Sample of College-Level Math Attributes in College Algebra

                                 Prepared by:       Nancy Priselac, Garrett College;

                                                            Bob Carson, Hagerstown Community College;

                                                            Debra Loeffler, Community College of Baltimore County

 

 

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 one-unit 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 one-unit 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.

 

Year

1996

1997

1998

1999

2000

2001

Consumption

3.5

3.8

4.2

4.6

4.9

5.5

 

  1. Create a scatterplot of the data, with year on the horizontal axis. Sketch a line on your scatterplot representing the linear trend.
  2. Write the equation of a linear function that models annual consumption, C, as a function of t, where t represents the number of years since 1995.
  3. According to your model, by how much is Americans’ bottled water consumption increasing each year?
  4. Use your model to find the value of C when t = 7. Explain what this means.

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(x-10)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      

x2 + 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

Hours of Sleep

5

6

7

8

9

Death Rate

1121

769

626

692

967

 

  1. Use the method of finite differences to explain why a quadratic model would be appropriate for the relationship between death rate and average hours of sleep per night.
  2. Write the equation of a quadratic function that models death rate, D, as a function of average hours of sleep, h.
  3. Use your model to find the amount of sleep per night (to the nearest tenth of an hour) that would correspond to the lowest death rate. What would the death rate be for this amount of sleep?
  4. Give a possible reason why death rate goes up as the amount of sleep per night increases beyond the optimal level.
  5. Would stating that men should avoid sleeping more than seven or eight hours a night in order to lower their chances of death be a valid conclusion from this study? Explain.

 

Given f and g described by f(x) = 8 – x and g (x) = √(2x + 3)

  1.  Find (f+g)(5)  and (f+g)(-4)
  2.  Do both exist as real numbers?  Explain.

(Keedy/Bittinger 1190)

 

 

Note Skills needed from Intermediate Algebra:

Parabolas, graphing non-linear 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 x-intercepts 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 y-intercepts.  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.

  1. How many seconds after the launch is the projectile 128 ft. above the ground?
  2. What is the projectile’s maximum height and when does it reach that height?
  3. How many seconds after the launch does the projectile return to the ground?

 

2

Ellipses: Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor axis.

9x2 + 16y2 = 144 and 2x2 + y2 = 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, 114-115)

 

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√ (x2 – 1)

 

Express h as a composition of two simpler functions f and g of the form f(x) = xn 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(n2-n), where n is the number of people (up to 5) attending the meeting.

 

  1. For this situation, what is the domain of L?
  2. Create a table with the headings Number of People and Length of Meeting.  Then complete the table by evaluating L.
  3. Although 1 is not a meaningful domain element in this situation, evaluate and interpret L(1).

(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.

 

  1. Write a function r to describe the length of the radius at time t (in seconds).
  2. Write a function A that describes the area enclosed by the ripple in terms of the radius r.
  3. Write a composite function f that gives the area of the ripple as a function of time t.
  4. To the nearest tenth, what is the area of the ripple after 2.5 seconds?

(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:

Y2 = 8x

16x2 + 25 y2 = 400

9y2 – 16x2 = 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) = x3

  1. g(x) = (x + c)3 for c = -4, 1, 2
  2. h(x) = -x3 + c for c = -8, -2, 3
  3. The graph of y = -x3 shifted left 4 units

(Hubbard/Robinson 214)

Families of functions such as  are introduced.  Symmetry about the x-axis, y- axis and origin are usually covered. 

1

Consider the expression x2 – 2x + c, where c is a number of your choosing.

 

  1. Produce the graphs of this expression for c-values that are less than 1.  Which points of the graphs represent solutions of the equation x2 – 2x + c = 0?
  2. Repeat this experiment for c = 1 and for c >1.
  3. From the results in parts a and b, what is your conjecture about the possible number of real number solutions of a quadratic equation?

(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 – x2, 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 one-to-one 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│ = u2

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 ˝ 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:

(x-3)ex=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:

  1. How many students will have heard the rumor after 3 days?
  2. How many days will have elapsed before 450 students have heard the rumor?

 

 

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.

 

  1. Write a function f to describe the distance traveled in t hours if the driver maintains a constant speed that exceeds the speed limit by 10 mph.
  2. Write a function s to describe the distance traveled in t hours if the driver maintains a constant speed at the speed limit.
  3. Evaluate and interpret (fs)(2).

(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 = mc2 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 10th 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 multi-step 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

x2 + 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: √(3x2)=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 (˝)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:

 

             (x˛ + 4)˝ - x˛(x˛ + 4)

                          x˛ + 4

 

 

(Sullivan 78)

 

Expand: (uv)5.

(Keedy and Bittinger 721)

 

Find the 8th 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 3rd 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:

 

[9x˛y1/3]˝

x1/3y

0

4

 

 

 

 

 

 

1,4

 

 

 

 

 

 

4

 

 

 

 

 

 

2,3

Find the horizontal and oblique asymptotes to

 

               H(x) = x4 + 2x˛ + 1

                           x˛ - x + 1.

 

(Sullivan 327)

 

Build a table and graph: f(x) =  1

                                                  x2

           

x

 

 

 

 

 

 

f(x)

 

 

 

 

 

 

(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)

 

  1. Determine the vertical, horizontal and oblique asymptotes, if they exist.
  2. Explain the reasons for your response to (a).
  3. Sketch a graph and illustrate the asymptote(s).

 

(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 i29

 

(Keedy/Bittinger, 91)