Sample of CollegeLevel Math Attributes in College Algebra
Prepared
by: Nancy Priselac,
Bob
Carson,
Debra
Loeffler, Community College of
Course Description:
Students will study the nature and scope of college mathematics through the study of real valued functions. Topics include graphing functions, equations and inequalities, polynomials and rational functions, inverse functions, and exponential/ logarithmic functions. Applications to real life are discussed.
Typical Outcomes:
Students successfully completing this course will be able to:
1. Gain facility in factoring, radicals, absolute values, literal equations, rationals, variation, exponential and logs. using algebraic/geometric skills.
2. Find solutions for linear, quadratic, cubic, quartic,, exponential, rational equations, and inequalities.
3. Identify types of relations/functions, e.g. polynomials, rationals, radicals, exponentials, logs, absolute value, and greatest integer.
4. Graph relations and functions using information about the functions, such as, zeros and properties, e.g. increasing, decreasing, asymptotes, intercepts, symmetry, shifts.
5. Perform operations on relations and functions, e.g. addition, subtraction, multiplication, division, composition, inverses.
6. Solve equations and inequalities containing two or more unknowns.
Several of the following topics will also be required.
1. Combinations, sequences, and series, e.g. expanding a binomial using the binomial theorem.
2. Partial Decomposition
3. Conics

Graphing & Function Notation 

Attributes

College Algebra 
Intermediate Algebra 

1 1,4 1,4 
Determine whether the points A(2,2), B(4,3), and C(2,2)
are vertices of a right triangle. Use
a figure to explain your answer mathematically using core concepts to
validate your conclusion. (Keedy and Bittinger 140). A
manufacturer
of custom windows uses rows and columns of oneunit panes, where the number
of rows is always 1 greater than the number of columns. The cost of a window is $45 per unit pane. a.
If a window has x columns, how is the number of rows
represented? Then what expression
represents the total number of oneunit panes? b.
Write a function C for the cost of a window with x
columns. c.
If a window can have no more than 6 rows, what is the domain of
function C? d.
Evaluate C to determine the total cost of a window with 4
columns. (Hubbard/Robinson
126) The table below gives the
total amount of bottled water consumed annually (in billions of gallons) by
Americans for each year from 1996 to 2001.
e. According to your model, in what year
will total bottled water consumption exceed
six billion gallons? What assumptions do you
need to make? Note
Skills needed from Intermediate Algebra: Linear
equations, solving linear equations for the dependent variable, graphing
lines and points, slope, etc. 
Topics
from Introductory Algebra are reviewed and extended. Distance between two points and the
midpoint of a line segment are introduced.
Numerically some of the problems involve points whose coordinates are
irrational numbers. Fitting lines to
data and correlation are introduced.
Application problems are more complicated than those of Introductory
Algebra. The hardest application problems for students are typified by the
following: If (2, 0) and (0, 5) are
points on the graph of y = mx + b, what are m and b? Each Sunday, a newspaper
agency sells x copies of a certain newspaper for $1.00 per copy. The cost to the agency of each newspaper is
$0.50. The agency pays a fixed cost
for storage, delivery, and so on of $100 per Sunday. a. Write the equation that
relates the profit P, in dollars, to the number x of copies sold. b. Graph your
equation. a.
What is the profit to the
company, if 5000 copies are sold? An economist wishes to
estimate a line, which relates personal consumption expenditures (C) and
disposable income (I). Both C and I
are in thousands of dollars. She
interviews 8 heads of households for families of size four and obtains the
following data: C 16 18
13 21 27
26 36 39 I 20 20
18 27 36
37 45 50 Let I represent the
independent variable. a.
Use a graphing utility to
draw a scatter plot. b.
Use a graphing utility to
fit a straight line to the data. c.
Interpret the slope. The slope of this line is called the marginal propensity to Income. d.
Predict the consumption of
a family whose disposable income is $42,000. 

