College Algebra

Exam File

Fall 2003

Exam 1

1.) Find an equation for the line that goes through the points (-2, 4) and (5, -2).

2.) Find the inverse of the function f(x) = 3 - 4x.

3.) Solve the following equation without using your calculator. Be sure to show all necessary work.

4.) Find an equation for the line on the point (1, 2) and perpendicular to the line

6x + 3y = 6.

5.) Find the domain of the function

6.) 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

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

a.) Determine if f has symmetry with respect to the x-axis? Show your work. ("My calculator told me so" is NOT sufficient.)

b.) Determine algebraically if f has symmetry with respect to the y-axis? Show your work. ("My calculator told me so" is NOT sufficient.)

c.) Determine algebraically if f has symmetry with respect to the origin? Show your work. ("My calculator told me so" is NOT sufficient.)

8.) Graph the following function.

9.) Suppose the graph shown is the graph of f(x). On the same set of axes, show the graph of g(x) = -2 f(x + 3) + 4.

10.) Graph f(x) = 3 (x + 2)² - 4. Explain in words how this graph is obtained from the graph of g(x ) = x².

11.) Suppose f(x) = 3x + 5 and g(x) = 2x² - 1.

a.) Find (f - g)(3).

b.) Find (f _° g)(2).

c.) Find (g _° g)(2).

12.) 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.

13.) 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 linear function that best fits the data.

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

d.) Does the line fit the data well? Give a reason for your answer.

Exam #2

1.) 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.

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

x² - 5x - 6 = 8

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

4.) Solve the following equation exactly.

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

6.) Solve the equation exactly.

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

-3x - 5 > 4

8.) Solve the following inequality. Give your answer in graph form.

|3x - 2| < 11

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

10.) Let f(x) = 3x² + x - 1. Find the vertex and all intercepts (Give exact answers only.). Graph the function.

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

12.) y varies jointly as x² and the inverse of z. If y = 4 when x = 3 and z = 9, find y when x = 9 and z = 3.

13.) Do the following.

a.) Find all of the roots of g(x) = x⁴ - 2x³ + 2x² - 2x + 1.

b.) Give the complete factorization of g(x).

c.) Use synthetic division to find g(3).

Exam #3

1.) Without using your calculator, solve the equation log(ln x) = 1. Exact answer only.

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

x-intercept(s)________________ y-intercept(s)________________

slant asymptote(s)________________ horizontal asymptote(s)________________

vertical asymptote(s)________________ local minima________________

local maxima________________ absolute minima________________

absolute maxima________________ intervals where increasing________________

intervals where decreasing________________ domain________________

range________________ x-axis symmetry________________

y-axis symmetry________________ origin symmetry ________________

GRAPH

3.) Find the maximum and minimum values of the function z = 4x - 3y in the quadrilateral with vertices (1, 2), (4, -1), (2, 9) and (7, 9).

4.) 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.

5.) The formula gives the average atmospheric pressure, , in pounds per square inch, at an altitude x miles 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?

6.) Give examples of two different physical phenomena that are modeled using logarithmic functions.

7.) Without using your calculator, solve the following system of equations. Exact answers only.

8.) Without using your calculator, solve the following system of equations by the elimination method. Exact answers only. SHOW ALL NECESSARY WORK!!!!

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

10.) Use your calculator to find both solutions to the equation 3^x/5 - log₇ (15x) = 0.

In #11 - 13, solve the given equation exactly. EXACT ANSWERS ONLY!!

11.) 4^x = 8 12.) log₂ x + log₂ (x - 3) = 2 13.) 5^2x = 70

14.) 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?

Exam #4

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

x – 2y + 3z = 7

2x + y + z = 4

-3x + 2y – 2z = -10

2.) Write the first three terms of the sequence defined below.

a_n = 2ⁿ/n!

3.) Find the value of the following using your calculator.

4.) A movie theater charges $9 for adults and $7 for senior citizens. On a day when 325 people paid an admission, the total receipts were $2495. How many who paid were adults? How many were seniors?

5.) Using your calculator, solve the following systems of equations using the inverse matrix. Show the inverse matrix. Where possible, have the calculator turn the answers into fractions.

6.) Given the matrices below, perform the given operations, if they exist. If an operation is undefined, write "does not exist" and tell why it does not exist.

a.) B + A

b.) 2A

c.) CB

d.) C^-1 (do not use your calculator)

7.) Find the first 3 terms and the 23-rd term of the sequence given by .

