Mathematics for the
Laboratory Sciences:
College Algebra, Precalculus,
and Up
Sheldon P. Gordon
gordonsp@farmingdale.edu
The Mathematics Curriculum
At most schools, the mathematics curriculum is
focused on moving students
up the mathematics pipeline:
either to become math majors or to serve the
traditional needs of engineering and physics
curricula.
But these students are only a small minority of
the students we see and whose needs we
should be serving.
Bachelor’s Degrees in Mathematics
In 2005,
P There were 1,439,264 bachelor’s degrees
P Of these, 14,351 were in mathematics
This is less than one percent!
(There are 23,000 degrees in recreation and leisure!)
The Needs of Our Students
The reality is that virtually none of the students
we face are going to be math majors.
They take our courses because of requirements
from other disciplines.
What do those other disciplines want their
students to bring from math courses?
Voices of the Partner
Disciplines
CRAFTY’s Curriculum
Foundations Project
Curriculum Foundations Project
A series of 11 workshops with leading
educators from 17 quantitative
disciplines to inform the mathematics
community of the current mathematical
needs of each discipline.
The results are summarized in the MAA
Reports volume: A Collective Vision:
Voices of the Partner Disciplines, edited
by Susan Ganter and Bill Barker.
What the Physicists Said
• Conceptual understanding of basic
mathematical principles is very important
for success in introductory physics. It is
more important than esoteric
computational skill. However, basic
computational skill is crucial.
• Development of problem solving skills is a
critical aspect of a mathematics education.
What the Physicists Said
• Courses should cover fewer topics and
place increased emphasis on increasing the
confidence and competence that students
have with the most fundamental topics.
What the Physicists Said
• The learning of physics depends less
directly than one might think on
previous learning in mathematics. We
just want students who can think. The
ability to actively think is the most
important thing students need to get
from mathematics education.
What the Physicists Said
• Students should be able to focus a
situation into a problem, translate the
problem into a mathematical
representation, plan a solution, and then
execute the plan. Finally, students should
be trained to check a solution for
reasonableness.
What the Physicists Said
• Students need conceptual understanding
first, and some comfort in using basic
skills; then a deeper approach and more
sophisticated skills become meaningful.
Computational skill without theoretical
understanding is shallow.
What the Biologists Said
• New areas of biological investigation have
resulted in an increase in quantification of
biological theories and models.
• The collection and analysis of data that is central
to biology inevitably leads to the use of
mathematics.
• Mathematics provides a language for the
development and expression of biological
concepts and theories. It allows biologists to
summarize data, to describe it in logical terms, to
draw inferences, and to make predictions.
What the Biologists Said
• Statistics, modeling and graphical representation
should take priority over calculus.
• The teaching of mathematics and statistics should
use motivating examples that draw on problems
or data taken from biology.
• Creating and analyzing computer simulations of
biological systems provides a link between
biological understanding and mathematical
theory.
What the Biologists Said
The quantitative skills needed for biology:
• The meaning and use of variables, parameters,
functions, and relations.
• To formulate linear, exponential, and logarithmic
functions from data or from general principles.
• To understand the periodic nature of the sine and
cosine functions.
• The graphical representation of data in a variety
of formats – histograms, scatterplots, log-log
graphs (for power functions), and semi-log
graphs (for exponential and log functions).
What the Biologists Said
Other quantitative skills:
• Some calculus for calculating areas and average
values, rates of change, optimization, and
gradients for understanding contour maps.
• Statistics – descriptive statistics, regression
analysis, multivariate analysis, probability
distributions, simulations, significance and error
analysis.
• Discrete Mathematics and Matrix Algebra –
graphs (trees, networks, flowcharts, digraphs),
matrices, and difference equations.
What the Biologists Said
• The sciences are increasingly seeing students who are
quantitatively ill-prepared.
• The biological sciences represent the largest science
client of mathematics education.
• The current mathematics curriculum for biology
majors does not provide biology students with
appropriate quantitative skills.
• The biologists suggested the creation of mathematics
courses designed specifically for biology majors.
• This would serve as a catalyst for needed changes in
the undergraduate biology curriculum.
• We also have to provide opportunities for the biology
faculty to increase their own facility with mathematics.
What the Chemists Said
• Six themes for the mathematical
preparation of chemistry students
emerged. Mathematicians are asked to
keep these six themes in mind as courses
in mathematics are redesigned.
