College Algebra

Test File

Fall 2002

Exam #1

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

2.) Solve the following equation. 2 (3x - 5) + 4 = x.

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

4.) Find the domain of the function

5.) Consider the function f(x) = (x - 1)(x + 1)(x - 2). Use your calculator to find the following.

a.) Find the local (relative) maximum.

b.) Find the local (relative) minimum.

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

a.) Does this graph have symmetry with respect to the x-axis? Show your work. ("My calculator told me so" is NOT sufficient.)

b.) Does this graph have symmetry with respect to the y-axis? Show your work. ("My calculator told me so" is NOT sufficient.)

c.) Does this graph have symmetry with respect to the origin? Show your work. ("My calculator told me so" is NOT sufficient.)

7.) Graph the following function.

8.) Suppose the graph shown is the graph of f(x). On the same set of axes, show the graph of -2f(x-3) + 1.

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

10.) 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). d.) Find the domain of

11.) Determine algebraically whether or not f(x) = 2x + 3 a one-to-one function.

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

13.) Does f(x) = |2x| have an inverse function? Give a reason for your answer.

14.) 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.

15.) Solve the following equation.

16.) In a basketball game, the Bulls scored 103 points and made three times as many field goals (2 points each) as free throws (1 point each). They also made eleven 3 point baskets. The coach wondered how many field goals they scored.

a.) Write an equation, in ONE variable, that can be used to solve this

problem.

b.) Use the equation in part a.) to find out how many field goals they had.

Exam #2

1.) Use your calculator to find all solutions for the following equation. DO NOT ROUND OFF AT ALL FROM WHAT THE CALCULATOR GIVES YOU!!!!

x³ + 3.2x² - 7.25x - 6.3 = 0

2.) Solve the following equation exactly. 2(3x - 5) + 4 = -x.

3.) Solve the following equation by factoring. Exact answers only. x² - 5x - 6 = 8

4.) Solve the following equation by completing the squares. Exact answers only.

x² - 6x - 6 = 8

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

6.) Solve the following equation exactly. 6x² + 11x - 35 = 0

7.) Solve the equation using your calculator. DO NOT ROUND OFF AT ALL FROM WHAT THE CALCULATOR GIVES YOU!!!!

8.) Solve the following equation exactly.

9.) Solve the following equation exactly.

10.) Find any point(s) of intersection between the following functions.

y = x² - 3x - 6 y = 2x + 1

11.) Let f(x) = 3x² + 2x - 1. Find the vertex and all intercepts. Graph the function.

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

x-intercept(s), y-intercept(s), horizontal asymptote(s), vertical asymptote(s), local minima, local maxima

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

-3x - 5 > 4

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

|3x - 2| < 11

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

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

16.) (4 - 5 i) (2 + 3 i)

17.)

18.) i^{85638111111111222222223333339}

Exam #3

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

2.) 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.

3.) 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?

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

5.) Solve each of the following for x. EXACT ANSWERS ONLY!!

a.) log₇ x = -2 b.) log_x 125 = 3

For #6 - 7, use log_b 2 = A, log_b 3 = B and log_b 5 = C to find the given quantities.

6.) log_b (2/3)

7.) log_b 20

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

9.) 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.

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

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.) The following scatter plots compares the number of car accidents to the number of fatalities in those accidents over a period of time.

a.) On the graph, draw the line that best fits the data. Don't try to use your calculator, just do it by looking and drawing.
b.) Assuming the line fits the data well (whether you think it does or not), would the line continue to fit the data well over a long period of time? Why or why not?
c.) What would you guess is the correlation coefficient, r, for your line? Is the line a good fit for the data?

15.) 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. Exact answers only.

2.) Solve the following system of equations by each of the methods indicated. Exact answers only. SHOW ALL NECESSARY WORK!!!!

a.) Substitution

b.) Elimination by hand

c.) Matrix elimination by hand (reduce to row reduced echelon form)

3.) Solve the following systems of equations by the calculator method indicated. Where possible, have the calculator turn the answers into fractions.

a.) Matrix elimination (rref)

b.) The inverse matrix

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

5.) 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).

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 + C b.) 2A c.) AB d.) A + B

7.) Consider the matrices below. AB is defined. Find the entry that would be in the second row and third column of AB.

8.) Find the first 3 terms and the 23-th term of the sequence given by .

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

a₁ = 2

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

10.) 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.

11.) 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.

12.) Consider the infinite geometric series whose first two terms are a₁ = 3 and a₂ = 1. Find the sum if it exists. If it does not exist, tell why not.

Final Exam

1.) Find an equation for the line that passes through the point (2, -5) and is parallel to the line 3x + 4y = 5.

2.) Consider the graph of f(x).	f(x)
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?

3.) Given f(x) = , find the following.

a.) f(3) b.) f(a + 1)

4.) Find the domain of the function from #3.

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

a.) (f _° g)(2) b.) (g _° g)(2)

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

7.) Suppose y varies directly with x and inversely with z. When x = 3 and z = 6, then y = 4. Find y when x = 6 and z = 3. Exact answer only.

8.) In a basketball game, the Bulls scored 103 points and made three times as many field goals (2 points each - yes, there are field goals in basketball!!) as free throws (1 point each). They also made eleven 3 point baskets. The coach wondered how many field goals they scored.

a.) Write an equation, in ONE variable, that can be used to solve this

problem.

b.) Use the equation in part a.) to find out how many field goals they had.

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

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

c.) Which function fits the data better. Why?

10.) 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.

11.) Consider the graph of f(x). Write, in inequality form, the solution to f(x) < 0.

f(x)

12.) Use the quadratic formula to find all solutions to the equation x² - 7x - 9 = 4. Exact answer only.

13.) Solve the following equation without using the quadratic formula. Exact answer only.

x² - 8x - 20 = 28

14.) Find the horizontal and vertical asymptotes of the following function.

15.) Solve the following exponential equation.

4^3x-2 = 8^2-x

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

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 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?

18.) Bob puts $4000 in a savings account that pays 4.5% interest compounded quarterly. If he takes out no money and adds no more money, how much money will he have after 5 years.

a.) Write the formula you are going to use.

b.) Give your answer.

19.) 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.) 2B + C b.) BA

20.) Find the maximum and minimum values of the function z = 3x - 5y in the triangle with vertices (1, -1), (5, 7) and (7, 3).

21.) Graph the solution to the system of inequalities.

22.) Solve the following systems of equations by the method of elimination (by hand - reduce to row reduced echelon form). Exact answers only. SHOW ALL NECESSARY WORK!!!!

23.) Solve the following systems of equations by using the calculator to perform the elimination (rref) method. Exact answers only.

24.) Find the first 3 terms and the 23-th term of the following sequence.

.

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

a₁ = 2

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