Math 2 Review
Name: ____________________ Period: _____
1. For a particular bolt, the amount of force (in pounds) F you must exert on the handle of a wrench to turn the
bolt is related to the length of the wrench L (in centimeters) by the rule FL = 1,800.
a. Express this relationship between the variables in a different but equivalent symbolic form.
b. Is this relationship an example of direct variation or inverse variation? Explain your reasoning.
c. What is the constant of proportionality for this relationship?
d. How does the amount of force needed change as the length of the wrench increases?
2. The surface area of a cube S is related to the length of a side of the cube
S=6 .
by the formula
a. Is this relationship an example of direct variation, inverse variation, or neither?
Explain your reasoning.
b. Calculate the surface area of a cube with side length 3 cm. Show your work.
c. What is the side length of a cube that has a surface area of 105.84 cm ? Show your work.
3. Shown below are graphs of power and inverse power models. The scales are the same on all graphs. Match
the graphs to the function rules and explain your reasoning in each case. Do not use any technological tools.
y=
y=
y=
y=
y=
y=
Rule: y =
Explanation:
Rule: y =
Explanation:
Rule: y =
Explanation:
Rule: y =
Explanation:
Rule: y =
Explanation:
4. A person’s weight in space S can be found by the formula S =
Rule: y =
Explanation:
, where r is the radius of Earth, w is the
person’s weight on Earth, and d is the person’s distance from the center of Earth.
a. How will a person’s weight in space change as the person’s distance from the center of
Earth increases?
b. Evelyn and Marissa are both in the space shuttle orbiting Earth. If Evelyn weighs more than Marissa when
they are on Earth, what can you say about their weights in space?
Explain your reasoning.
c. Choose the correct word to fill in each blank in the sentence below.
A person’s weight in space varies (directly or inversely) with a person’s
weight on Earth and (directly or inversely) with a person’s distance from the
center of Earth.
d. Write a rule that gives d as a function of S, r, and w.
e. Write a rule that gives w as a function of S, r, and d.
5. Draw a graph of 5x + 6y = 30.
6. Eli has just started a new catering business. He can place an ad in the local newspaper for $120 or on the local
radio station for $30.
a. The amount of money M that Eli will spend on advertising depends on the number of ads he
places in the newspaper n and the number of ads he places on the radio r. Write a rule
expressing M as a function of n and r.
b. Suppose that Eli has budgeted $1,200 for advertising. Write an equation that represents the
combinations of newspaper and radio advertisements that he can buy for $1,200.
c. If Eli buys 20 radio ads, how many newspaper ads should he buy in order to use his
entire budget?
d. If Eli decides he is only going to buy radio ads, how many should he buy in order to use his
entire budget?
e. Rewrite your equation from Part b so that it expresses r as a function of n. Then describe what the slope and
y-intercept of the graph of this function would tell you.
Equation:
Meaning of slope:
Meaning of y-intercept:
7. The Taylor Nuts and Chocolate store sells large and small gift baskets. The small gift baskets contain 3
pounds of chocolate and 1 jar of nuts. The large gift baskets contain 4 pounds of chocolate and 3 jars of nuts.
They have 58 pounds of chocolate and 36 jars of nuts that they want to use to make gift baskets. Suppose that
the store asked you to help them decide how many of each size gift basket they should make so that they use
all of the chocolate and all of the nuts.
a. Write an equation that expresses the relationship between the pounds of chocolate used to make x small
baskets and y large baskets and the amount of chocolate available.
b. Write an equation that expresses the relationship between the number of jars of nuts used in making x small
baskets and y large baskets and the number of jars of nuts available.
c. Use the substitution method or the elimination method to determine how many of each size
basket should be made. Show your work.
Number of small baskets: ________
Number of large baskets: ________
Work:
8. Below is the graph of the following system of equations:
Madison looked at the graph and said that the solution to the system is (1.5, 7). Is Madison correct? Explain
your reasoning or show your work.
Is Madison correct?
