Curve Fitting

Linear Function

Consider a function of the form y = a1x + a0.

Two Points

If two distinct points (x1,y1) and (x2,y2) are known, then the coefficients a0 and a1 are given by

a0 = x2y1 - x1y2
    ------------
       x2 - x1

a1 = y2 - y1
    --------
     x2 - x1

One Point and a Slope (Gradient)

If one point (x1,y1) is known, along with the slope m, then the coefficients a0 and a1 are given by

a0 = y1 - mx1

a1 = m

Quadratic Function

Consider a function of the form y = a2x2 + a1x + a0.

Three Points

If three distinct points (x1,y1), (x2,y2) and (x3,y3) are known, then the coefficients a0, a1 and a2 are given by

a0 = y1x2x3(x3 - x2) + y2x1x3(x1 - x3) + y3x1x2(x2 - x1)
    -------------------------------------------------
               (x2 - x1)(x3 - x1)(x3 - x2)

a1 = y1(x22 - x32) + y2(x32 - x12) + y3(x12 - x22)
    -------------------------------------------
            (x2 - x1)(x3 - x1)(x3 - x2)

a2 = y1(x2 - x3) + y2(x1 - x3) + y3(x2 - x1)
    ---------------------------------------
           (x2 - x1)(x3 - x1)(x3 - x2)

Two Points and a Slope (Gradient)

If two distinct points (x1,y1) and (x2,y2) are known, along with the slope z3 at x = x3, then the coefficients a0, a1 and a2 are given by

a0 = y1x2(x2 - 2x3) - y2x1(x1 - 2x3) + z3x1x2(x2 - x1)
    -----------------------------------------------
                (x2 - x1)(x1 + x2 - 2x3)

a1 = 2y1x3 - 2y2x3 + z3(x22 - x12)
    ----------------------------
      (x2 - x1)(x1 + x2 - 2x3)

a2 = -y1 + y2 + z3(x1 - x2)
    -----------------------
    (x2 - x1)(x1 + x2 - 2x3)

Note that these results are invalid if x3 = (x1 + x2)/2.

One Point and Two Slopes (Gradients)

If one point (x1,y1) is known, along with the slopes z2 at x = x2 and z3 at x = x3 (x2 and x3 being distinct values), then the coefficients a0, a1 and a2 are given by

a0 = 2y1(x2 - x3) - z2x1(x1 - 2x3) + z3x1(x1 - 2x2)
    ---------------------------------------------
                      2(x2 - x3)

a1 = -z2x3 + z3x2
    -------------
        x2 - x3

a2 =  z2 - z3
    ----------
    2(x2 - x3)

Note that these results reduce to a linear function if z2 = z3.