8.) Find the second, third and fourth term of the recursively defined sequence below.

a₁ = 2

a_n+1 = 2a_n + 3, for n > 1

9.) Consider the arithmetic sequence whose first two terms are a₁ = 3 and a₂ = 7.

a.) Find the 23-rd term.

b.) Find the sum of the first 23 terms.

10.) Consider the geometric sequence whose first two terms are a₁ = 3 and a₂ = 6.

a.) Find the 23-rd term.

b.) Find the sum of the first 23 terms.

11.) Consider the infinite geometric series whose first two terms are a₁ = 1 and a₂ = 2. Find the sum if it exists. If it does not exist, tell why not.

12.) A restaurant serves 5 different kinds of appetizers, 6 different kinds of salad, 23 different kinds of entrée, 8 different kinds of dessert and 12 different kinds of drinks. If you order one item from each category, how many different meals could you order?

13.) A mathematician has 30 books he wants to put on a shelf. Unfortunately the shelf will only hold 21 books. How many different sets of books could he put on the shelf?

Final Exam

1.) Find an equation of the line through the point (2, 3) that is perpendicular to the line 3x - 5y = 0.

For #2 - #4, consider the following functions.

f(x) = x² + 2x + 3 g(x) = 2x - 3

2.) Find the following

a.) f(3) b.) (f + g)(2) c.) (f g)(a)

d.) (g / f)(2) e.) domain of f / g f.) (f ◦ g)(2)

3.) Multiple Choice - Evaluate . Circle your answer.

a.) b.) 2x + 2h + 2

c.) d.) h + 2

4.) Find the inverse function for g(x), if it exists. If it does not exist, tell why not.

5.) Suppose y varies directly with x and inversely with the square of z. If y = 20 when x = 5 and z = 1, find y when x = 18 and z = 3.

Problems #6 and #7 refer 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

6.) 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.

c.) Find an equation for the logarithmic function that best fits this data.

7.) Answer the following questions.

a.) Which function from #6 fits the data better? Give a mathematical reason for your answer.

b.) Which function from #6 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 logarithmic model, what would the population of Rogers have been in 1900?

8.) Use your calculator to find all real number solutions to the following equation.

x⁵ + 3.15x⁴ - 15.11x³ + 3.15x² - 16.12x = 0

9.) Solve this equation by hand. Exact Answer Only!!!

37x - 19 = 48

10.) Solve this equation by hand. Exact Answer Only!!!

x² + 2x - 3 = 32

11.) Solve the following inequality.

7 - 5x < 10

12.) Without using your calculator, find all solutions to the following equation. BE SURE TO SHOW ALL WORK!!!

x⁵ - 3x⁴ + x³ + x² + 4 = 0

13.) In the 1990s, the sales of bowling equipment in the United States increased and then decreased according to the model C = 0.165x³ - 7.16x² + 100.6x - 303.1, where x = 8 corresponds to the beginning of 1988.

a.) During what year were sales at a maximum?

b.) What was the maximum?

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

a.) Draw a graph illustrating this model. Use xmin = ymin = 0 and xmax = ymax = 10.

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

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

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

9^2-3x = 27^2x-1

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

log₆ (x - 2) + log₆ (x + 3) = 1

17.) Cesium-136 has a half-life of 13.1 days. Suppose you have 14 grams of Cesium-136. How much will you have after one week?

18.) Use your calculator to find all solutions to the following system of equations. Give at least 4 decimal places.

ln(2x + 5) + 2 = y

3^x+2 = y

19.) Solve the following system of equations by hand. Exact Answers Only!!!

x² + 2x + 1 = 4y

y = x + 4

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

x² + y < 5

3x + y > 1

21.) Solve the following system of equations using a matrix. You may use your calculator but show the matrix you used.

x - 2y + 3z = 13

2x + 3y - z = 5

2x + 2y - 5z = -7

22.) Given a_n = (-1)ⁿ n / (2n - 1), find the following.

a.) a₁

b.) a₂

c.) a₂₃

23.) Use your calculator to find the following.

24.) Given an arithmetic sequence with a₁ = 2 and a₃ = 8, find the following without using your calculator.

a.) a₂

b.) a₁₅

c.) S₁₅

25.) Solve the following inequality.

x² - 5x - 6 > 18

26.) WITHOUT USING YOUR CALCULATOR, solve for x. Show ALL work.

log₇ (log₃ x) = 1

27.) Find the apparent general term of the sequence beginning with 2, 5, 10, 17, 26.

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

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

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