MA112 – 1.3 Linear Functions, Slope and Applications
Today:
 quiz on finding domain
 1.3: lines: constant slope
Announcements
 Homework 1.3 assigned, due Wed
 Module 1 quiz next Wed/Fri
A. Line: Simplest Relationship Between x and y
The simplest possible relationship between x and y is that one depends upon the other in a
constant ratio.
Example: The number of eggs y given a certain
number of cartons x.
y=12x (the # of eggs = 12 times the # of cartons)
x
0
1
2
3
4
y
0
12
24
36
48
In general, an equation in which y and x have an unchanging relationship (unchanging
slope) is called a line and has form y=mx+b.
Line:
y = mx+b
B. Slope
A slope is



rise
.
run
If the rise is shallow, the
slope is small.
If the rise if steep, the
slope is big.
If the rise drops, the
slope is negative.
Constant slope.
Slope Between Two Points
The slope between two points P= x1 , y1  , Q= x2 , y2  ,=
Graph with small and large,
positive and negative slopes.
rise y 2  y1
=
.
run x 2  x1
A line is an equation with constant slope.
Since the slope never changes, there are never any bends or curves.
Two Special Cases
Generally, y = mx+b describes a line.
There are two special cases, when the line is horizontal (m=0) and the line is vertical
(m∞).
Notice for every point on the line, x changes but y =-3.
horizontal line:
thus the line is described by:
y = -3
Notice this makes sense considering the slope is 0 and
the y-intercept is -3:
y=mx+b
y = 0 x + (-3)
y = -3
Notice for every point on the line, y changes but x = 4.
vertical line:
by analogy with the horizontal line, the line is described
by:
x=4
This is harder to see in relation to the equation y=mx+b
because the slope here is undefined and there is no yintercept!
no ‘run’ … x+0y = 4