What the Chemists Said
• Multivariate Relationships: Almost all
problems in chemistry from the lowly ideal
gas law to the most sophisticated applications
of quantum mechanics and statistical
mechanics are multivariate.
• Numerical Methods: Used in a host of
practical calculations, most enabled by the
use of computers
What the Chemists Said
• Visualization: Chemistry is highly visual.
Practitioners need to visualize structures and
atomic and molecular orbitals in three
dimensions.
• Scale and Estimation: The stretch from the
world of atoms and molecules to tangible
materials is of the order of Avogadro’s
number, about 1024. Distinctions of scale,
along with an intuitive feeling for the
different values along the scales of size, are of
central importance in chemistry.
What the Chemists Said
• Mathematical Reasoning: Facility at
mathematical reasoning permeates most of
chemistry. Students must be able to follow
algebraic arguments if they are to
understand the relationships between
mathematical expressions, to adapt these
expressions to applications, and to see that
most specific mathematical expressions can
be recovered from a few fundamental
relationships in a few steps.
What the Chemists Said
• Data Analysis: Data analysis is a widespread
activity in chemistry that depends on the
application of mathematical methods. These
methods include statistics and curve fitting.
Health-Related Life Sciences
• “Many participants put special emphasis on the use
of models.” “Models are a way of organizing
information for the purpose of gaining insight and
providing intuition into systems that are too
complex to understand any other way”.
• “Students should master a higher level interface,
e.g.: spreadsheet, symbolic/numerical
computational packages( e.g. Mathematica, Maple,
Matlab), statistical packages.
• BE FLEXIBLE: package topics creatively thru
long-term interaction between mathematics and the
life sciences.
What Business Faculty Said
• Courses should stress problem solving,
with the incumbent recognition of
ambiguities.
• Courses should stress conceptual
understanding (motivating the math with
the “why’s” – not just the “how’s”).
• Courses should stress critical thinking.
• An important student outcome is their
ability to develop appropriate models to
solve defined problems.
What Business Faculty Said
Mathematics is an integral component of the business
school curriculum. Mathematics Departments can help
by stressing conceptual understanding of quantitative
reasoning and enhancing critical thinking skills.
Business students must be able not only to apply
appropriate abstract models to specific problems but
also to become familiar and comfortable with the
language of and the application of mathematical
reasoning. Business students need to understand that
many quantitative problems are more likely to deal
with ambiguities than with certainty. In the spirit that
less is more, coverage is less critical than
comprehension and application.
What Business Faculty Said
• Courses should use industry standard
technology (spreadsheets).
• An important student outcome is their
ability to become conversant with
mathematics as a language. Business
faculty would like its students to be
comfortable taking a problem and casting
it in mathematical terms.
What the Majority of Students Need
• Conceptual Understanding, not rote manipulation
• Realistic applications and mathematical
modeling that reflect the way mathematics
is used in other disciplines and on the job
• Fitting functions to data
• Statistical reasoning
• Recursion and difference equations – the
mathematical language of spreadsheets
Conceptual Understanding
Everybody talks about emphasizing Conceptual
Understanding, but
• What does conceptual understanding mean?
• How do you recognize its presence or absence?
• How do you encourage its development?
• How do you assess whether students have
developed conceptual understanding?
What Does the Slope Mean?
Comparison of student response to a problem on the final
exams in Traditional vs. Reform College Algebra/Trig
Brookville College enrolled 2546 students in 1996 and 2702 students
in 1998. Assume that enrollment follows a linear growth pattern.
a. Write a linear equation giving the enrollment in terms of the year t.
b. If the trend continues, what will the enrollment be in the year 2016?
c. What is the slope of the line you found in part (a)?
d. Explain, using an English sentence, the meaning of the
slope.
e. If the trend continues, when will there be 3500 students?
Responses in Reform Class
1. This means that for every year the number of students
increases by 78.
2. The slope means that for every additional year the number of
students increase by 78.
3. For every year that passes, the student number enrolled
increases 78 on the previous year.
4. As each year goes by, the # of enrolled students goes up by 78.
5. This means that every year the number of enrolled students
goes up by 78 students.
6. The slope means that the number of students enrolled in
Brookville college increases by 78.
7. Every year after 1996, 78 more students will enroll at
Brookville college.
8. Number of students enrolled increases by 78 each year.
Responses in Reform Class
9. This means that for every year, the amount of enrolled
students increase by 78.