Reasoning or work:
9. Determine if each system of equations has exactly one solution, an infinite number of solutions, or no
solutions. Provide algebraic reasoning for each choice.
a. 12x + 18y = 36
8x + 12y = 24
exactly one solution
Reasoning:
an infinite number of solutions
no solutions
an infinite number of solutions
no solutions
an infinite number of solutions
no solutions
b. x + y = 10
x=8-y
exactly one solution
Reasoning:
c. 9x + 3y = 27
3x - 9y = 27
exactly one solution
Reasoning:
10. Willie Mays, Orlando Cepeda, and Willie McCovey were great hitters on the San Francisco Giants baseball
team in the 1950s, 60s, and 70s, respectively. The matrices below show the number of doubles, triples, and
home runs they each hit in their first three full seasons as Giants.
a. Use the information in these matrices to complete the table below.
b. The sum of a baseball player’s doubles, triples, and home runs is a statistic called extra-base hits. Complete
the matrix below so that it gives the extra-base hits of each of the three players for each of his fi rst three full
seasons on the Giants. Describe the matrix operations that you could use to get the extra-base hit matrix.
Description of matrix operations:
c. In baseball, a player’s hits is the sum of his singles (first-base hits) and his extra-base hits. The matrix on
the left below gives the hits for each player in each of his first three years. Complete the matrix on the right so
that it gives the singles of each of the three players for each of his first three years. Describe the matrix
operations that you could use to get the singles matrix. Label the rows and columns of the singles matrix.
Description of matrix operations:
d. Baseball players are rated as hitters according to various statistics including the number of hits they make,
the number of extra-base hits, and the number of home runs. According to these statistics, put Mays, Cepeda,
and McCovey in order from best hitter to worst hitter in their first three full seasons on the Giants. Give
reasons for your ordering.
Best:________________
Second:__________________
Third:________________
Reasons:
11. Gloria runs a small hobby shop. She has four employees (labeled A through D). The matrix below gives the
number of hours each employee worked during each full week of December.
a. Because of the workload during the holiday rush, Gloria paid different wages each week. She paid $8 per
hour in week 1, $11 per hour in week 2, $12 per hour in week 3, and $9 per hour in week 4. Complete the
hourly wage matrix below.
b. Use matrix multiplication with the two matrices above to find a third matrix that shows the total amount
each employee earned in December. Explain how these matrices can be multiplied to get the December
Wages matrix. If the matrices must be in a particular order, explain that as well.
Label the row and columns of the December Wages matrix.
Explanation:
12. Let A =
and B =
. Consider the matrix that is obtained when you multiply matrix A
by matrix B.
a. What is the size of A
B? How can you determine this without using a calculator or computer?
Size:___________________
Explanation:
b. Explain how you compute the entry in the second row and second column of matrix A
B.
13. For a concert given by the school orchestra, 300 tickets were sold for total revenue of $1,336. Tickets cost $3
for students and $5 for the general public. The student music organization managed the sales. For future
planning, they would like to know how many student and general public tickets were sold. Represent the
number of student tickets sold by s and the number of general public tickets sold by g.
a. Write and solve a matrix equation to determine how many of each type of ticket were sold. Show your
work.
Student tickets: ________________
General public tickets: ______________
b. Use another method to solve this system of equations. Show your work and be sure that you get the same
solution that you got in Part a.
14. Provide an example of two matrices A and B such that A + B is possible but A
why A + B is possible and why A B is not possible.
A=
B is not possible. Explain
B=
Why is A + B possible?
Why is A
B not possible?
15. Identify one property of real numbers that is not a property of matrices. Provide an example that illustrates
that the property does not hold for matrices.
Property:
Example:
16. Triangle ABC has vertex matrix
a. Sketch
.
ABC on the coordinate grid below.
b. Is ABC an equilateral triangle (that is, are all three sides the same length), an isosceles triangle (that is,
are exactly two sides the same length), or a scalene triangle (no sides the same length)? Explain your
reasoning.
c. Find the coordinates of the midpoint of
another? Explain your reasoning.
. Call the midpoint D. How are
and
related to one
D: (___, ___)
Relation and explanation:
17. Is quadrilateral ABCD as shown below with vertices A, B, C, and D a parallelogram? Explain your reasoning
and show your work.
18. Consider the circle represented by the equation
a. What is the radius of the circle?
b. What are the coordinates of the center of the circle?
c. Draw the circle on the grid below.
.
19. Give the coordinates of the image of the point (-3, 7) under each indicated transformation .
20. Consider the rectangle ABCD shown below with vertex matrix
.
a. Transform ABCD by first reflecting it across the y-axis and then applying a size transformation of
magnitude 3 centered at the origin. Draw the final image on the above grid and label it WXYZ. Provide
coordinates of points W, X, Y, and Z.
W __________
X __________
Y __________
Z __________
b. Write a symbolic rule that describes the composite transformation in Part a. Describe how you obtained
your rule.
