Name: ______________________________________________

Practice Integrated II Final Exam Review The questions below represent the types of questions you will see on your final exam. Your final will be all multiple choice however, so if you are able to answer the questions below you will be just fine . Remember to use your toolkit entries to help you if you!!

1. To answer the following, refer to the equation z =

where x, y, and z are all positive.

a. If x is held constant and y increases, how does z change? What do we call this relationship?

b. If y is held constant and x increases, how does z change? What do we call this relationship?

c. Write an equivalent rule that shows x as a function of y and z.

d. Write an equivalent rule that shows y as a function of x and z. 2. The time required to complete a 100-mile bike race is inversely proportional to the average speed

that the rider maintains. a. Write a rule that expresses the relationship between average speed s and race time t.

b. What is the constant of proportionality for this situation?

c. Tina took 5 hours and 15 minutes to complete the race. What was her average speed?

d. Gregory maintained an average speed of 16 miles per hour. How long did it take him to complete the race?

3. Towne Sporting Goods establishes a selling price S for an item based on the cost C that it paid the manufacturer and the rate R of markup that it charges in order to cover its expenses and make a profit. These variables are related by the following equation: S = C(1 + R) a. Towne Sporting Goods gets a pair of in-line skates from the manufacturer at a cost of $80. If

Towne uses a 28% markup, what is the selling price of the skates to the nearest dollar?

b. Use the equation S = C(1 + R) to write an equivalent equation that gives C as a function of S and R.

c. After the holidays, Towne Sporting Goods had a sale during which it sold all items in the store for a markup of only 10%. The sale price of a tennis racket was $32. To the nearest dollar, how much did it cost Towne Sporting Goods to buy the racket from the manufacturer?

4. Draw a graph of the equation 3x + 4y = 24.

5. Consider the following system of equations:

y = 3x – 12 6x + 4y = 15

a. Use an algebraic method to solve this system of equations. Show your work.

b. How does the solution you found in Part a relate to the graphs of the two equations? 6. Joe looked at the following system of equations and announced that the system had infinitely many

solutions. Is Joe correct? Describe how you can determine this just by looking at the equations.

6x + 8y = 24 9x + 12y = 24

7. The Student Council sponsored a talent show to raise money. They charged $5 admission for each

adult and $2 for each student. A total of 248 people attended the show and they made $715. How many students and how many adults attended the talent show? Show work.

8. A system of linear equations can have 0, 1, or infinitely many solutions.

a. Write a system of equations that has no solution. Explain how you know the system does not have a solution.

b. Write a system of equations that has exactly one solution. Explain how you know it has exactly one solution.

c. Write a system of equations that has an infinite number of solutions. Explain how you know there are an infinite number of solutions.

9. Refer to the following system of linear equations.

3x + 2y = 15 x - y = 2

a. Graph this system of equations and estimate the (x, y) pair that solves it.

b. Check your estimate in Part a algebraically.

10. The graph to the right shows the height (in meters) of a baseball in flight as time (in seconds) passes

and y = h(x).

a. Why is it correct to say that height of a baseball is a function of time in flight?

b. Is time in flight of a baseball a function of the height of the baseball? Explain your reasoning.

c. What does the equation h(1.2) = 16.8 tell about the flight of the ball?

d. What is the value of h(3) and what does it tell about the flight of the ball?

e. Estimate the values of x that satisfy the equation 10 = h(x), and what do those values tell about the flight of the ball?

f. Identify the practical domain and range of h(x).

g. The maximum value of the graph is 20 and the x-intercepts are (0, 0) and (4, 0). Find a function rule for h(x).

h. Identify the theoretical domain and range of h(x).

11. The graph of a particular quadratic function has one of its x-intercepts at (4, 0) and a minimum point

of (0, -16). a. What is the other x-intercept of the function? Explain your reasoning.

b. Write a function rule for this quadratic function.

c. If possible, find a function rule for another quadratic function that has the same x-intercepts as this function but has a different y-intercept. If not possible, explain why not.