2
1
1,4 2,3 
To determine when a forest
should be harvested, forest managers often use formulas to estimate the
number of board feet a tree will produce.
A board foot equals 1 square foot of wood, 1 inch thick. Suppose that the number of board feet y
yielded by a tree can be estimated by y=f(x)=15+0.004(x10)^{3}
where the diameter of the tree in inches measured at a height of 4 feet above
the ground. Graph y=f(x) for
10≤x≤25. Partial
Fractions: Decompose
into partial fractions 5x + 7 x^{2} + 2x  3 The data in the table below
gives the results of a study that was conducted to determine the relationship
between average hours of sleep per night and death rate per 100,000 males.
Death Rate related to Sleep
Given f and g described by f(x)
= 8 – x and g (x) = √(2x + 3)
(Keedy/Bittinger 1190) Note
Skills needed from Intermediate Algebra: Parabolas,
graphing nonlinear functions, solving quadratic equations, etc. 
Graphing
of quadratic equations is extended to include recognizing when a quadratic
equation has complex solutions.
Students should be asked to recognize the shape of other polynomial
function, in particular cubic and quartic equations, and identify its maximum
number of roots. In addition, they may
be asked to find the xintercepts for some cubic and quartic equations by
factoring. Graphing calculators may be
used to estimate intercepts and max/min points. Graphing of rational functions shall be
introduced along with the concept of asymptotes. Use of a calculator to fit a quadratic,
cubic or quartic equation to a data set may be required. The hardest application problems for
students are typified by the following: Graph the following
functions, finding approximate and exact values (if possible) for the x and
yintercepts. Determine the
multiplicity of the roots, the power function the graph resembles for large
values of x, the number of turning points and any asymptotes. Estimate all local maxima and minima. _{}, _{}, and _{} Explain how you tell from
its equation that a polynomial is a parabola. The height H, in feet, of
a projectile with an initial velocity of 96 ft./sec launched from 120 ft.
above ground level is given by the equation _{}, where t = time in seconds. Sketch the graph of this function and find
the following.


2

Ellipses: Sketch the graph of each
equation, find the coordinates of the foci, and find the lengths of the major
and minor axis. 9x^{2}
+ 16y^{2} = 144 and 2x^{2} + y^{2} = 10 
Graphing
of circles and finding their center and radius are introduced. In addition, parabolas, ellipses, and
hyperbolas may be introduced. The
hardest application problems for students concerning circles are typified by
the following: Find the center and radius
of the following circle and sketch its graph: _{}. 

3,5 3,5 
How
far can you see to the horizon through an airplane window at a height of
30,000 ft? A
person can see 144 miles to the horizon from an airplane window. How high is the airplane? (Keedy
and Bittinger, 114115) Use a graphing
calculator to estimate the real solutions to the following equations and, if
possible, find the exact solutions algebraically.
_{} 
Equations
containing radicals are often introduced in Intermediate Algebra. The hardest problems for students
concerning equations containing radicals are typified by the following: Distance
to the horizon. The formula V = 1.2 √h
can be used to approximate the distance V, in miles, that a person can see to
the horizon from a height h, in feet. 

2 2 2 
Find
the
domain N(x) = 1/ ^{3}√ (x^{2} – 1) Express h as a composition of two
simpler functions f and g of the form f(x) = x^{n} and g(x) = ax + b
where n is a rational number and a and b are integers. H(x) =
(4/√x) +3 A time management consultant finds that the length L
of a meeting (in minutes) can be modeled by the function L(n) =
10(n^{2}n), where n is the number of people (up to
5) attending the meeting.
(Hubbard/Robinson
121) When a stone is dropped into a
pond, the radius of the circular ripple increases at a rate of 1.5 feet per
second.
(Hubbard/Robinson
292) 
Functions
are defined, along with function notation.
The concept of domain and range are introduced. Composition of functions is also
defined. The hardest problems for
students concerning function notation are typified by the following: Find the domain and range
of the function _{}. Find _{}if, _{} 

2 1,4 1 
Graph each equation. Explain if a function, describe any
similarities and differences: Y^{2}
= 8x 16x^{2}
+ 25 y^{2} = 400 9y^{2}
– 16x^{2} = 144 The table gives the total
number of stock funds and bond funds in selected years. (Source: Investment
Company Institute.) Year Number of funds 1991
244 1993 653 1994 756 1996
541 a.
By examining a scatterplot of the data, decide what type of model is
appropriate. b.
Let x represent the number of years since 1990 and determine a
quadratic regression equation to model the data. (Round coefficients to the nearest
integer.) c.
Use the model to estimate the year(s) in which the number of funds is
approximately 400. d.
Estimate and interpret the vertex of the graph of the model function. (Hubbard/Robinson
189) A function f is
given. In parts a and b, produce the
graphs of the associated functions in the same coordinate system and describe
the graphs in comparison to the graph of f. In part c, write a function whose graph is
described. f(x) = x^{3}
(Hubbard/Robinson
214) 
Families
of functions such as _{} are introduced. Symmetry about the xaxis, y axis and origin are usually covered. 