10. Student enrollment increases by an average of 78 per year.
11. For every year that goes by, enrollment raises by 78
students.
12. That means every year the # of students enrolled increases
by 2,780 students.
13. For every year that passes there will be 78 more students
enrolled at Brookville college.
14. The slope means that every year, the enrollment of students
increases by 78 people.
15. Brookville college enrolled students increasing by 0.06127.
16. Every two years that passes the number of students which is
increasing the enrollment into Brookville College is 156.
Responses in Reform Class
17. This means that the college will enroll .0128 more students
each year.
18. By every two year increase the amount of students goes up
by 78 students.
19. The number of students enrolled increases by 78 every 2
years.
Responses in Traditional Class
1. The meaning of the slope is the amount that is gained in years
and students in a given amount of time.
2. The ratio of students to the number of years.
3. Difference of the y’s over the x’s.
4. Since it is positive it increases.
5. On a graph, for every point you move to the right on the xaxis. You move up 78 points on the y-axis.
6. The slope in this equation means the students enrolled in 1996.
Y = MX + B .
7. The amount of students that enroll within a period of time.
8. Every year the enrollment increases by 78 students.
9. The slope here is 78 which means for each unit of time, (1
year) there are 78 more students enrolled.
Responses in Traditional Class
10. No response
11. No response
12. No response
13. No response
14. The change in the x-coordinates over the change in the ycoordinates.
15. This is the rise in the number of students.
16. The slope is the average amount of years it takes to get 156
more students enrolled in the school.
17. Its how many times a year it increases.
18. The slope is the increase of students per year.
Understanding Slope
Both groups had comparable ability to calculate the slope of a
line. (In both groups, several students used x/y.)
It is far more important that our students understand what
the slope means in context, whether that context arises in a
math course, or in courses in other disciplines, or eventually
on the job.
Unless explicit attention is devoted to emphasizing the
conceptual understanding of what the slope means, the
majority of students are not able to create viable
interpretations on their own. And, without that understanding,
they are likely not able to apply the mathematics to realistic
situations.
Further Implications
If students can’t make their own connections with a concept as
simple as the slope of a line, they won’t be able to create
meaningful interpretations and connections on their own for
more sophisticated mathematical concepts. For instance,
• What is the significance of the base (growth or decay factor) in
an exponential function?
• What is the meaning of the power in a power function?
• What do the parameters in a realistic sinusoidal model tell
about the phenomenon being modeled?
• What is the significance of the factors of a polynomial?
• What is the significance of the derivative of a function?
• What is the significance of a definite integral?
Further Implications
If we focus only on developing
manipulative skills
without developing
conceptual understanding,
we produce nothing more than students
who are only
Imperfect Organic Clones
of a TI-89
Should x Mark the Spot?
All other disciplines focus globally on the entire universe of a
through z, with the occasional contribution of  through .
Only mathematics focuses on a single spot, called x.
Newton’s Second Law of Motion: y = mx,
Einstein’s formula relating energy and mass: y = c2x,
The Ideal Gas Law: yz = nRx.
Students who see only x’s and y’s do not make the connections
and cannot apply the techniques when other letters arise in
other disciplines.
Should x Mark the Spot?
Kepler’s third law expresses the relationship between the
average distance of a planet from the sun and the length
of its year.
If it is written as y2 = 0.1664x3, there is no suggestion of
which variable represents which quantity.
If it is written as t2 = 0.1664D3 , a huge conceptual
hurdle for the students is eliminated.
Should x Mark the Spot?
When students see 50 exercises
where the first 40 involve solving for x,
and a handful at the end involve other letters,
the overriding impression they gain is that x is the only
legitimate variable and the few remaining cases are just
there to torment them.
Some Illustrative
Examples and Problems
for
Conceptual Understanding
and Mathematical Modeling
Identify each of the following functions (a) - (n) as linear, exponential,
logarithmic, or power. In each case, explain your reasoning.
(g) y = 1.05x
(h) y = x1.05
(i) y = (0.7)x
(j) y = x0.7
(k) y = x(-½)
(l) 3x - 5y = 14
(m)
x
y
(n)
x
y
0
0
3
5
1
1
5.1
7
2
2
7.2
9.8
3
3
9.3
13.7
For the polynomial shown,
(a) What is the minimum degree? Give two different
reasons for your answer.
(b) What is the sign of the leading term? Explain.
(c) What are the real roots?
(d) What are the linear factors?
(e) How many complex roots does the polynomial have?