(x, y)
(__________, __________)
Explanation:
c. How do the lengths of
and
compare? Explain your reasoning.
d. How does the area of WXYZ compare to the area of ABCD? Explain your reasoning.
21. Triangle ABC and
A´B´C´ are shown below. The scale on each axis is 1.
Since A´B´C´ is bigger than ABC, Alex thinks that A´B´C´ can be obtained by applying a size
transformation centered at the origin to ABC. Do you agree or disagree with Alex? Explain your reasoning.
22. Water and coffee are two of the most common drinks throughout the world. The table below ranks 12
countries according to their per capita coffee consumption and also provides the relative ranking of those
countries in terms of per capita bottled water consumption.
a . Will the rank correlation of this data be positive or negative? Explain how
you can determine this by just looking at the scatterplot.
b. Use the formula
to calculate the rank correlation for this set of data.
= _________
23. The scatterplot matrix below indicates the mean annual amounts of three different pollutants
(measured in micrograms per cubic meter) in 11 selected European cities. The pollutants
considered are suspended particulates, sulfur dioxide, and nitrogen dioxide. Each of the points
corresponds to one city.
a. Which pair of variables has the strongest association? Explain your reasoning.
b. The circled point in each scatterplot corresponds to Athens, Greece. Use the scatterplot matrix to estimate
the amount of each type of pollutant in Athens, Greece.
Suspended Particulates: _______________
Sulfur Dioxide: _______________
Nitrogen Dioxide: ________________
c. On the (suspended particulates, nitrogen dioxide) scatterplot, the point with coordinates
(77, 248) is an outlier. Identify what type of outlier it is. Explain your reasoning.
d. Give possible coordinates for a point that would be an outlier for both variables for the
(suspended particulates, sulfur dioxide) data.
24. The scatterplot below shows the relationship between engine size (in liters) and highway miles per gallon for
many small cars.
a. Describe the shape of the scatterplot.
b. Add a point to the scatterplot that would be an outlier only when the engine size and mpg are considered
jointly. Label your point C. Explain why it is an outlier only when both variables are considered together.
25. The scatterplot below indicates the birth rates (per 1,000 people) and the death rates (per 1,000 people) for 11
different states in the western United States. The regression line is shown on the plot and has equation y =
–0.334x + 12.5.
a. Which of the following is the best estimate of the correlation between birth rate and death rate for these 11
states? Explain your reasoning.
r = -0.82
r = -0.32
r = 0.32
r = 0.82
b. The circled point is the point for Washington. It has coordinates (13, 7.4). On the scatterplot, draw in the
line segment representing the residual for that point.
c. Use the equation for the regression line to calculate the residual for Washington, and then
explain what it tells you.
d. On the scatterplot, circle the point that is an outlier. Then, describe how the slope of the
regression line and the correlation would change if that point were deleted from the data set.
Slope of regression line:
Correlation:
26. The eighth-graders at Claremont Middle School determined their best times (in minutes) for
running both a quarter mile Q and one mile M. The regression equation for the line of best fit is M = 5.25Q +
0.18.
a. Explain the meaning of the slope of the regression line.
b. Circle the vertex-edge graph that you think best illustrates the relationship between the two
running times. Explain your reasoning.
c. The mean of the quarter-mile running times was 2.8 minutes. Find the mean running time for the one-mile
run.
d. Suppose that the times are changed from minutes to seconds. How will the correlation for this new set of
data compare to the correlation for the original set of data? Explain your reasoning.
27. Consider the following summary of a story reported on the radio.
a. What are the explanatory and response variables identified in the news report?
Explanatory variable: _____________________
Response variable: _______________________
b. Do you agree or disagree with the statement “that encouraging people to brush more could help prevent
obesity”? Explain your reasoning.
c. Identify the lurking variable mentioned in the report.
28. Jenny needs to build a box that has a square base and a volume of 900 cm . The height of the box is related to
the length of one side of the base by the function h(s) =
.
a. What value of y satisfies the equation y = h(10). What does this tell you about the box?
b. What value(s) of s satisfy the equation 100 = h(s)? Do all of your solutions make sense in this context?
Explain your reasoning.
c. Describe the theoretical domain and range for h(s).
d. What are reasonable practical domain and range for h(s)?
29. a. Complete the table of values below so that y is not a function of x. Explain.
Explanation:
b. True or False: If x is a function of y, then y is a function of x. Explain your reasoning.
30. Write a rule for a quadratic function with a graph that has x-intercepts (–2, 0) and (8, 0) and
y-intercept (0, 8).