1 
Consider the expression x^{2}
– 2x + c, where c is a number of your choosing.
(Hubbard/Robinson
90) 
In
intermediate algebra students are often asked to solve quadratic and rational
inequalities of the type _{}, and to graph inequalities of the type _{} and _{} for functions such
as: _{}. Note:
sometimes these skills may be asked for indirectly in problems such as ones
asking student to find the domain and range of the functions, expressing
their answers in interval notation. 

1
1

Find the inverse of f(x) = 4x
– x^{2}, x ≥ 2. Graph
f, f^{1} and y = x in the same coordinate system. Let f(x) =
√(x + 2). Determine a
rule for f^{1} (Hubbard/Robinson
301) Find the inverse,_{}, of the function _{}. Use interval
notation to state the domain and range of both _{}. In a short paragraph,
explain the procedure you should use to find the inverse of a function. 
Students
should be able to recognize onetoone functions, find the graph of their
inverse as the reflection across the line y
= x, and find their inverse algebraically. Students should also understand the
relationship between the domain and range of a function, and that of its
inverse function. The hardest problems
for students concerning inverse
functions are typified by the following: 

2
2 3,5 
Find
a
piecewise function of f (x) that does not involve the absolute value
function. Sketch the graph, and find
the domain and range and any points of discontinuity. Graph:
f(x) = │x│/x Solve: │3u  2│ = u^{2} 
In
Intermediate Algebra, students may be asked to solve both graphically and
algebraically absolute value equations or inequalities, such as, _{}. The greatest
integer function is sometimes introduced in Intermediate Algebra. 

2
2
3,5

Bacterial Growth: If bacteria in a
certain culture double every ½ hr, write an equation that gives the number of
bacteria N in the culture after t hours, assuming culture has 100
bacteria at the start. Graph the equation
for 0≤t≤5. Solve each equation: (x3)e^{x}=0 Find
the domain of _{} (Sullivan,
426) Solve
_{} (Sullivan,
427) Note
Skills needed from Intermediate Algebra: Finding,
graphing and determining exponential functions, using logarithms as an
inverse to exponential functions, etc. 
Exponential
functions are usually introduced in Intermediate Algebra. Log functions are defined as the inverse of
the exponential function; however, the laws of logarithms are not usually
stressed. Students are expected to
recognize both types of functions and to graph them. The hardest problems for students
concerning exponential and log functions are typified by the following: A model for the number of
people N in a community college who have heard a certain rumor is N
= Pe^{0.15d}, where P is the total
population of the community college and d is the number of days that have
elapsed since the rumor began. In a
community of 1000 students, find the following:
The model year and prices
of used Honda Accords are given in the table below (not shown). Draw a scatter plot of this data and use
your calculator to find the equation of the curve that best fits this
data. (Note to the reader: the data
fits an exponential function.) 

SOLVING
EQUATIONS AND INEQUALITIES





Intermediate Algebra 

3,5 
Going into the final exam,
which will count as 2/3 of the final grades, Mike has test scores of 86, 80,
84, and 90. What score does Mike need
on the final in order to earn a B, which requires an average score of
80? What does he need to earn an A,
which requires an average of 90? (Sullivan
103) Suppose that the speed limit on a
long stretch of interstate is 70 mph.
(Hubbard/Robinson
292) 
The
topics from Introductory Algebra are reviewed and extended in a condensed
format. The types of problems given
are of a more complex nature. Work
problems with more in depth reasoning are focused on. Typical examples: In a class of 62 students, the number of
females is one less than twice the number of males. How many females and how
many males are there in the class? A sum of $10,000 is split
between 2 investments, one pays 9%, another pays 11%. If the return of the 11% investment is 60
dollars more per year than the 9%, how much is in each fund? 

