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