Emma Willard School
Troy, New York 12180
MATHEMATICS PLACEMENT TEST 2017-2018
Updated 4/4/2017
Purpose: The tests in this booklet are to help determine proper mathematics placement and minimize the
need for course changes after the start of the academic year. It is important that the student work
independently so that the test will give us a fair representation of her current knowledge and skills. The
tests are for placement purposes only. They do not affect admissions decisions in any way. It is important
to answer questions to the best of your ability in order for the mathematics department to place you
properly.
Date student received test:_________________
Name: (please print)_______________________________Date test taken:________________
Circle grade you are entering at Emma Willard School: 9
10
11
12
PG
Phone number (with area code): _________________________________________________
E-mail address: _____________________________________________ (print legibly please)
Name of most recent school attended and the city and state/country where it is located.
______________________________________________________________________________
What math course did you take this year, and what is your average grade at the time you are taking
this test?
______________________________________________________________________________
Have you taken a full-year course called Geometry? Circle your answer.
YES
NO
If you were remaining in your current school district or at your current school, what would be the
name of the course you would take next year?
______________________________________________________________________________
How do you view yourself as a mathematics student?
Please read the following carefully:
Directions to Parents: Please see that your daughter has a quiet place to complete the test in one
sitting. The test has four levels. Each level is designed to be completed in about an hour to an
hour and a half. Calculators may be used (EXCEPT on the Level One test), but texts and notes
should not be used. We do not recommend extensive review prior to taking the test. It is meant to
reflect your daughter’s accessible knowledge of her retained mathematics knowledge. It is not to
your daughter’s advantage to obtain help on this test since proper placement is contingent on
accurate assessment of her current knowledge of mathematics.
Directions to the Student: This booklet contains four tests, Level One, Level Two, Level Three,
and Level Four. Do as much of all four parts of the tests as you can. The sooner you complete
the tests and return them to the school, the sooner we can properly place you and start the
process of creating your schedule for the fall. Print out the levels you wish to complete. Be sure
to check that all diagrams and problems have printed correctly. Please mail in all parts that you
have completed as soon as possible after you complete them. You may use a calculator except on
Level One. It is important that you give these tests your serious consideration as they will be
the main factor in determining the math course in which you are enrolled. Please show your
work neatly next to the problem (including scratch work) as it is useful in our evaluation of your
methods and skills. Do not use extra paper, and simplify all answers. Showing your work helps
us see where your mistakes were and adds to proper assessment of your understanding and hence
proper placement.
Calculator Use: It is important to note that all Emma Willard mathematics students will use the
Texas Instruments TI-84 PLUS Graphing Calculator in all of our courses. These may be
purchased at cost in our school store. While we teach students to use their calculators
proficiently, we also stress the need to recognize problems that do not need a calculator, and may
require students to solve those problems without one. For this reason, on these placement tests
we ask that you do as many problems as possible without the use of a calculator. On the Level
One test, NO calculators are allowed.
Please complete as many questions as you can on all levels of the four tests. For example, if
you are an entering freshman and have only completed an eighth grade math course, and
are only capable of completing a few problems on the Level One part, THIS IS FINE. If
you are an entering junior, and hope to have proper placement, complete as much as you
can of all four tests.
The purpose of these tests is not to make a judgment about your mathematical ability. It is
to assess how well you have been prepared for the sequence of courses at Emma Willard.
We strive to place new students in the course in which they will find the most appropriate
challenge.
Level One Test
(total points = 108)
In all questions, SHOW YOUR WORK in the space under the question and place your
final answer on the line provided to the right. Do NOT use a calculator on this test.
1. A coat that is regularly sold for $220 is now advertised on sale
at 40% off. What is the sale price of the coat?
1. __________________
(3 pts)
2. A school’s ratio of girls to boys is 4:3. If there are 350 students, 2. __________________
(3 pts)
how many boys attend the school?
3. Evaluate 4𝑥 2 − 2𝑥 − 5 when 𝑥 = −1.
3. __________________
(3 pts)
In 4-5, simplify the expression.