3,5 
Solve for r, if _{} (Sullivan 94) 
Equations
that contain more than one variable are referred to as literal
equations. Sometimes referred to as
formulas. Practice is given in solving
for a given variable, i.e., solving E = mc^{2} for m. Sample application problems would be: How many gallons of a 15%
salt solution need to be mixed with a 35% salt solution to obtain 8 gallons
of 30% salt solution? 

3,5

The force F (in newtons) required to maintain an
object in a circular path varies jointly with the mass m (in kilograms) of
the object and the square of its speed v (in meters per second) and inversely
with the radius r (in meters) of the circular path. The constant of proportionality is 1. Write an equation relating F, m, v, and r. A motorcycle with mass 150 kilograms is
driven at a constant speed of 120 kilometers per hour on a circular track
with a radius of 100 meters. To keep
the motorcycle from skidding, what frictional force must be exerted by the
tires on the track? (Sullivan,
198) 
Most
texts have a section or unit on direct and inverse variation or proportionality
between two variables, and they discuss the constant of proportionality
application problems. They include
problems from all areas of science.
Typical examples of problems students find the hardest are: A varies jointly as b
& h. If A = 120, when b = 6 and h = 5, find A when b = 12 and h = 10. The volume of gas (V)
varies directly as temperature (T) and inversely as pressure (p). Set up equation and evaluate the constant
of proportionality of V = 48 when T = 320 and P= 20. 

1

You are the manager of a
clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the
regular price, you plan to have a sale giving 40% off the original selling
price. However, you still want to make
a profit of $4 on each shirt at the sale price. What should you price the shirts at
initially to ensure this? If, instead
of 40% off at the sale, you give 50% off, by how much is your profit reduced? (Sullivan
105) 
The
focus of each method is reviewed. A
discussion of the discriminate to predict the number of solutions is
generally gone over and the applications go into more depth. Quadratic equations with complex number
solutions are usually introduced. The
most difficult application problem encountered are typified by: A 62’ wire that makes an
angle of 60 degrees with the ground is attached to a telephone pole. Find the distance from the base of the pole
to the point on the pole where the wire is attached. Express your answer to the nearest 10^{th}
of a foot. 

1
1 
A truck firm wants to
purchase a maximum of 15 new trucks that will provide at least 36 tons of
additional shipping capacity. Model A
truck holds 2 tons and costs $15,000 and model B truck holds 3 tons and costs
$24,000. How many trucks of each model
should the company purchase to provide the additional shipping capacity at
the minimum cost? What is the minimum
cost? In
your
Economics 101 class, you have scores of 68, 82, 87, and 89 on the first four
of five tests. To get a grade of B,
the average of the first five test scores must be greater than or equal to 80
and less than 90. Solve an inequality
to find the range of the score that you need on the last test to get a B. (Sullivan
135) 
A
review of the basic techniques is covered. All topics are extended and they
also involve absolute value, quadratic and rational inequalities. Occasionally linear inequalities in two
variables are covered. Interval
notation is usually introduced. The
graphing calculator is often used to aid in finding the solutions. These problems may be the ones that
students find the most difficult, because they combine many separate skills
into one multistep problem. They
require perhaps the greatest amount of synthesis of any problem requiring
only manipulative skills. The most
difficult problems encountered are typified by: Solve the following
inequalities, expressing your answer in interval notation. (x + 2)(x – 7) < 0 x^{2} + 2x – 7
> 0 7x – 6 < 22 _{} 

2
3,5 
Safety Research: If a person driving
a vehicle slams on the brakes and skids to a stop, the speed v in
miles per hour of the vehicle at the same time the brakes are applied is
given approximately by v=f(x)=C=√x where x is the length in feet
of the skid marks and C is a constant that depends on the road conditions and
the weight of the vehicle. On the same
set of axis, graph v=f(x), 0<=x<=100, for C=3,4, and 5. Solve: √(3x^{2})=8
√(2x)4√2 
What
has been covered in introductory algebra is reviewed, and the techniques for
solving equations containing radicals are furthered by investigating some
more complicated situations. Typical
problems: _{} _{} Solve for w: _{}. 

