College Algebra

Exam File

Fall 2004

 

Exam #1

 

1.)        Find an equation for the line that goes through the points (-2, 1) and (4, -3).

2.)        Find an equation for the line on the point (1, 2) and perpendicular to the line      2x + 3y = 6.

3.)        Find the following for f(x) = 2x2 - 5x - 4

a.)        f(3)                  b.)        f(a + 3)                        c.)        f(x2)

4.)        The height of a ball thrown off the top of a tall building is given by the quadratic function s(t) = -16t2 + 45t + 500, where height is in feet and time is in seconds. 

a.)        How long would it take for the ball to hit the ground?

b.)        At what time(s) is the ball 510 feet high?

c.)        What is the maximum height attained by the ball and how long does it take for it to get to that height?

d.)        Bob, who is 6 foot 6 inches tall, is directly under the ball.  How long after the ball is thrown will it hit Bob on the head?

5.)        Without using your calculator, put the following quadratic function in vertex form and give the coordinates of the vertex.

 

f(x) = x2 - 6x - 13

 

 

6.)        Consider the graph of f(x).

 

f(x)

[image]

a.)        What is f(3)?

 

 

b.)        What is the slope of the line?

 

 

c.)        What is the y-intercept of the line?

 

 

d.)        What is an equation for the line?

 

 

 

7.)        Find the domain of the function

 

8.)        Consider the function f(x) = x3 - 2x2 - 2x + 3.  Use your calculator to find the following.

 

a.)        any local (relative) maxima.             b.)        any local (relative) minima.

c.)        any x-intercepts                                             d.)        any y-intercepts

 

Problems #9 and #10 refer to the following table which gives population data for Rogers, Arkansas.  Use x = 0 to represent 1900, x = 10 to represent 1910, etc.

 

Year

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

Population

2,820

3,318

3,554

3,550

4,962

5,700

11,050

17,429

24,692

38,829

 

9.)        Do the following

 

a.)        Find an equation for the linear function that best fits this data.

 

b.)        Find an equation for the quadratic function that best fits this data.

 

10.)     Answer the following questions.

 

a.)        Which function from #9 fits the data better?  Give a mathematical reason for your answer.

 

b.)        Which function from #9 is a better model for predicting past and future values for the population of Rogers?  Give a good reason for your answer.

 

c.)        Based on the quadratic model, what would the population of Rogers have been in 1900? 

 

d.)        Is the answer from c.) a reasonable answer?  Why or why not?

 

Exam #2

1.)        Solve the following equation by factoring.  Exact answers only.

 

x2 - 5x - 6 = 8

 

2.)        Solve the following equation exactly.         

 

3.)        Solve the equation exactly. 

 

4.)        Find all of the x-intercepts of g(x) = x4 - 2.3x3 - 4.2x2 + 5.3x + 1.7.

 

5.)        A sugar molecule is made up of hydrogen, oxygen and carbon.  It has twice as many atoms of hydrogen as it does oxygen and one more atom of carbon than oxygen.  The sugar molecule has a total of 45 atoms.

 

a.)        Write an equation, in ONE variable, that can be used to solve the problem of how many of each kind of atom there are in the molecule.

b.)        Use the equation in part a.) to find how many of each kind of atom there are in the molecule.

 

6.)        Suppose y varies directly as x.  If y = 3, when x = 9, find y when x = 4.

 

7.)        Solve the following inequality.  Give your answer in interval form.

 

-3x - 5 > 4

 

8.)        Solve the following inequality.  Give your answer in inequality form.  You may use your calculator.

 

9.)        Solve the following inequality.  Give your answer in inequality form.  Give the exact solution only.

2x2 - 5x - 5 > 0

 

10.)     Solve the equation exactly.  |5x - 13| = 112

 

11.)     Solve the inequality |8x2 - 7x| > 3.  You may use your calculator to solve it.

 

12.)     Define a polynomial function that could have the following graph.  (Hint:  Be sure to pay attention to the y-intercept.)

 

 

13.)     Graph the following function.

