Curve Fitting

Discussion

order general form name
0   y = a   constant
1   y = a + bx   linear
2   y = a + bx + cx2   quadratic
3   y = a + bx + cx2 + dx3   cubic
4   y = a + bx + cx2 + dx3 + ex4   quartic
5   y = a + bx + cx2 + dx3 + ex4 + fx5   quintic
     ⋮           ⋮     
n  
n
y = a0x0 + a1x1 + a2x2 + a3x3 +… =  Σ aixi
i = 1
  nth order polynomial
Polynomial relationships summarized

exponential growth

Description:

General form.

y = anbx

A suitable conclusion statement from such a relationship would be that…

The ratio of successive iterations is a constant. The quantity is multiplied by a fixed amount at regular intervals.

Appearance: asymptotic with negative x-axis, followed by runaway expansion

Example(s): unrestricted population growth, the magic of compound interest

exponential decay

Description:

General form.

y = anbx

A suitable conclusion statement from such a relationship would be that…

The ratio of successive iterations is a constant. The quantity is divided by a fixed amount at regular intervals.

Appearance: large initial value followed by abrupt collapse, approaches positive x-axis asymptotically

Example(s): radioactive decay, discharging a capacitor, de-energizing an inductor

exponential approach

Description:

General form.

y = a (1 − nbx) + c

A suitable conclusion statement from such a relationship would be that…

Appearance: asymptotically approaches a horizontal line

Example(s): charging a capacitor, energizing an inductor, teaching (half the students get it, then half of the remaining students get it, then half of the remaining students get it, and so on…)

periodic

Description:

General form.

y = a sin (bx + c)

A suitable conclusion statement from such a relationship would be that…

Appearance: A sine curve is the prototypical example, not the only example. Any curve that repeats itself is periodic.

Example(s): Any daily (diurnal), monthly (lunar), yearly (annual, seasonal), or other periodic change.