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