How to find an equation of the function $f$ which has a specific tangent line to that function's graph, at a specific point? E.g. how to find an equation of the function $f$ which has a specific tangent line to that function's graph e.g.: the equation of the tangent line to the function $f$ could be $y=\frac{1}{2}−\frac{3x}{2}$; and there's a point which lies both on the $f$'s graph and the graph of that tangent line, i.e.:
$\exists (1,−1) \in y \wedge \exists (1, -1) \in f$ | $y$ is a tangent line to non-linear $f$ at $(1, -1)$ where $y=\frac{1}{2}−\frac{3x}{2}$. How to find the equation of $f$?
Another example would look like this:
example
We're looking for the equation of the purple function; the red line is the purple's function tangent line and the point at which they meet is the specific point.
 A: A line with slope $m$ that passes through point $(h,k)$ will be tangent to any function of the form
$$
f(x) = k + m(x-h) + (x-h)^2 g(x).
$$
If you're given the line in the form $y = b + mx$ and want $f$ to be tangent at $x = h$, this can be written equivalently as
$$
f(x) = b + m x + (x-h)^2 g(x).
$$
This works because $(x-h)^2 g(x)$ is always zero and has zero derivative at $x = h$, so $f(x)$ will have the same value and slope as the line there. So for the line $y = 1/2-3/2 x$, a function tangent at $x = 1$ will have the form
$$
f(x) = \frac{1}{2} - \frac{3}{2} x + (x-1)^2 g(x)
$$
for some function $g(x)$.
Note that $g(x)$ can be any function that doesn't blow up at $x = h$. There might be other restrictions, but they're technicalities it's probably not worth getting into.
A: A tangent line is a straight line which has degree 1.
A tangent line to another straight line is not possible. It is only a parallel line with minimum intersection.
For a second or higher degree curve say $y= ax^2+bx+c$:
In order to find the equation of tangent you have to equate its derivative to the slope of the given equation
$$2ax+b= \frac{-3}{2}$$
so that there are two coinciding points of intersection.
