Name: ______

Algebra 2 – Final Exam Review

Rational Functions

1.Suppose an alcohol solution consists of 5 gallons of water mixed with 3 gallons of alcohol. An additional x gallons of alcohol is added to the mixture.

  1. In an alcohol mixture, the concentration was given by. For what value(s) of x is the function equal to zero? Explain what your answer means about the mixture.
  1. Find equations for all vertical and horizontal asymptotes of given in part d. Explain what each asymptote means about the mixture.
  1. For what value of x is the concentration of alcohol in the second mixture (part d) equal to 60% or 0.60? Show or explain your work.

2.Given the following equations of rational functions, find any zeroes, vertical asymptotes, horizontal asymptotes, oblique asymptotes, removable discontinuities, and end behavior.

a. b.

3.Write the equation for a rational function that has a graph with two real zeroes, vertical asymptotes of x = –5 and x = 4, and a horizontal asymptote of .

4.Sketch a graph of the rational function. Indicate any zeroes, vertical asymptotes, and horizontal asymptotes.

5.Determine a possible equation for the following graph. The scales on both axes are one. Explain how the characteristics of the graph help you determine the equation.

Simplify the following

1.2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.

Classify the following rational functions fully

22. 23. 24.

25. 26. 27.

28. 29. 30.

31. 32. 33.

You should also be able to write a rational function given information about zeroes, vertical asymptotes, horizontal asymptotes, end behavior, and/or removable discontinuities.

Trigonometric Functions

(ROUND ALL SIDE LENGTHS AND AREAS TO THE NEAREST TENTH)

Use trigonometric ratios (SOH-CAH-TOA) to find the unknown side lengths in the following four right triangles. Show your work in order to receive full credit.

1.2.3.4.

5. This graph shows the graph of number of hours of daylight versus the month for Grants Pass, Oregon. Write an equation that models this graph.

6. The pattern of population change of Woodchucks in Michigan is shown on the graph below.

  1. Identify the following:

Max: ______Min: ______Amplitude: ______Period: ______Vertical Shift: ______

Domain: ______Range: ______

  1. Write a function rule P(x), the population of Woodchucks based on the time, x, in months.

7. Write an equation for a sine function that has been translated up 5 units, has a period of 10, and has amplitude 3.

8. The daily average temperature in Nashville is approximately related to the time of year as graphed. The first month, January, is given as x = 0.

  1. Write an equation that models this situation.

Amplitude: _____Period: _____Vertical Shift: _____

Equation: ______(Hint: There are several correct equations)

  1. Which other month has an average daily temperature most like the month of April?
  2. Suppose a city in South America has average daily temperatures very much like those of Nashville, except that fall and spring are reversed, as are winter and summer. Sketch a graph of the average daily temperatures for a year in this city.

Find the equations for the following graphs:


Probability and Statistics

  1. Connecticut has license plates that have 2 letters then 3 numbers and then 2 more letters. The first two letters cannot be repeated. How many different license plates are possible? Show your work.
  1. Phone numbers have 10 digits. The first three digits (the area code) of a phone number in the Lancaster, PA area are 717. How many different phone numbers are possible in the Lancaster area? Show your work.
  1. How many possible numbers end in 11 in the Lancaster area? Show your work.
  1. Student council is holding an election for the homecoming court. 16 boys and 14 girls have been nominated to be part of the election. The court will consist of 4 boys and 4 girls. How many different courts are possible?
  1. At a school with a six-hour day, how many possible schedules can be created if the class options are English, biology, algebra 2, English literature, gym, study hall, ceramics, drama, concert band, and choir? Show your work.
  1. How many ways are there to dealt 4 aces in a five-card hand from a standard 52 card deck? Show your work.
  1. A pair of regular dice is rolled. Find the probability of getting a sum of 6 on the dice. Show your work.
  1. A coin is tossed and a die is rolled. What is the probability that the coin shows tails and the die shows 4?
  1. What is the probability of drawing a red ace and then a spade, with replacement, from a deck of cards?
  1. A drawer contains 2 red socks, 10 white socks, and 8 blue socks. Without looking, you draw out a sock, return it, and draw out a second sock. What is the probability that the first sock is red and the second sock is purple?
  1. What is the probability of rolling doubles or a sum of 8 and then rolling a sum of 10 in two rolls of a pair of dice?
  1. A bag contains 5 red marbles and 6 purple marbles. One marble is drawn at random and not replaced. Then a second marble is drawn at random. What is the probability that the first marble is purple and the second one is red?
  1. Of 100 students, 30 are taking Calculus, 35 are taking French, and 15 are taking both Calculus and French. If a student is picked at random, what is the probability that the student is taking Calculus or French?
  1. There are 30 people on student council – 7 freshmen, 7 sophomores, 7 juniors, and 9 seniors. How many ways can you:
  1. Choose a committee of 4 members?
  2. Choose a committee of 1 senior, 1 junior, 1 sophomore, and 1 freshman?
  3. Choose a President, a Vice-President, a Treasurer, and a Secretary?
  4. Choose a President, a Vice-President, a Treasurer, and a Secretary if they all must be seniors?
  5. Line all the members up for a photo?
  1. There are 52 guys on a football team, from whom 10 will be randomly chosen for drug testing.
  1. How many ways can you choose 10 players for testing?
  2. How many of these groups contain the star quarterback?
  3. What is the probability that a randomly chosen group will contain the star quarterback?
  1. From a standard deck of 52 cards,
  1. How many different 7-card gin hands can be dealt?
  2. How many of these hands contain exactly three kings?
  3. How many of these hands contain at least three kings?
  1. There are 10 students in a class to be assigned to 4 different groups – 2 to group ‘A’, 3 to group ‘B’, 3 to group ‘C’, and 2 to group ‘D’. How many different ways can this be done?
  2. Bishop High School’s schedule has 6 periods each day. The principal, Ms. Cheney, is planning the daily schedule of classes for the coming school year.
  1. There are 6 elective classes for seniors. Ms. Cheney schedules one elective class in each of the 6 time periods. In how many different orders can she do this?
  2. Chester is senior who worked hard during his first 3 years of high school and completed all required courses. He only needs to take 4 electives during his senior year. How many different sets of 4 classes might Chester take from among the 6 different elective classes that are offered?
  3. Eight different classes must be scheduled for juniors. Room 222 is one classroom in which junior classes will be scheduled. It will be used for a different class during each of the six periods of the day. In how many orders can Ms. Cheney schedule classes for juniors in room 222?