31. Write each product in equivalent ax + bx + c form.
a. (x - 8)(x - 4)
b. (3x + 2)(x + 5)
c. (x + 9)
d. (x + 4)(x - 4)
32. Find an equivalent factored form for each quadratic expression.
a. x - 10x + 21
b. x - 100
c. x + x - 56
d. x + 8x + 16
e. 1 - 2x + x
f. 14x + 18x
g. 2x + 10x + 8
33. Solve each quadratic equation. Use factoring at least once and use the quadratic formula at least once. Show
your work.
a. x + 9x + 18 = 0
b. 3x - 17 = -8
c. x - 3x + 3 = 31
d. 2x - 3x - 1 = 2
e. 5x = 25x
34. a. Write a quadratic equation in the form ax + bx + c = 0 that has solutions of x = -5 and x = 8.
b. Is the equation that you wrote in Part a the only quadratic equation of that form that has the
indicated solutions? Explain your reasoning.
35. The Hillsdale Rock the Vote committee is organizing a community concert and voter registration drive. The
cost for running the concert is $2,470. They need to decide what the minimum admission charge for the
concert should be so that they do not lose any money.
a. If they charge $5 admission, how many people would need to attend the concert in order for the income
from admission charges to equal the cost of running the concert?
b. Write a rule that indicates how the number of people that would need to attend the concert N depends on
the admission charge c.
c. The formula A = 640 - 40c indicates the number of people A likely to attend the concert if the admission
charge is c dollars. Will they make money or lose money if they charge $4? Explain your reasoning or show
your work.
d. Write and solve an equation that will identify the minimum charge in order for the concert to attract enough
people to cover its costs.
36. Consider this system of equations:
y = 3x + 25
and
y = 2x + 14x - 15
a. Use graphs or tables of values to solve this system of equations. Explain your reasoning and
include sketches of graphs or tables of values showing the solution.
b. Solve the system of equations by reasoning with the symbolic form
37. Solve this equation by reasoning with the symbols.
=x+1
38. Explain how it is possible for a system of equations involving one linear equation and one quadratic equation
to have no real number solutions.
39. Without using your calculator, determine if each of the following statements is true or false. Explain your
reasoning for each part.
a. log 10,000 = 4
b. log 0.01 =
c. There is no value of x such that 10 = 54.
40. Without using your calculator, determine which of the following statements is true. Explain how you decided
the statement as true.
Statement I
900
log 925
1,000
Statement II
2
log 925
3
Statement III
3
log 925
4
Statement IV
9
log 925
10
41. Solve each equation without using tables or graphs. Show your work.
a. 10 = 71
b. 10 + 1 = 4,170
42. Recall that a sound with intensity 10 watts/cm has a decibel rating of 10x + 120. The sound
intensity of a popping balloon is 5,011 watts/cm . What is the decibel rating for a popping balloon?
43. The amount of light that is able to pass through the water in a lake depends on the clarity of the water.
Suppose that in one lake, the function rule for light intensity (measured in lux) at a depth of d meters is I(d) =
60,000
.
a. What is the light intensity at a depth of 1 meter?
b. Determine the depth at which the light intensity will be 3,000 lux. Show your work.
44. Consider the line that contains the origin and the point with coordinates (3, 4).
a. Sketch this line and find the equation of the line.
b. Label the angle formed by the line and the positive x-axis
express the following in ratio form.
tan
=
cos
=
sin
. Without using your calculator,
=
45. Triangle ABC is shown below.
a. Compute the length of
AC =
.
b. Without using your calculator, find the following. Express your answers in ratio form.
sin A =
cos A =
tan A =
46. Usually, it is not possible to measure the heights of very tall structures, like towers or flagpoles, with a tape
measure. Surveyors use right triangle trigonometry, measuring lengths that are accessible with a tape measure
(or other tool) and angles between two lines of sight with a transit.
a. A flagpole and transit are pictured below. Sketch and label the vertices of a right triangle with an angle that
could be measured by the transit, a side that could be measured with a tape measure, and a second side that is
(nearly) the height of the flagpole. From your sketch, name the angle and sides that fit the above description.
Angle measured by transit:______________
Side measured by tape measure:______________
Side that is (nearly) flagpole height:______________
b. Rita uses the transit, which is 1.6 m high and 12 m from the flagpole (on level ground), to sight to the top
of the flagpole. The angle of elevation to the top of the flagpole measures 58. Label these measures on the
sketch below.
c. Find the height of the flagpole to the nearest tenth of a meter. Explain your reasoning.
