Patterns in Graphs and Equations
If you have ever heard someone attempt to justify their position on a controversial
issue or heard a news reporter explain the cause of a recent event and were not satisfied
with their assessment, it may be because you have seen evidence in the past that
conflicted with their assumptions. If the ultimate intent of debate or news coverage is to
clarify understanding and perhaps arrive at a consensus view of the truth, then the typical
process in which we engage these issues could use some revision.
Such revisions should start with a clearly stated question. The original question
may be of a philosophical nature such as:
Is it acceptable to use all the nonrenewable natural resources so they are not available for
future generations?
Should the US Government be allowed to borrow money?
Is our current approach to justice the best one?
Such questions, while potentially interesting for a debate, do not often lead to a
consensus between people who originally had opposing opinions. Along the way
however, it is possible that someone will mention a number. For example, someone
might say that there were originally an estimated 3 trillion barrels of oil on the planet or
that the US government has a debt in excess of 15 trillion dollars or that we currently
have 2.5 million people in prison. As such, these numbers do not provide much useful
information because they lack a point of reference. Consider how different these
numbers would be if there was a comparison that could be made. Would your
interpretation of the amount of oil used be different if we started with 3 trillion barrels of
oil and there were currently 2.5 trillion barrels remaining or if there were currently 0.5
trillion barrels remaining? Would your view of the current national debt be different if
the debt 10 years ago was 6 trillion or if it was 21 trillion?
The best way to have numbers make sense is to ask questions that would allow
the philosophical arguments to be more concrete. Examples that would correspond to the
original philosophical questions are:
How long will the nonrenewable resource last, given the past, present and future rates of
extraction?
What is the national debt and how does it compare to the size of the economy?
What has been the change in the number of incarcerated people in the US?
The best way to understand numbers is through graphs known as behavior-overtime graphs or time series graphs.
World oil production history and projections are shown in the following graph
produced by the US Energy Information Administration (EIA) and are available on the
website:
http://www.eia.gov/pub/oil_gas/petroleum/presentations/2000/long_term_supply/sld001.
htm. These projections were made in 2000, so that we can now see if they are on track
by looking at data that has been obtained between the time of the predictions and 2010.
World Oil Supply (Billion BBL/year)
Data From http://www.eia.gov/cfapps/ipdbproject/IEDIndex3.cfm
World Oil Supply (Billion BBL/year)
35
30
25
20
1975
1980
1985
1990
1995
2000
2005
2010
2015
A comparison of the actual data from 2010 to the projected value shows
approximate agreement. Consequently, confidence in the model is increased since the
model’s prediction has been confirmed 10 years after the prediction.
One of the biggest political issues is the size of the national debt. Democratic
Presidents have a reputation of increasing the national debt whereas Republicans are seen
as more fiscally conservative and in favor of smaller government. The graph below
shows the national debt through 2008, the year President Obama was elected. The solid
line shows the national debt.
A close look at the national debt may lead to the question of whether there is a
way to model the debt and therefore to be able to predict what happens in the future.
Such a prediction could indicate whether the current President and Congress are being
consistent with past trends or exceeding these expectations.
The national debt has been modeled with a dashed line. The shape of this dashed
line is nearly horizontal on the left side of the graph and approaching vertical on the right
side of the graph. While there is variation from this line for each President, the variation
is not substantial and the model looks reasonable. President Obama appears to be
following the pattern of national debt growth that was established by previous Presidents
from both parties.
US Federal Debt
http://www.usgovernmentspending.com/index.php
100
14000
80
12000
Obama
10000
8000
60
40
6000
4000
20
Gross Public Debt-fed pct GDP
120
Bush
Clinton
Bush
Reagan
Nixon
Ford
Carter
Eisenhower
Kennedy
Johnson
Truman
Roosevelt
Hoover
Harding
Coolidge
16000
Wilson
Gross Public Debt-fed $billion
18000
Taft
20000
140
Roosevelt
22000
2000
0
0
-2000
1900
-20
1920
1940
1960
Year
Gross Public Debt - fed $ billion(L) (1900-2008)
Gross Public Debt-fed pct GDP(R) (1900-2008 )
1980
2000
Gross Public Debt-fed $billion(L) (2009 -)
Gross Public Debt - fed pct GDP(R)(2009-)
Gross Public Debt-fed $billion = 1.90933142E-66*exp(0.0800364413*x)
The size of the US prison population has several implications. There are issues of
cost during tight budget times, safety of the American public, and how the “land of the
free” can have the highest incarceration rate of any country in the world.
(http://bjs.ojp.usdoj.gov, by George Hill & Paige Harrison, 2000)
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800
700
600
500
400
300
200
100
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Prisoners per 100,000
Top Ten Countries with Highest
Incarceration Rates
Top Ten Countries
The change in the US prison population is shown in the graph below. The data
comes from http://bjs.ojp.usdoj.gov, by George Hill & Paige Harrison, 2000.
