Rate of Change
The slope of a function that describes real, measurable quantities
is often called a rate of change. In that case, the slope refers to
the change in one quantity (y) per unit change in another quantity
(x).
Converting rates of change is fairly straight forward so long as
one remembers the equations for rate (i.e., the equations for
slope).
Example 1:
Andrea has a part time job at the local grocery store. She saves for her
vacation at a rate of $15 every week. Express this rate as money saved
per day and money saved per year.
Rate per day:
$15
∙
1 week
1 week
7 days
$15
5
= 7 days = 7 dollars per day
≈ $2.14 per day
Rate per year:
$15
∙
1 week
52 weeks
1 year
= $15 ∙
52
year
= $780 per year
Modified from Introductory Algebra, by Andrew Gloag and Anne Gloag, CC-BY 2012, CK-12 Foundation,
www.ck12.org. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0)
A graph is used to visualize a rate of change. As a rule, the
vertical axis represents the quantity listed in the numerator, and
the horizontal axis represents the quantity listed in the
denominator. A scale is selected which allows for plotting the
information on a reasonably sized graph.
Example 2:
A candle has a starting length of 10 inches. Thirty minutes after lighting
it, the length is 7 inches. Determine the rate of change in length of the
candle as it burns. Determine how long the candle takes to completely
burn to nothing.
Since the candle length is a function of time, the horizontal axis is time in
minutes, and the vertical axis is the candle length in inches. As indicated
on graph, the moment the candle is lit (time = 0) the length of the candle
is 10 inches. After 30 minutes (time = 30), the length is 7 inches.
Modified from Introductory Algebra, by Andrew Gloag and Anne Gloag, CC-BY 2012, CK-12 Foundation,
www.ck12.org. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0)
Rate of Change = Slope =
y2 − y1
x2 − x2
=
=
(7 inches)−(10 inches)
(30 minutes)−(0 minutes)
−3 inches
30 minutes
= -0.1 inches per minutes
A negative rate of change indicates that the quantity is decreasing with
time. This is what would be expected with a burning candle.
The rate can also be converted to inches per hour:
−0.1 inches 60 minutes
1 minute
∙
1 hour
=
−6 inches
1 hour
= -6 inches per hour
To find the point when the candle reaches zero length, according to the
graph, the time is 100 minutes. This can be verified algebraically:
rate × time = length burned
-0.1 × 100 = 10
the original length
Modified from Introductory Algebra, by Andrew Gloag and Anne Gloag, CC-BY 2012, CK-12 Foundation,
www.ck12.org. Licensed under a Creative Commons Attribution 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0)