How do you find the equation for a curve between two points?

Question

Isabella Ackerman · Accepted Answer

In general, the equation of a curve in 2-d is of the form f(x, y; a, b, c, d, ..) = 0, where a, b, c, d, ... are parameters that take specific values for a particular curve. See explanation.

Two points are enough for determining the parameters m and c of the straight line f(x, y; m, c) = y-mx-c=0, If this passes through (0, 0) and (1, 1) then c = 0 and m = 1. The line is #y - x = 0# or for that matter any circle through the origin given by #f(x, y; a, b) = x^2+y^2+2ax+2by=0# If the circle passes though (0, 1) and (1, 0), a = b = #-1/2#, and the circle is given by #x^2+y^2-x-y=0#.. If the the number of parameters is more than two, two points are not sufficient. For example, consider #f(x,y;a, b, c)=y-a sin ( bx + c )=0# This has three parameters. So, three points are required to determine this sine curve,

Paisley Johnson · Answer

undefined To find the equation for a curve between two points, follow these steps:

1. Determine the coordinates of the two points on the curve: (x₁, y₁) and (x₂, y₂).
2. Use the slope formula to find the slope (m) of the line passing through these two points: m = (y₂ - y₁) / (x₂ - x₁).
3. Once you have the slope, you can use it to find the equation of the line in slope-intercept form (y = mx + b), where 'm' is the slope and 'b' is the y-intercept.
4. Plug the coordinates of one of the points into the equation and solve for 'b'.
5. Substitute the values of 'm' and 'b' into the equation to obtain the equation of the line.
6. If the curve is not a straight line but a curve, then you'll need to use an appropriate curve-fitting method or polynomial regression to find an equation that best fits the data points between the given points.

These steps will give you the equation for the curve passing through the two specified points.