The following table shows world-wide wind power
generating capacity, in megawatts, in various
years.
Year
1980 1985 1988 1990 1992 1995 1997
1999
Wind
power
10 1020 1580 1930 2510 4820 7640 13,840
15000
10000
5000
0
1980
1985
1990
1995
2000
(a) Which variable is the independent variable and which is
the dependent variable?
(b) Explain why an exponential function is the best model to
use for this data.
(c) Find the exponential function that models the relationship
between power P generated by wind and the year t.
(d) What are some reasonable values that you can use for the
domain and range of this function?
(e) What is the practical significance of the base in the
exponential function you created in part (c)?
(f) What is the doubling time for this exponential function?
Explain what does it means.
(g) According to your model, what do you predict for the total
wind power generating capacity in 2010?
A Temperature Experiment
An experiment is conducted to study the rate at which
temperature changes. A temperature probe is first
heated in a cup of hot water and then pulled out and
placed into a cup of cold water. The temperature of the
probe, in ̊C, is measured every second for 36 seconds
and recorded in the following table.
Time 1
42.3
31
8.78
2
3
4
5
6
7
8
36.03 30.85 26.77 23.58 20.93 18.79 17.08
32
8.78
33
8.78
34
8.78
35
8.66
Find a function that fits this data.
36
8.66
A Temperature Experiment
The data suggest an
exponential decay function,
but the data do not decay to 0.
To find a function, one first
has to shift the data values
down to get a transformed set
of data that decay to 0.
y = T – 8.6 = 35.439(0.848)t
Temperature (degrees C)
45
40
35
30
25
20
15
10
5
time (1 - 36 seconds)
Then one has to fit an exponential function to the
transformed data. Finally, one has to undo the
transformation by shifting the resulting exponential
function. T = 8.6 + 35.439(0.848)t.
Modeling the Decay of a Drug
Every drug is washed out of the bloodstream,
usually by the kidneys, but at different rates.
For example, in any 24-hour period, about
25% of any Prozac in the blood is washed out,
leaving 75% of the amount.
This suggests an exponential decay function.
If the initial dosage is 80 mg. then the model is
D(t) = 80 (0.75)t, t in days.
Predictive Questions
What will the level of Prozac (or any drug) be
after 7 days (or any given number of time
periods)?
How long will it take until the level of Prozac is
down to 10 mg (or to any given level)?
What is the half-life of Prozac in the blood?
Repeated Doses of a Drug
• 25% of the Prozac in the blood is
washed out each day, leaving 75%
• Typical dose is 40 mg each day
Level of Prozac
D0 = 40
D1 = .75(40) + 40 = 30 + 40 = 70
D2 = .75(70) + 40 = 92.5
D3 = .75(92.5) + 40 = 109.375
{40, 70, 92.5, 109.375, 122.031, 131.523, …}
Level of Prozac
D0 = 40
D1 = .75 D0 + 40
D2 = .75(D1 ) + 40
D3 = .75(D2) + 40
D4 = .75(D3) + 40
In general, after any number of days n,
we have the difference equation:
Dn+1 = .75 Dn + 40
Difference Equation Model
Dn+1 = .75 Dn + 40
D0 = 40
D1 = .75(D0 ) + 40 = 70
D2 = .75(D1 ) + 40 = 92.5
D3 = .75(D2 ) + 40 = 109.375
D4 = .75(D3 ) + 40 = 122.031
Solution to the Difference Equation:
{40, 70, 92.5, 109.375, 122.031, 131.523, …}
Solution of the Difference Equation
200
Level of Prozac
160
120
80
40
0
0
5
10
15
Number of Days
20
What Happens if an Overdose?
Level of Prozac
D0 = 400
D1 = .75(400) + 40 = 340
D2 = .75(340) + 40 = 295
D3 = .75(295) + 40 = 261.25
400
360
320
280
240
200
160
120
80
40
0
0
5
10
Number of Days
15
20
Finding the Maintenance Level
Dn+1 = 0.75 Dn + 40
Assume that Dn+1 = L and Dn = L
L = 0.75L + 40
L = 40 /0.25 = 160 mg.
Creating a Formula for the Solution
n
0
1
Dn 40 70
160-Dn 120 90
2
92.5
67.5
3
109.4
50.6
4
122.0
38.0
160 - Dn = 119.99961 (0.7500021) n
Dn = 160 - 119.99961 (0.7500021) n
5
131.5
28.5
How Well Does it Fit?