12. Write each quadratic expression in equivalent factored form.

a. x2 + 2x – 35

b. x2 - 8x + 16

c. 2x2 + 16x + 30

d. 3x2 - 22x + 7

13. Use algebraic reasoning to solve each equation. Show your work. a. x2 + 7x + 15 = 3 e. log x = 2 b. x2 - 100 = 0 f. 10x = 83

c. 3x + 5 = x2 - 6x + 9 g. x + 10 =

d. 3( ) = 456

14. It is important to understand logarithms as well as be able to use them to solve equations.

a. Explain why log 0.1 = -1.

b. Explain how you could determine without using a calculator that log 1,863 is between 3 and 4.

c. Why do you get an error message when you type “log(-10)” into a calculator? 15. The number of E. coli bacteria that are present in some food t hours after contamination can be

modeled by the function N(t) = 50. a. Identify the theoretical domain and range of N(t).

b. b. Evaluate N(0) and explain what it tells you about the number of E. coli bacteria present.

c. The solutions to 5,000 = 50 (101.806t) answer a question about the number of bacteria present. What is that question?

d. Algebraically solve the equation in Part c. Show your work.

16. The tractor-pulling contest is one of the most popular contests each year at the Johnson County Fair. Based on data from previous years, the organizers can expect that income I(p) and expenses E(p) both depend on the price of admission. They predicted that:

I(p) = -6x2 + 100x and E(p) = 15x + 230.

a. Use algebraic reasoning to determine the ticket price(s) for which income will equal expenses.

b. Write a rule that gives predicted profit F(p) as a function of the admission price.

c. Use your profit function to determine the maximum predicted profit.

d. What admission price should they charge in order to get the maximum predicted profit?

17. The scatterplot to the right shows information

about newspaper delivery for 13 different people. It shows the number of newspapers on each route n and the amount of time t each person spends delivering the newspapers. The least squares regression line is drawn on the scatterplot and has equation t = 0.6n + 6.3. The correlation for this data is r = 0.65. a. The circled point is an outlier. Describe

what type of outlier it is and how this person’s newspaper delivery is different from the others.

b. If the circled point was deleted from the data set, how would the slope of the regression line change?

c. If the circled point was deleted from the data set, how would the correlation change? Explain your reasoning.

d. Add a point to the scatterplot that would be considered an outlier with respect to the number of newspapers only. Give the coordinates of your point. Explain your reasoning.

18. Examine the scatterplots below. The axes all have the same scale.

a. Which plot shows the weakest correlation between its two variables? Explain your choice.

b. Do any of the plots illustrate a negative correlation? If so, which plot? Explain your response.

c. Which plot shows the strongest correlation? Explain your choice. 19. The scatterplot below gives the number

of hours studied last week s and the number of hours of television watched last week t by each student in an eleventh-grade classroom. The equation of the regression line is given below, and the value of the correlation is –0.5462. (Each “2” indicates a single point that represents two students.)

t = 8.00 ・ 0.341s

a. What are the coordinates of the point that appears to be an influential point? If this point was removed from the data set, would the absolute value of the correlation increase, decrease, or remain about the same?

b. Briana is the student who studied 12 hours last week and watched 2 hours of television. Graph the regression line on the scatterplot. Draw in the line segment that represents the residual for Briana. Compute the residual for Briana.

c. The slope of the regression line is –0.341. Interpret this slope in the context of the data.

d. What might explain the negative correlation between the number of hours these students studied and number of hours they spent watching television?

e. The mean number of hours these students spent watching television was 5.28. Describe two ways that you could determine the mean number of hours they spent studying.

