with whole number coordinates) but you can approximate with good results. (This also holds true for experiments -- if you are looking for a polynomial model for a set of data, try finding finite differences to see what degree you should use)

The graphs of `y=4x^2,y=x^4` :

Sources:

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