Height of flagpole =______________
47. James needs to attach a stabilizing wire to a tall tower. The wire is 200 feet long and should be attached to the
tower at a height of 100 feet. Assume that the ground around the tower is level and that the entire length of the
wire is used.
a. Draw a sketch of this situation and label any known lengths.
b. Determine the angle that the wire will make with the ground.
c. Find the distance from the tower to the point where the wire is attached to the ground.
48. Melissa and Rosa are golfing on a beautiful summer day. The ninth hole is 380 yards. Melissa
hooked (hit to the left of the correct direction) her drive on hole #9 as sketched below.
How far is Melissa’s ball from the hole? Explain or show your work.
49. A triangular region has sides measuring 25, 35, and 15 feet. Find the measure of the angle opposite the 35
foot long side.
50. Suppose that you know the measures of all three angles in
and that NP = m and
Determine whether or not each of the following would give the correct length for
.
a. MP =
b. MP =
c. MP =
MN = p.
d. MP =
51. To know how much paint is needed for a barn, a farmer is estimating the total surface area of the barn. One
part of the surface is triangular as sketched below.
a. The darkened sides in the figure are the edges of the roof. This trim will be painted white. Find the length
of each of these two sides of the triangle. Explain or show your work.
b. The triangular surface needs to be painted red. Find the area of the triangle. Explain or show your work.
52. Suppose that you are randomly selecting a person from a town with a population of 25,000.
Consider the following events:
• Person is a female.
• Person has a driver’s license.
• Person owns a car.
• Person is younger than 12 years old.
• Person is older than 21 years old.
• Person has read Hamlet.
For each part, identify events A and B from the list above that will make the equation true. Then explain your
reasoning. You may use events more than once.
a. P(A or B) = P(A) + P(B)
Event A ____________________
Event B ________________________
b. P(A and B) = P(A) • P(B)
Event A ____________________
Event B ________________________
c. P(A and B) = 0
Event A ____________________
Event B ________________________
d. P(A) P(A | B)
Event A ____________________
Event B ________________________
53. In Colorado, approximately 20% or 1 in 5 people over the age of six belong to a gym.
(Source: USA Today)
a. Suppose that you randomly select two people who are over
the age of six and live in Colorado. Draw an area model and
use it to determine the probability that only one of the two
people belongs to a gym.
Area Model
b. Suppose you randomly select three people who are over the
age of six and live in Colorado. Find the probability that all
three of them belong to a gym.
c. Suppose Austin randomly selects one person over the age of six who lives in Colorado. Since about half of
the people over the age of six in Colorado are male, he determines the probability of getting a male who
belongs to a gym by doing the following calculation:
P(male and belongs to a gym) =
=
Is Austin’s calculation correct? Explain your reasoning.
54. The table below indicates the number of physicians in Minnesota in 2005 by age and gender.
Suppose that you randomly choose one of these physicians to interview. Find each of the following
probabilities. (Source: Minnesota Physicians Facts and Data 2005)
a. P(female)
b. P(under 65)
c. P(female and under 65)
d. P(female or under 65)
e. P(female | under 65)
f. P(under 65 | female)
g. Are gender and age of physicians in Minnesota in 2005 independent? Support your reasoning using
information from the table.
55. Approximately 12% of the U.S. population wears contact lenses.
a. If you randomly select 200 people, how many would you expect to wear contact lenses?
b. If you randomly select 70 people how many would you expect to wear contact lenses?
56. The Taylor Art Association is planning to have a fund-raiser each month at their monthly art show. People
will pay to spin the spinner below and will win a gift certificate with the indicated value.
a. Complete the probability distribution table for the outcome of one spin .
b. If the Art Association charges $20 for one spin, should they expect to make money, lose money, or break
even over the long run? Show your work or explain your reasoning.
c. What is the fair price to charge for a spin of this spinner? Show your work.
d. Over the course of the year, they expect to have 500 people pay to spin the spinner. What should they
charge for a spin if they want to have at least $2,000 to provide scholarships to Art Camp? Explain your
reasoning.
57. The prizes in a raffle are five $25 gift certificates to a local restaurant, three weekend getaways each worth
$650, and the grand prize of a trip for two to Washington, D.C. worth $2,375. Exactly 3,000 raffle tickets will
be sold.
a. If you buy one raffle ticket, what is the probability that you will win a weekend getaway?
b. What is the fair price to charge for one raffle ticket? Show your work.