20. Match the correct approximate correlation to each plot.

I. r = –2.0 II. r = –0.60 III. r = 0.20 IV. r = 0.50 V. r = 0.80

21. Each time Carly filled up the gas tank

in her car, she recorded the number of miles she had driven m since the last fill-up and the number of gallons of gas g required to fill her tank. Her data is shown in the scatterplot below. The equation of the linear regression line is

g = 0.057m ・0.27.

a. Describe the shape of this distribution. Include the direction of the relationship, the strength of the relationship and whether the strength varies.

b. What is the approximate slope of the regression line? Interpret the slope in the context of these data.

c. Circle the point that could be considered an outlier. Describe the type of outlier that your indicated point is.

d. Would the outlier be considered an influential point? Explain your reasoning.

e. Which of the following is the correlation for these data?

–0.87 –0.27 0.057 0.27 0.87

22. Determine whether each statement is True or False. If False, explain your reasoning. a. You can add any two matrices together.

b. For all matrices A and B that can be added together, A + B = B + A.

c. For all matrices A and B that can be multiplied together, A B = B A.

d. If A = , then .

e. For any 2 2 matrix A, it is always possible to find a matrix B such that A B = I where I is the identity matrix.

f. When you add two matrices together, you add the corresponding entries. 23. The Burlington-Edison School District athletic director is writing

a funding proposal to the school board for new athletic equipment for the five middle schools (I–V) in the district. She decides that the schools need a variety of equipment including baseball bats (BB), volleyballs (VB), and football helmets (FH). The matrix, Equipment Request by School, summarizes her request. a. The equipment can be ordered at different levels of quality and cost. The matrix below gives the

cost of each piece in dollars.

Use a matrix operation to complete the total-equipment-cost matrix below. Show how you obtained this answer.

b. If school V orders average-quality equipment instead of high-quality equipment, how much will the school save? Show your work.

c. Find the total savings if the athletic director orders average-quality equipment for all schools.

d. The athletic director decides to order high-quality football helmets for safety reasons but only average-quality baseball bats and volleyballs.

i. Construct a 1 3 cost matrix that gives the cost of each piece of equipment that is requested. Be sure to properly label your matrix.

ii. Then find a matrix for total cost per school. Indicate the matrices to be multiplied. Be sure to properly label your total cost per school matrix.

24. Nwanko’s Lawn Service offers three types

of lawn care contracts. For an average-sized lot, monthly rates are $30, $50, and $75 for economy, standard, and deluxe contracts, respectively. The table to the right shows the number of contracts serviced last summer in the communities of Hastings and Dover. a. Construct a matrix that shows the price for each type of lawn care contract.

b. Create a contract matrix, and use the price and contract matrices to find how much money Nwanko’s Lawn Service received per month in each of these two communities. Describe your method.

c. Before this summer started, Nwanko did a lot of advertising and got new customers. The table below shows the number of new contracts that Nwanko received. Use an operation on appropriate matrices to construct a matrix showing the total number of each type of contract Nwanko now has.

d. In which community will Nwanko make more money during this summer?

25. Complete the following matrix operations.

a. d . If , then ___________.

b. . If , then ___________.

c. If and , then N =___________.

26. A plan for a new Amusement Park is sketched on a grid below. One unit on the grid is equivalent to 10 meters. The main gate is located at point G. The entrances to some rides are marked: the Ferris wheel is F, the roller coaster is R, and the Tilt-a-Whirl is T. Complete the following tasks about the plan. Show or explain your work for each part.

a. Main Street is planned to run directly from G to F. Find an equation in the form y = ax + b of the

line representing Main Street.

b. The haunted house H is to be built on Main Street, and it has the same x-coordinate as the roller coaster. Mark H on the map. What are the coordinates of H?

c. A concession stand N is planned midway between the gate and the Tilt-a-Whirl T. Mark N on the map, and find its coordinates.

d. The planners want the concession stand to be within 100 meters of the roller coaster. Does its present location, found in Part c, satisfy this condition?