200
160-D n
150
100
50
0
0
5
10
Number of days
15
20
Finding the Solution in General
Diff. Eqn: Dn+1 = 0.75 Dn + 40
Solution: Dn = 160 - 120(0.75) n
In general:
Dn+1 = b Dn + C
Dn = L - (L - D0) @ bn
Proving the Solution in General
Dn+1 = b Dn + C
Dn = L - (L - D0) @ bn
LHS: Dn+1 = L - (L - D0) @ bn+1
RHS: b Dn + C = bL - b(L - D0) bn + C
= bL - (L - D0) bn+1 + C
= bL - (L - D0) bn+1 + C
= bL - (L - D0) bn+1 + L(1 - b)
= L - (L - D0) @ bn+1
The Species-Area Model
Biologists have long observed that the larger the area of a
region, the more species live there. The relationship is best
modeled by a power function. Puerto Rico has 40 species of
amphibians and reptiles on 3459 square miles and Hispaniola
(Haiti and the Dominican Republic) has 84 species on 29,418
square miles.
(a) Determine a power function that relates the number of
species of reptiles and amphibians on a Caribbean island to its
area.
(b) Use the relationship to predict the number of species of
reptiles and amphibians on Cuba, which measures 44218
square miles.
Island
Area
N
Redonda
1
3
Saba
4
5
Montserrat
40
9
Puerto Rico
3459
40
Jamaica
4411
39
Hispaniola
29418
84
Cuba
44218
76
Number of Species
The accompanying table and associated scatterplot give
some data on the area (in square miles) of various
Caribbean islands and estimates on the number species of
amphibians and reptiles living on each.
100
80
60
40
20
0
0
15000
30000
Area (square miles)
45000
(a) Which variable is the independent variable and which is
the dependent variable?
(b) The overall pattern in the data suggests either a power
function with a positive power p < 1 or a logarithmic function,
both of which are increasing and concave down. Explain why a
power function is the better model to use for this data.
(c) Find the power function that models the relationship
between the number of species, N, living on one of these islands
and the area, A, of the island and find the correlation
coefficient.
(d) What are some reasonable values that you can use for the
domain and range of this function?
(e) The area of Barbados is 166 square miles. Estimate the
number of species of amphibians and reptiles living there.
The Dead Body Problem
Mr. Jones' body was found in his kitchen at 9
am by the police who noted that the body
temperature was 77.3̊ F and that the room
temperature was 70̊. An hour later, the
medical examiner found the body
temperature was 76.1̊. You may presume that
Mr. Jones' body temperature was the normal
reading of 98.6̊ at the time of death. At what
time was he murdered?
Write a possible formula for each of the following
trigonometric functions:
The average daytime high temperature in New York as a
function of the day of the year varies between 32F and
94F. Assume the coldest day occurs on the 30th day and
the hottest day on the 214th.
(a) Sketch the graph of the temperature as a function of
time over a three year time span.
(b) Write a formula for a sinusoidal function that models
the temperature over the course of a year.
(c) What are the domain and range for this function?
(d) What are the amplitude, vertical shift, period,
frequency, and phase shift of this function?
(e) Predict the most likely high temperature on March 15.
(f) What are all the dates on which the high temperature is
most likely 80?
Balancing Chemical Reactions
Photosynthesis
Carbon dioxide (CO2) plus Water (H2O) produces
Glucose (C6H12O6) plus Oxygen (O2).
How many molecules of each are needed?
If Glucose is the “target” molecule, then
x CO2 + y H2O - z O2 = 1 C6H12O6
Carbon:
1x + 0y – 0z = 6
Oxygen: 2x + 1y - 2z = 6
Hydrogen: 0x + 12y - 0z = 2
Need to solve this linear system of 3 equations in 3 unknowns
Farmingdale’s Math & Bio Project
Almost all math and bio curriculum
projects start at the calculus level or above.
But the overwhelming majority of
beginning biology students, both majors
and especially non-majors, typically are at
the college algebra or precalculus level.
Most of these students have avoided math
as much as possible.
The Mathematical Needs of Biology
• In discussions with biology faculty, it
became clear that most courses for nonmajors (and even those for majors in
some areas) use almost no mathematics
in class.
• Mathematics arises almost exclusively in
the lab when students have to analyze
experimental data and then their weak
math skills become dramatically evident.