           

 

 

 

14.)     The following table gives seven different National League leading home run totals.  Use your calculator's regression capabilities to find a fourth degree polynomial that best fits this data.  Use "Year" as the x-value (with 1910 as x=0) and "Home Runs" as the y-value.

 

Year

Player (Team)

Home Runs

1999

Mark McGwire (STL)

65

1991

Howard Johnson (NYM)

38

1982

Dave Kingman (NYM)

37

1965

Willie Mays (SFG)

52

1949

Ralph Kiner (PIT)

54

1939

Johnny Mize (STL)

28

1920

Cy Williams (PHI)

15

 

            a.)        Write the fourth degree polynomial function that best fits the data.

 

b.)        Does the function fit the data well?  Give a mathematical reason for your answer.

 

c.)        Based on the function from part a.), what would the home run leading totals from 1964 and 2004 be?

 

15.)     Consider the function f(x) = x2 - x + 4.

 

a.)        Determine if f has symmetry with respect to the x-axis?  Show your work. 

b.)        Determine algebraically if f has symmetry with respect to the y-axis?  Show your work.

c.)        Determine algebraically if f has symmetry with respect to the origin?  Show your work.

 

Exam #3

1.)        Suppose the graph in the left hand grid is f(x).  On the right hand grid, graph

g(x) = -2 f(x-3) + 1.                                                    

 

 

 

 

 

 

 

2.)        Graph the following function.  Fill in all of the following.

 

 

x-intercept(s)_________________________     y-intercept(s)_____________________

does it have slant asymptote_____________      horizontal asymptote(s)_____________

vertical asymptote(s)___________________      local minima______________________

local maxima_________________________     absolute maximum_________________

absolute minimum_____________________     domain__________________________

range_______________________________    does it have x-axis symmetry_________

does it have y-axis symmetry_____________     does it have origin symmetry_________

intervals where increasing_______________      intervals where decreasing__________

 

3.)        Suppose f(x) = 3x2 + 2 and g(x) = x - 1.

a.)        Find (f - g)(3).             b.)        Find (f ° g)(2).

c.)        Find (g ° g)(2).                       d.)        Find the domain of

4.)        Does f(x) = |x| have an inverse function?  Give a reason for your answer.  If it has an inverse, find it.

 

5.)        Suppose f is a one-to-one function.  Suppose that (2, 5), (5, 7) and (7, 14) are all points on the graph of f.  Suppose that g is the inverse function of f.  Find all of the following that are possible.  If one is not possible to find based on the information given, write "not enough information."

 

            a.)        g(2)                                         b.)        g(5)

 

            c.)        g(7)                                         d.)        g(14)

 

6.)        Write each of the following as complex numbers in standard form.

 

a.)        (1 - 3 i) +  (2 + 2 i)                 b.)        (1 - 3 i) - (2 + 2 i)

c.)        (1 - 3 i)(2 + 2 i)                       d.)       

e.)        i856381116549873213571513456337839

 

7.)        Graph f(x) = 2x+3 - 4.  Be sure to include the asymptote(s).

 

8.)        The formula  gives the average atmospheric pressure, , in pounds per square inch, at an altitudemiles above sea level. 

 

a.)        Find the atmospheric pressure for an altitude for 2.8 miles.   Round your answer to the nearest hundredth.

 

b.)        If the atmospheric pressure around you is 4.6 pounds per square inch, how far above sea level are you?

 

9.)        Solve the given equation exactly.  No calculator.  All work must be shown.

 

42x-3 = 85x+1

 

10.)     Solve the equation using your calculator.

 

52x-4 = 70

 

11.)     Consider the following table of values.

x

11

13

15

16

17

19

22

24

y

8

9

10

12

18

34

72

121

           

            a.)        Draw a scatter plot of the data. 

 

b.)        Write the exponential function that best fits the data.

 

c.)        Draw the function from b.) on your scatter plot from a.).

 

d.)        Write the third degree polynomial function that best fits the data.

 

e.)        Draw the function from d.) on your scatter plot from a.).

 

f.)         Which function, the one from b.) or the one from d.) fits the data better.  Why?

 

g.)        Based on the exponential function, what will y be when x = 25?