27. A parallelogram ABCD has vertex matrix ABCD =

a. Write an equation for the line containing the side of the parallelogram determined by (4, 0) and (6, 3). Show your work.

b. Is the parallelogram a rhombus? Justify your response.

c. Find equations of the lines containing the diagonals of the parallelogram. Show your work.

d. Are the diagonals of the parallelogram perpendicular? Explain your reasoning.

e. Find the midpoint of each diagonal. What do these results tell you about the diagonals?

28. Is the quadrilateral ABCD with vertices A(-3, -2), B(-1, 2), C(4, 3), and D(3, -1) a parallelogram? Provide a mathematical argument that supports your answer.

29. Quadrilateral ABCD is a rhombus. a. Determine the coordinates of point C. Show your work

and explain your reasoning.

b. Prove that .

c. Prove that bisects . 30. Shown below is a circle with radius 4 and center at the origin. Identify the coordinates of two points

that are on the circle and are not on the x- or y-axis. Show your work or explain your reasoning.

31. Triangle ABC = is sketched to the right.

a. appears to be a right triangle. Check by

finding the slopes of and . Is a right triangle? Explain your reasoning.

b. Find the lengths of , , and . Explain how to use these lengths to determine whether

is a right triangle.

c. Write an equation for . Show your work.

d. Draw the image of under each transformation given below. Label it A’B’C’.

e. Describe the effect on of the transformation represented by

f. Find the area of . Show or explain your work.

g. Consider A’B’C’, the size transformation of magnitude 2 centered at the origin of that

you sketched in Part d. Explain how to find the area of A’B’C’ from the area of without

computing the lengths of any sides of A’B’C’.

32. Each graph on the next page shows and its image under one of the six transformations provided below. Identify the correct transformation for each graph and write your choice on the line provided. Then, in the space provided to the right of the graph, demonstrate how you can accomplish the indicated transformation using matrix multiplication or a coordinate rule.

I. 90° counterclockwise rotation

II. 270° counterclockwise rotation III. Reflection across the x-axis IV. Reflection across the y-axis V. Reflection across the line y = x VI. Reflection across the line y = -x

a. ______

b. _______

c. _______

33. Consider the transformation that is a composite of these two transformations in the order given.

Transformation 1: (x, y) (3x, 3y)

Transformation 2: (x, y) (x + 2, y + 1)

a. Describe each transformation.

b. Suppose that ABC = . What is the image of ABC under this composition of transformations?

c. Write a rule (x, y) (__, __) that describes this composition of transformations.

d. The perimeter of ABC is 12 units. What is the perimeter of the image of ABC? Explain your reasoning or show your work.

e. The area of ABC is 6 square units. What is the area of the image of ABC? Explain your reasoning or show your work.

f. How if at all would any of your answers to Parts b–e above change if the transformations were applied in the opposite order? Explain your reasoning

34. Refer to the right triangle ABC.

a. If measures 27° and c = 13 cm, find a and b to the

nearest hundredth of a centimeter and the measure of to

the nearest hundredth of a degree. Show or explain your

work.

a =__________ b =__________ m __________

b. If a = 12 cm and b = 7 cm, find c to the nearest hundredth of a centimeter and the measures of

and to the nearest hundredth of a degree. Show or explain your work.

35. The Great Pyramid of Cheops in Egypt has a square base 230 meters on each side. Each face of the

pyramid makes an angle of approximately 52 with the ground. A sketch of the pyramid is shown

below.

a. Find the height of the pyramid.

b. Find the area of one triangular face of the pyramid.

36. Suppose four stars in the Milky Way are labeled A, B, C, and D. Star A is 2.1 light years from star B,

3.2 light years from star C, and 1.7 light years from star D.

a. To the nearest tenth of a light year, how far is star B from star C if m = 71°.

b. Find the measure of to the nearest degree if star D is 4.2 light years from star C.

37. The pitcher’s mound on a softball field is 40 feet from home plate, and the distance between bases

is 60 feet. As shown in the diagram below, m = 45°. (Note: is not a right angle.)

a. How far is the pitcher’s mound P from first base F?

b. Find m .

c =__________ m __________ m =__________

38. Find the indicated measures for both triangles below.

a. b.

