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\$.