1.4 Types of Functions and Their Rates of
Change
 the graph of f(x) = 2x + 1 is a straight line
 true for any linear function f(x) = mx + b, where m and
b are any real numbers
 note: f(1) = 3, f(2) = 5, f(3) = 7, f(4) = 9, etc.
 as the input increases by 1 unit, the outputs 3, 5, 7, 9,
etc. always increase by 2 (= m) units:
 as the input increases by 2 units, the output increases by
4 units
 the output always increases by the same amount for
equal increases in the input
 a function like f(x) = 3 (= 0x + 3) has m = 0, and is
called a constant function
 m is called the slope of the line
increase in output y

 and is the ratio
as you move
increase in input x
from any point on the line to another
Slope of a line
large
slope
(steep)
another way to think of slope:
small
slope
(gradual)
vertical rise
y
=
horizontal run
x
1.4-1
Computing slope, given two points
In general, given two points (x1, y1) and (x2, y2), the slope
of the line passing through them is
y2 - y1
m=x -x
(slope formula)
2
1
Kinds of slope
1.4-2
Slope as rate of change
In 1998, the value of an investment was $2540. It is
projected to in increase by $100 per year ($100/year)
 notice the “per” - that’s the key word for a rate
o think of “50 miles per hour” - it’s a rate (of speed)
 the $100/year is referred to as a rate of change

slope = rate = 100 $/yr



100



1




x
1.4-3
Average rate of change on an interval
 a non-linear (curved graph) function has no fixed slope
 therefore no constant rate of change
 but we can still talk about average rate of change (on a
given interval)
 consider traveling in a car that goes at a variable speed
o slow at first
o then gradually increasing in speed
 here’s the graph of distance vs. time (the solid line):
d



(3, 100)











(0, 0)




t

the car is at the 100 mile mark in 3 hours
what is its average speed (average rate of change)
on the interval 0 to 3 hrs?
r = d/t = 100/3 = 33 1/3 mph
and it is just the slope of the dotted line:
m = (100 - 0)/(3 - 0) = 33 1/3 mph
 for different intervals, the average rate of change would
be different
1.4-4