Consider the linear function $f(x) = mx + b$.
This form for a linear function is usually described as the "slope- Y-intercept" form of $f$.
The number $m$ is referred to as the "slope" and the value of $f(0) = b$ gives the second coordinate of the Y-intercept, $(0,b)$ for the graph of $f$
There are one other very useful form for the linear function $f$ that uses $m$, the slope of $f$, and the value of $f$ at any input. This form is called the "point -slope" form of $f$.
If we assume the slope of $f$ is $m$ and $f(x_0) = y_0$, then the point $(x_0,y_0)$ will be on the graph of $f$ and one can find $f(x)$ by using the equation
$f(x) = y_0 + m(x-x_0)$.
Expanding the last equation gives $f(x) = y_0 +mx -mx_0 = mx + (y_0 -mx_0) = mx +b$ where $ b = y_0 -mx_0$.
These two forms, as well as how to find the value of $m$ from two values for $f$, are covered thoroughly in most algebra texts.

In this subsection we explore how mapping diagrams for linear functions help visualize these forms and the interpretation of $m$ as a magnification factor and a rate.
These help in understanding further the meaning of the X- and Y- Intercepts of the graph of $f$ through using mapping diagrams.

Example  LF.FORM.1 : Illustrating the slope = $m=2$ and the Y-intercept is $3$.

Example LF.FORM.2 : Illustrating the slope $=m=2$ and $f(-1) = 3$.

Example LF.FORM.3 : Illustrating a linear function with $f(1)  = 2$ and $f(-1) = 4$.

Example LF.FORM.4 : Illustrates a linear function as the composition of the three core linear functions  $f_{+ k}\circ f_{*m}\circ f_{ -h}$.

Example LF.DFORM.0 Dynamic Visualization of Slope and Point-Slope Form for a Linear Function: Graphs and Mapping Diagrams

To be added... A dynamic example to visualize finding $m$ and the point slope form of a linear function with a mapping diagram of $f$  and the lines in the graph of $f$.