 

12.)     Suppose you invest $8000 at an interest rate of 5.4 percent.  Find the balance after 4 years if the interest is compounded quarterly.  Be sure to write down the formula you are using.

 

13.)     Radium-224 has a half-life of 3.66 days.  Suppose you have 25 grams of Radium-224.  How much will you have after two weeks?

 

14.)     In 1904, Bubbaville had a population of 1200.  In 1913, the population was 2400.  Assuming the growth is exponential and the percentage growth rate remains the same, what is the population in 2004?

 

           

15.)     Make up your own problem and solve it.  (Hint:  You get no extra points for making it beyond first grade level.)

 

Exam #4

 

1.)        Suppose logb 2 = A, logb 3 = B and logb 5 = C.  Find the given quantities.

 

a.)        logb (2/3)                    b.)        logb 20                        c.)        logb 2.5

 

2.)        Use your calculator to solve the following equation.

 

 

3.)        Without using your calculator, solve following the equation.

 

log3 4 + log3 x = 2

 

4.)        Without using your calculator, solve for x in each of the following.

            a.)        log2 8 = x                                b.)        log3 x = 4                    c.)        logx 4 = 1/2

 

5.)        Solve the following system of equations (no calculator) using the method of elimination.

2x + y = 8

3x - 2y = 12

 

6.)        Use your calculator to solve the following system of equations.  Explain how you used your calculator.

5.4x - 3.75y = 14.2

6.21x + 4.8y = ep

 

7.)        Solve the following system of equations (no calculator) using the method of substitution.

x2 + y2 = 20

x + 2y = 0

 

8.)        Use your calculator to solve the following system of equations.  Explain how you used your calculator.

y + 3 = x log x

(x - 4)2 + y2 = 2

 

9.)        Graph the solution to the following system of inequalities

2x - y £ 5

x + y £ 4

 

10.)     Find the maximum and minimum values of the objective function z = 4x - 5y over the quadrilateral with vertices (0, 0), (4, 0), (0, 7) and (9, 5).

 

11.)     Suppose an = (2n + 1)/n2.  Find the following:

            a.)        a1                       b.)        a2                       c.)        a20                     d.)       

12.)     Suppose we define the following sequence.

 

a1 = 4

an = 3an-1 - 4n, n ³ 2

 

            Find the following:

            a.)        a1                           b.)        a2                       c.)        a3

 

13.)     Consider the arithmetic sequence with a1 = 2 and a3 = 8 (pay CLOSE attention to the subscripts).  Find the following:

            a.)        a20                     b.)         

14.)     Consider the geometric sequence with a1 = 4 and a2 = 8.  Find the following:

 

            a.)        a20                     b.)       

 

15.)     For each of the following infinite geometric series, determine whether or not the sum exists (be sure to state your reason).  If the sum exists, find it.

 

            a.)        a1 = 4 and a2 = 8

 

            b.)        a1 = 4 and a2 = -3 

 

Final Exam

 

1.)        Suppose f(x) = 3x2 + 2 and g(x) = x - 1.

a.)        Find (f - g)(3). b.)        Find (g ° g)(2).

 

c.)        Find the domain of

 

2.)        This problem refers to the following table which gives population data for Rogers, Arkansas.  Use x = 0 to represent 1900.

 

year

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

y

2,820

3,318

3,554

3,550

4,962

5,700

11,050

17,429

24,692

38,829

 

            Do the following

 

a.)        Find an equation for the exponential function that best fits this data.

 

b.)        Is it a good fit?  Why or why not?

 

c.)        Based on the exponential model, what would the population of Rogers have been in 1900?

 

3.)        Without using your calculator, put the following quadratic function in vertex form and give the coordinates of the vertex.

 

f(x) = x2 - 6x - 13

 

4.)        Solve the following quadratic equation.     x2 + 5x - 14 = 0

 

5.)        Solve the following equation by the quadratic formula.  3x2 - x - 7 = 8.

 

6.)        Without using your calculator, solve the equation exactly. 