It might be useful to model the prison population to determine what can be
expected in the future. First degree, second degree and third degree polynomial models
are considered. Which one appears to be the best fit and would lead to the best
predictions?
Linear (first degree) Model
Quadratic (second degree) Model
Cubic (third degree) Model
There are many issues that affect the public, which, in particular, should be
understood by college-educated individuals. A clearer understanding of these issues can
be gained by a quantitative analysis of historical data. By modeling this data with an
appropriate equation, the general patterns of change can be distinguished from the
random variation that naturally occurs. As you can see in the above graph, there are a
variety of shapes of lines that were used to model the data. Perhaps you noticed the
actual equation written into the titles. Our objective in this section is to recognize the
different types of equations that are common to find in community college level
mathematics and understand the graphic patterns that these equations make. Each
different type of equation produces a particular pattern for a graph. We will categorize
each type as one of the families of functions. A function is an equation for which every x
value produces exactly one y value. The families of functions you should be able to
identify by both the equation form and the graphic form are linear (first degree),
quadratic (second degree), exponential, radical, rational (inverse) and absolute value.
The key to identifying each of these functions and graphic examples of them are provided
below.
Family: Linear (first degree polynomials) Functions.
Linear are the most common function to encounter. They were studied in the
Math 60, Introduction to Algebra class. They reflect a constant rate of change between
points (slope). Linear equations are usually provided in standard form ax + by = c or
slope-intercept form y = mx + b. You can recognize a linear equation because it contains
an equal sign, both variables do not have an exponent that is shown (because it is 1) and
both variables are in the numerator. Examples of linear equations include:
1
3
x  3 , y  3 x  5 and y = x. As a contrast, the equation y   2 is NOT a
2
x
linear equation because the x is in the denominator. Use your knowledge of slopes and
intercepts to match each equation with the appropriate line.
y
Note: The equations for these lines are written using function notation, which we
will learn about later. The notation f(x) is read “f of x”. The function f(x) = (1/2)*x –3 is
1
equivalent to y  x  3 .
2
Family: Quadratic (second degree polynomials) Functions
While we will spend considerable time working with quadratic functions, they are
not used as frequently as linear functions. Quadratic equations are recognized because
they have an equal sign and the equation contains an x2 term, but does not contain any
higher exponent. All variables are numerator terms. Examples of quadratic equations
are y = x2 + 2x – 3 and y = -x2 + 3x + 4. As a contrast, x2 + 5x – 6 is a polynomial
expression that can be factored, but it is NOT a quadratic equation.
One observation you may make is that one of these parabolas opens up and the
other opens down. The one that opens up has a positive x2 term and the one that opens
down has a negative x2 term. This is not coincidental.
Family: Exponential
Exponential equations are very important functions, although we will only spend
a little time with them at the end of the quarter. Both the oil production graph and the
national debt graph were modeled with exponential models. These models are based on
geometric growth which means they grow by a constant percent. Exponential equations
are recognized because the x is the exponent, not the base. Of course they also have an
x
equal sign. Examples include y = 2x and y  0.1  3 . The equation y = 2x starts nearly
horizontal and transitions to becoming nearly vertical. It is an exponential growth curve.
The other graph starts nearly vertical and transitions to nearly horizontal. It is an
exponential decay graph. The –3 in the equation is what causes it to be shifted down 3
units.
Family: Radical
Radical equations are recognized because they have an equal sign and by the use
of a radical sign (e.g. square root symbol). Examples are y  x and y  2 x  3 .
This is the only family of graphs that we will study that cannot be drawn as a
border-to-border graph. It will have a distinct starting point but then can continue to a
border.
Family: Rational Functions
These can also be called inverse functions. These functions have an equal sign
and are distinguished from other functions by the x being in the denominator of a
1
fraction. The most basic example of a rational function is y  but they can be more
x
2
x 1
complex than that, for example: y 
. Notice these graphs have two parts. What
x 1
separates the parts is the location where the x value would cause the denominator to be 0.
1
This location is called the asymptote. For the equation y  , the asymptote is the y
x
axis. For the other equation, the asymptote occurs at x = 1.
Function: Absolute Value
The last family you should be able to identify is absolute value functions. They
are recognizable by the equal sign and the x variable enclosed by absolute value signs.
Examples include y = |x| and y = -|2x|-3. The pattern an absolute value function makes is
a V.
Expressions and Inequalities
All the functions the have just been shown are equations. You should also be able
to distinguish these from expressions and inequalities, which may look similar but do not
contain an equal sign. Expressions do not contain any symbol that shows a relationship
(=, <,>,,). Inequalities contain one of the inequality symbols for showing relationships
(<,>,,).
Equations
y = 3x + 2
y = 3x
y = |5x – 2|
Expressions
3x + 2
3x
|5x – 2|
Inequalities
y < 3x + 2
y  3x
y > |5x – 2|