1

A
5 horsepower (hp) pump can empty a pool in 5 hours. A smaller, 2 hp pump empties the same pool
in 8 hours. The pumps are used
together to begin emptying this pool.
After two hours, the 2 hp pump breaks down. How long will it take the larger pump to
empty the pool? (Sullivan,
105) 
This
topic is usually reviewed in Intermediate Algebra as part of a general review
to solving equations that usually includes linear, quadratic and rational
equations together. Further synthesis
may be stressed by investigating what happens as _{}in the equation _{}. Typical examples: _{} 

3,5

Solve algebraically and
graphically: 2x – 3 = x + 5 5 – 6x ≤ 10 + 7x (Keedy
and Bittinger, 233) 
A
much fuller discussion of absolute value equations and inequalities are dealt
with here on this level. This Distance
from 0 idea is usually the major application.
Typical examples: 2x + 1 > 1 x + 4<0 

1

Katy, Mike, Danny, and Colleen
agreed to do yard work at home for $45 to be split among them. After they finished, their father
determined that Mike deserves twice what Katy gets, Katy and Colleen deserve
the same amount, and Danny deserves half of what Katy gets. How much does each one receive? (Sullivan
896) 
Usually
the basic techniques are reviews and then expanded to 3 x 3 systems. Some intermediate algebra courses use
linear algebra (matrices and determinants) approaches to solutions. Typical example: Solve by any method: _{} 

3,5
3,5 
Solve each equation: _{} _{} 
Exponential
equations are often introduced in Intermediate Algebra. Logs are used to find solutions for
variables that are in the exponent of an exponential equation. In most application problems the
exponential equation is explicitly given.
The hardest application problems are typified by: The
annual profit P of a company due to the sales of a particular item after it
has been on the market for x years is determined to be P = $100,000  $60,000
(½)^{x} a.
What is the profit after 5
years? (10 years?) b.
When will the profit be
$80,000? c.
What is the most profit
that the company can expect from this product? 

Manipulative
Skills not Implied by the Previous Topics





Intermediate Algebra 

3,5 6 6 6 
Write
the
expression as a single quotient in which only a positive exponent and/or
radical appears: (x² + 4)^{½}  x²(x²
+ 4)^{½} x² + 4 (Sullivan
78) Expand: (u – v)^{5}. (Keedy
and Bittinger 721) Find the 8^{th} term
in the expansion of (2x – 5y)^{6}. (Keedy
and Bittinger 723) At one point in a recent
season, Darryl Strawberry of the Los Angeles Dodgers had a batting average of
0.313. Suppose he came to bat 5 times
in a game. The probability of his
getting exactly 3 hits is the 3^{rd} term of the binomial expansion
of (0.313 + 0.687)^{5}. Find
that term and use your calculator to estimate the probability. (Keedy
and Bittingeer 725) 
Facility
with exponents is assumed of students entering Intermediate Algebra, so this
topic is usually reviewed as it is needed in other problems. If negative exponents are not covered in
Introductory Algebra, they are covered here.
Problems tend to be more complex and rational exponents are usually
included. The complexity of the
problems is typified by: [9x²y^{1/3}]^{½} x^{1/3}y _{}0 

4 1,4 4 2,3 
Find the horizontal and
oblique asymptotes to H(x) = x^{4
}+ 2x² + 1 x²  x +
1. (Sullivan
327) Build a table and graph: f(x)
= 1
x^{2}
(Keedy
and Bittinger 271) Determine the vertical asymptote
and explain your reasoning f(x) = 3x – 2____ x(x –
5)(x + 3) (Keedy
and Bittinger 273 Given 3x – 2____ x(x – 5)(x
+3)
(Keedy/Bittinger,
273) 
In
Intermediate Algebra polynomial long division is often reviewed as part of a
more complex task, such as finding oblique asymptotes for rational functions.
Typical example: Find the horizontal and
oblique asymptotes to_{}. 

2

Find the complex zeros of each
polynomial function and write f in factored form. _{} (Sullivan
386) _{} (Sullivan
389) Solve
the equation: _{} (Sullivan
389) 
Factoring
of trinomials is usually assumed of students entering Intermediate
Algebra. It is usually reviewed in the
context of other problems, such as, solving quadratic equations. 

2,3
2 3 
Show graphically 2 + 2i and 3
– i. Show also their sum. (Keedy/Bittinger,
543) Find
3 + 4i (Keedy/Bittinger,
544) Find i^{29} (Keedy/Bittinger,
91) 