 

7.)        Solve the following inequality.              7 - 5x < 10

 

8.)        Solve the following inequality.          x2 - 4x - 5 > 16

 

9.)        Solve the following equation.

|3x - 2| = 11

 

10.)     Find an equation for the line passing through the points (2, -5) and (4, 7)

 

11.)     Graph the following function.  f(x) = (x - 1)2 (x + 2) 2 (x - 3) 5 (x - 5) 4 (x + 4) 3

 

12.)     The height of a ball thrown off the top of a tall building is given by the quadratic function s(t) = -16t2 + 45t + 500, where height is in feet and time is in seconds.   

 

a.)        How long would it take for the ball to hit the ground?

 

b.)        At what time(s) is the ball 510 feet high?

 

c.)        What is the maximum height attained by the ball and how long does it take for it to get to that height?

 

d.)        Bob, who is 6 foot 6 inches tall, is directly under the ball.  How long after the ball is thrown will it hit Bob on the head?

 

13.)     Graph the following function.  Fill in all of the following.

x-intercept(s)_________________________     y-intercept(s)_____________________

horizontal asymptote(s)_________________      vertical asymptote(s)_______________

local minima______________________      _     local maxima_____________________

 

14.)     Use algebraic methods to determine whether or not f(x) = x/(x2 + 1) has symmetry with respect to:

 

            a.)        the x-axis        b.)        the y-axis        c.)        the origin

 

15.)     Suppose the graph in the left hand grid is f(x).  On the right hand grid, graph

g(x) = f(x+3) - 1.                                                        

 

 

 

 

 

 

 

 

 

16.)     Suppose you invest $8000 at an interest rate of 5.4 percent.

 

a.)        Find the balance after 4 years if the interest is compounded quarterly.  Be sure to write down the formula you are using.

 

b.)        Find the balance after 4 years if the interest is compounded continuously.  Be sure to write down the formula you are using.

 

17.)     The formula  gives the average atmospheric pressure, , in pounds per square inch, at an altitudemiles above sea level. 

 

a.)        Find the atmospheric pressure for an altitude for 1.5 miles.   Round your answer to the nearest hundredth.

 

b.)        If the atmospheric pressure around you is 5.2 pounds per square inch, how far above sea level are you?

 

18.)     Polonium-218 has a half-life of 3.05 minutes.  If you have 15 grams of it at noon, how much will you have left at 5:15 p.m. the same day?

 

19.)     Suppose logb 2 = A, logb 3 = B and logb 5 = C.  Find the given quantities.

 

a.)        logb (10/3)      b.)        logb 45            c.)        logb 0.6

 

20.)     Using your calculator, solve the equation log7 (x2 +1) - log5 2x - 1 = 0.

 

21.)     Solve the following system of equations (no calculator) using the method of elimination.

4x + y = 7

8x - y = 5

 

22.)     Consider the following table of values.

 

x

11

13

15

16

17

19

22

24

y

8

10

13

14

15

15

16

17

 

            a.)        Draw a scatter plot of the data. 

 

b.)        Write the logarithmic function that best fits the data.

 

c.)        Draw the function from b.) on your scatter plot from a.).

 

d.)        Write the third degree polynomial function that best fits the data.

 

e.)        Draw the function from d.) on your scatter plot from a.).

 

f.)         Which function, the one from b.) or the one from d.) fits the data better.  Why?

 

23.)     Solve the following system of equations (no calculator) using the method of substitution.

2x2 + y2 = 21

x + y = 0

 

24.)     Solve the following system of equations.  You may use your calculator.

 

24.55x + 34.98y = -2.107

px - epy = 4.752

 

25.)     Use your calculator to solve the following system of equations.  Explain how you used your calculator.

 

y + 3 = x log x

(x - 4)2 + y2 = 2

 

26.)     Solve the following equation by hand.  Exact Answer Only!!!    92-3x = 272x-1

 

27.)     Solve the following equation by hand.  Exact Answer Only!!!

log6 (x - 1) + log6 (x) = 1

 

27.)     Suppose 0 = 2.7x3 - 3.25x2 - 5.1x +2.4.  Use your calculator to find one positive and one negative solution to the equation.  Give every decimal place that your calculator gives.

 

28.)     Use your calculator to solve the following equation.         

 

29.)     Suppose we define the following sequence.