Consider the linear function $l(x) = mx +b$.

This form for a linear function is usually described as the "slope,  Y-intercept" form of $l$.

The number $m$ is referred to as the "slope" and the value of $l(0) = b$ gives the second coordinate of the Y-intercept, $(0,b)$ for the graph of $l$

The Point Slope Form: There is  another very useful form for the linear function $l$ that uses $m$, the leading coefficient of $l$, and two number $x_0$ and $y_0$ to determine $l(x)$.
In this form, $l$ is evaluated by $l(x) = m(x-x_0) +y_0 $.

This form is called the "point slope" form of $l$.  The value $l(x_0) = y_0$ gives the correspondence of the pair $(x_0,y_0)$ as being  a point on the graph of $l$ and the arrow $<x_0,y_0>$ as an arrow on the mapping diagram of $l$.

Connection between the Standard and Vertex Forms.
Expanding the point slope form equation gives  $l(x) = m(x-x_0) +y_0 = mx - mx_0 +y_0$.
From this expression we can see the connection between $b$, $m$, and the pair $(x_0,y_0)$, namely, $b = y_0 - mx_0$.

These two forms 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