Normal Distributions

The height of men is approximately normally distributed with a mean of 68.8” and a standard deviation of 2.6”.

1.Sketch the graph of the distribution of height below, making sure to label the horizontal axis accurately:

2.What range of heights represents 95% of men?

3.What is the percentile of Jim, who is 6’2” tall?

4.What proportion of men is between 65” and 70” tall?

5.What percent of men are either taller than 6’1” or shorter than 5’4”?

6.How tall is a man who is in the top 15% of men in height?

7.Jerry is 5’10’ tall and Jason is in the 65th percentile for height. Who is taller and why?

8.Most graduate schools of business require applicants for admission to take the Graduate Management Admission Council’s GMAT examination. Scores on the GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112.

  1. What is the probability of an individual scoring above 500 on the GMAT?
  2. How high must an individual score on the GMAT in order to score in the highest 5%?

9.The length of human pregnancies from conception to birth approximates a normal distribution with a mean of 266 days and a standard deviation of 16 days.

  1. What proportion of all pregnancies will last between 240 and 270 days (roughly between 8 and 9 months)?
  2. What length of time marks the shortest 70% of all pregnancies?

10.The average number of acres burned by forest and range fires in a large New Mexico county is 4,300 acres per year, with a standard deviation of 750 acres. The distribution of the number of acres burned is normal.

  1. What is the probability that between 2,500 and 4,200 acres will be burned in any given year?
  2. What number of burnt acres corresponds to the 38th percentile?

11.The Edwards’s Theater chain has studied its movie customers to determine how much money they spend on concessions. The study revealed that the spending distribution is approximately normally distributed with a mean of $4.11 and a standard deviation of $1.37.

  1. What percentage of customers will spend less than $3.00 on concessions?
  2. What spending amount corresponds to the top 87th percentile?

12.The amount of mustard dispensed from a machine atThe Hotdog Emporiumisnormallydistributedwith a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times,approximatelyhow many times will it be expected to dispense 1 or more ounces of mustard?

13.Professor Halen has 184 students in his college mathematics lecture class. The scores on the midterm exam arenormally distributedwith a mean of 72.3 and a standard deviation of 8.9. How many students in the class can be expected to receive a score between 82 and 90? Express answer to thenearest student.

14.A machine is used to fill soda bottles. The amount of soda dispensed into each bottle varies slightly. Suppose the amount of soda dispensed into the bottles is normally distributed. If at least 99% of the bottles must have between 585 and 595 milliliters of soda, find the greatest standard deviation, to thenearest hundredth, that can be allowed.

15.Battery lifetime isnormally distributedfor large samples. The mean lifetime is 500 days and the standard deviation is 61 days. To thenearest percent,what percent of batteries have lifetimes longer than 561 days?

16.A shoe manufacturer collected data regarding men's shoe sizes and found that the distribution of sizes exactly fits thenormal curve. If the mean shoe size is 11 and the standard deviation is 1.5, find:

  1. The probability that a man's shoe size is greater than or equal to 11.
  2. The probability that a man's shoe size is greater than or equal to 12.5.

17.Five hundred values arenormally distributedwith a mean of 125 and a standard deviation of 10.

  1. What percent of the values lies in the interval 115 - 135, to thenearest percent?
  2. What percent of the values is in the interval 100 - 150, to thenearest percent?
  3. What interval about (above and below) the mean includes 95% of the data?
  4. What interval about (above and below) the mean includes 50% of the data?

18.A group of 625 students has a mean age of 15.8 years with a standard deviation of 0.6 years. The ages arenormally distributed. How many students are younger than 16.2 years? Express answer to thenearest student?

  1. Assume blood-glucose levels in a population of adult women are normally distributed with mean 90 mg/dL and standard deviation 38 mg/dL.
  2. Suppose the “abnormal range” were defined to be glucose levels outside of 1 standard deviation of the mean (i.e., either at least 1 standard deviation above the mean, or at least 1 standard deviation below mean). Individuals with abnormal levels will be retested. What percentage of individuals would be called “abnormal” and need to be retested? What is the normal range of glucose levels in units of mg/dL?
  1. Suppose the abnormal range were defined to be glucose levels outside of 2 standard deviations of the mean. What percentage of individuals would now be called “abnormal”? What is the normal range of glucose levels (mg/dL)?
  1. Suppose a random sample of 100 12-year-old boys were chosen and the heights of these 100 boys recorded. The sample mean height is 64 inches, and the sample standard deviation is 5 inches. You may assume heights of 12-year-old boys are normally distributed. Which interval below includes approximately 95% of the heights of 12-year-old boys?