Farmingdale’s Math & Bio Project
Our original plan was to develop the first
stages of a new mathematics curriculum to
serve the needs of biology students, both
the bioscience majors and the non-majors
who take introductory biology courses.
This would also impact the level of
quantitative work in the biology courses.
Farmingdale Math & Bio Project
Our first step was to develop an alternative
to our modeling-based precalculus course
that would focus almost exclusively on
biological applications.
The course would feature a lab component
taught by the biology faculty, so that each
week’s primary math topic would be
accompanied by an experiment requiring
the use of that mathematical method.
Course Topics
Week 1 Behavior of Functions
Week 2 Families of Functions,
Linear Functions
Week 3 Linear Functions
and Linear Regression
Week 4 Exponential Growth
and Decay Functions
Week 5 Exponential Regression
and Power Functions
Week 6 Power Functions and
Polynomials
Intro to Measurements
and Measuring
Linear Growth – part 1
Linear Growth – part 2
Exponential Growth
Exponential Decay
part 1
Exponential Decay –
part 2
Course Topics
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Week 13
New functions from Old
Logistic and Surge
Functions
Matrix Models and
Linear Systems
Sinusoidal Functions
and Periodic Behavior
Periodic Functions –
part 2
Probability Models
Probability Models and
Difference Equations
Power Function growth
Logarithmic Functions
Logistic Growth
Surge Functions
Polynomial growth
Periodic Behavior
Probability Model
(genetics)
What Happened Next
To accommodate the lab component, we
had to change the precalculus course from
four to five credits.
Because of that and conflicts with other
courses (intro chemistry), the biology
students did not register for the course and
it did not run.
What We’ve Done Instead
All the labs in the introductory biology
course are being changed to dramatically
increase the level of quantitative experience
– the new labs will incorporate most of the
experiments that were to be part of the
precalculus course.
What We’ve Done Instead
The math department has created a new
four credit precalculus course to serve the
needs of the biology students – basically,
the same math course without the lab.
The focus is on conceptual understanding,
data analysis, statistical reasoning, and
mathematical modeling, not on developing
algebraic skills (other than in a few special
cases where algebra is needed to solve
problems that arise naturally in context).
What We’ve Done Instead
The math department has also created a
new two-semester calculus sequence for
biology students – it also emphasizes
concepts over manipulation and stresses
biological applications.
The math department has created a onesemester post-precalculus course on
mathematical modeling in the biological
sciences for bioscience majors and applied
math majors.
The Next Challenge
Based on the Curriculum Foundations
reports and from discussions with
faculty in the lab sciences (and most
other areas), the most critical
mathematical need of the other
disciplines is for students to know more
about statistics. How do we integrate
statistical ideas and methods into math
courses at all levels?
The Curriculum Problems We Face
• Students don’t see traditional precalculus or
college algebra courses as providing any useful
skills for their other courses.
• Typically, college algebra is the prerequisite for
introductory statistics.
• Introductory statistics is already overly
crammed with too much information.
• Most students put off taking the math as long as
possible. So most don’t know any of the statistics
when they take the courses in bio or other fields.
Integrating Statistics into Mathematics
• Students see the equation of a line in pre-
algebra, in elementary algebra, in intermediate
algebra, in college algebra, and in precalculus.
Yet many still have trouble with it in calculus.
• They see statistics ONCE in an introductory
statistics course. But statistics is far more
complex, far more varied, and often highly
counter-intuitive, yet they are then expected to
use a wide variety of the statistical ideas and
methods in their lab science courses.
Integrating Statistics in Precalculus
Data is Everywhere! We should capitalize on it.
1. A frequency distribution is a function – it can be
an effective way to introduce and develop the
concept of function.
2. Data analysis – the idea of fitting linear,
exponential, power, polynomial, sinusoidal and
other functions to data – is already becoming a
major theme in some college algebra courses. It
can be the unifying theme that links functions,
the real world, and the other disciplines.
Integrating Statistics in Precalculus
3. The normal distribution function is
N ( x) 
1
 2
e
 ( x   )2 / 2 2
It makes for an excellent example involving
both stretching and shifting functions and
a function of a function.
Integrating Statistics in Precalculus
4. The z-value associated with a measurement x
is a nice application of a linear function of x:
z
x

Integrating Statistics in Precalculus
5. The Central Limit Theorem is another
example of stretching and shifting functions
-- the mean of the distribution of sample
means is a shift and its standard deviation
x 

n
produces a stretch or a squeeze, depending on
the sample size n.