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.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
$f$.