Apologies in advance, I'll try my best to explain clearly but I'm not versed in the communication of mathematical ideas. I'm also not sure whether my title is an accurate reflection of what I'm looking for.
Take a function $f(x) = y$.
Let's say we have two scenarios where $y$ has a different value, call them
$$f(x_1) = y_1$$
$$f(x_2) = y_2$$
Now what if I wanted to express $y_2$ in different terms?. Instead of expressing it in terms of its correspondent $x$ value, I wanted to express it in terms of the value that $y$ used to be $y_1$ and the change in the $x$ value
$$f(y_1, \Delta x) = y_2$$
I feel like the function above would just involve the $\Delta x$ value being plugged into the initial $f(x)$ function to determine the change in $y$, and then added to $y_1$ to determine $y_2$.
So my questions are
- Does this make any mathematical sense?
- If so, is it a pointless/non-beneficial rearrangement?
- And if not, does this form already exist in use/have any name or purpose?
Thanks
No worries! This is a well-written question. As it turns out, there is a function of the form:
$f(y_{1}, \Delta x) = y_{2}$
Where $\Delta x = x_2 - x_1$. It makes perfect mathematical sense and is not pointless. In Algebra, the measure of steepness between two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is known as slope:
$m = (y_2 - y_1)/ \Delta x$
Now we can rearrange this like so:
$y_2 = m \Delta x + y_1 = f(y_{1}, \Delta x)$
Which is the function you were looking for. This is important for creating a linear interpolation of two points, which is a form of curve fitting. You use curve fitting when you have a series of data points and attempt to find an analytical solution that describes them.