AB = ___________ AC = ___________ ___________ m ___________

MK = ___________

39. Taylor High School surveyed its juniors and seniors about their experiences at school. One question

asked if the student had been challenged to do his or her best work during that school year. The results are in the table below.

Suppose you randomly pick one of the students in order to gather follow-up information. Find each probability.

a. P(student is a junior)

b. P(student is a junior or student was challenged to do best work)

c. P(student is a junior and student was challenged to do best work)

d. P(student is a junior |student was challenged to do best work)

e. Are grade level and whether or not the student was challenged to do his or her best work independent? Explain your reasoning.

40. The choir booster club is selling scratch-off cards to raise money. The probability of winning each prize and the prize values are given in the table below. a. What is the probability of winning nothing?

b. If they charge $25 for each scratch off card, will they make or lose money? Explain your reasoning.

41. In a roulette game at a Las Vegas Charity Night for the Special Olympics, a horizontal wheel with 38

slots numbered 0, 00, and 1-36 is spun. A ball is dropped onto the wheel, rolling around until it comes to rest in a numbered slot. It is equally likely that the ball will land in any one of the 38 slots. Suppose a player bets on the number 8. If the ball lands in the slot numbered 8, the player wins 35 coupons. The price to play is 1 coupon. (Assume the ball always lands on the wheel.) a. What is the probability of the ball landing in the slot numbered 8?

b. Is one coupon a fair price to play this version of roulette? If so, show why. If not, find the fair price.

c. If a player plays 100 games (at one coupon per game), how many coupons can the player expect to win or lose?

42. Each time a spinner is spun by a player in a board game, it is equally likely to stop on any of the

numbers 1 through 10. The rules of the game require getting a 1, 5, or 10 on a spin before a player’s marker may be moved out of the Start position. If the player fails to get a 1, 5, or 10, the following players try in turn. After getting a 1, 5, or 10, the player spins again on that turn and moves the marker the number of spaces indicated on that spin. (The player does not move any spaces on the spin in which the 1, 5, or 10 appears.) a. What is the probability that a player gets his or her marker out of Start on the first turn? Explain

your answer.

b. Out of every 100 times you play this game, how many times would you expect to get started on your first turn? Your second turn?

First Turn:__________ Second Turn:__________

c. Complete the following theoretical probability distribution table for this situation.

d. Using the values in your table, approximate the expected number of turns needed to get your marker out of Start. What is the theoretical expected number of turns? Explain how you get each.

Approximated mean from table:__________ Expected value:__________

e. Why is the approximated expected number of turns lower than the theoretical expected

number of turns?

f. What is the probability that on your second turn you end up ten spaces beyond Start? Explain. 43. In Mathematics II at Bedford High School, 20% of students got an A on the first exam and 15% got an

A on the second exam. Is it reasonable to estimate that (0.2)(0.15) = 0.03, or 3%, of students got an A on both exams? Why or why not?

44. The average probability of a baseball batter getting a hit against Nolan Ryan was about 0.2. Assume for these questions that 0.2 is the independent probability that each batter would get a hit against this pitcher. a. Describe a method using your calculator or a table of random digits to simulate the situation of

finding the number of the batters until a batter gets a hit in a game against the pitcher.

b. What is the probability that the first hit in a game against Nolan Ryan was by the first batter? By the second batter? By the third batter? Explain your reasoning.

P(first batter):__________ P(second batter):__________ P(third batter):________

c. Describe the general shape of the distribution of the number of batters until a batter gets a hit in a game against Ryan.

d. Find the theoretical expected number of batters until a batter gets a hit against Ryan in a game. Explain or show your work.

e. For a pitcher to pitch a no-hitter (that is, a game in which no opposing batter makes a safe hit), he must get 27 consecutive batters out (ignoring walks, errors, and so forth). What is the probability that Ryan will get 27 consecutive batters out in a particular game? Explain or show your work.