Math Is Fun Forum

To construct a general function/method based on two sets of minimum/maximum data point constraints with a fixed starting point (y-axis intercept) and x-range, while maintaining a zero rate of change over time, you can use a piecewise linear function. This approach will allow you to vary the function as little as possible within the given constraints.

Here's an outline of the method:

Define your starting point, which is the y-axis intercept.
Determine the x-range over which your function will operate.
Divide the x-range into segments based on the constraints you have. Each segment will correspond to a different rate of change.
Calculate the slope for each segment, which represents the rate of change for that segment. The slope should be such that the segment stays within the given minimum and maximum constraints.
Create linear equations for each segment using the slope and the corresponding x-values.
Combine all the linear equations to form a piecewise function that satisfies the constraints and has zero overall rate of change.
Here's an example to illustrate the process:

Let's say we have the following constraints:

Starting point: (0, 10)
x-range: [0, 10]
Minimum constraint: (2, 8)
Maximum constraint: (8, 12)
To construct the function, we'll divide the x-range into two segments: [0, 2] and [2, 10].

For the first segment, the slope can be calculated as:
slope1 = (8 - 10) / (2 - 0) = -1

The linear equation for the first segment becomes:
y1 = -x + 10

For the second segment, the slope can be calculated as:
slope2 = (12 - 8) / (8 - 2) = 0.8

The linear equation for the second segment becomes:
y2 = 0.8x + 2.4

Combining both segments, we get the piecewise function:
f(x) = {
-x + 10, 0 <= x <= 2
0.8x + 2.4, 2 < x <= 10
}

This function satisfies the constraints, has zero overall rate of change, and varies as little as possible within the given constraints.

You can modify this approach to accommodate different sets of constraints by adjusting the number of segments and calculating the slopes accordingly.