4. −8𝑥 + 2 − 4𝑥 − 9
4. __________________
(3 pts)
5. 2(3𝑥 − 5) − 𝑥(𝑥 − 4)
5. __________________
(3 pts)
In 6-8, simplify the expression.
6. 8𝑐 4 + 5𝑐 6 − 3𝑐 4
7.
6. __________________
(3 pts)
𝑤 8 𝑤 10
7. __________________
𝑤2
(3 pts)
8. (2𝑥)3
8. __________________
(3 pts)
In 9-12 solve the equation.
9. −4𝑐 + 2 = −10
9. _________________
(3 pts)
10. 28𝑥 + 5 = 2𝑥 − 4
10. _________________
(3 pts)
11. The formula F  95 C  32 converts Celsius temperature (C) to
Fahrenheit (F). What is the Fahrenheit equivalent of 15℃?
11. _________________
(3 pts)
9
12. Solve for C in the formula F  C  32 .
5
12. _________________
13. Multiple Choice. To rent a truck for a day, a driver pays a $25
fee. She pays an additional 22 cents for each mile she drives. If
the total cost in dollars is c and she drives d miles, then
13. _________________
(3 pts)
(3 pts)
A) 𝑑 = .22𝑐 + 25
B) 𝑑 = 25𝑐 + .22
C) 𝑐 = .22𝑑 + 25
D) 𝑐 = 25𝑑 + .22
14. Solve the equation for y: 6𝑥 − 2𝑦 = 12
14. _________________
(3 pts)
15. Is (3, 1) a solution of 𝑦 = 3𝑥 − 7? (Support your answer with
work below.)
3
16. Graph the line 𝑦 = 4 𝑥 − 2 using slope & y-intercept. Do not
use a table of values. (1 square = 1 unit)
15. _________________
(3 pts)
16. (3 pts)
y
x
17. Given the line AB graphed to the right,
a. Find its numerical slope.
17.
a. __________________
(3 pts)
b. Write an equation to represent the line.
b. __________________
(3 pts)
18. A line passes through the points (-7, 3) and (2, -4).
a. Calculate the slope of the line. Show work.
18.
a. __________________
(3 pts)
b. Write an equation for this line.
b. __________________
(3 pts)
3
19. Given the slope of a line is and the point (2, -9) is on the
7
line. Write an equation for the line.
19. _________________
20. Solve the system, algebraically, showing your work:
𝑥 + 2𝑦 = 1
{
5𝑥 − 4𝑦 = −23
20. _________________
(3 pts)
(3 pts)
21. Multiply: (4𝑥 − 3)(2𝑥 + 5)
21. _________________
(3 pts)
22. Simplify: (𝑥 + 8)2
22. _________________
(3 pts)
In 23-26, factor (using integers) the polynomial expression.
23. Factor: 7𝑎4 𝑏 2 − 28𝑎𝑏 8
23. _________________
(3 pts)
24. Factor: 𝑥 2 − 64
24. _________________
(3 pts)
25. Factor: 𝑥 2 − 2𝑥 − 15
25. _________________
(3 pts)
26. Factor: 8𝑥 2 − 2𝑥 − 3
26. _________________
(3 pts)
27. Solve for x: (𝑥 + 7)(𝑥 − 6) = 0
27. _________________
(3 pts)
28. Solve the equation:
7
4
=
9
28. _________________
𝑤−5
(3 pts)
29. Rewrite √28 in simplified radical form.
29. _________________
30. Solve for x: 𝑥 2 − 5 = 0
30. _________________
(3 pts)
(3 pts)
31. Solve for x: 6𝑥 2 + 3 = 27
31. _________________
(3 pts)
32. Simplify: (−4√2)(2√6)
32. _________________
33. Simplify: 7√3 + 6√3
33. _________________
(3 pts)
(3 pts)
b  b 2  4ac
34. Use the quadratic formula, x 
to solve the
2a
equation: 3𝑥 2 − 7𝑥 − 6.
34. _________________
(3 pts)
Level Two Test
In all questions, SHOW YOUR WORK in the space under the question and place your final
answer on the line provided to the right. You may use a scientific or graphing calculator.
1. (2 pts) Measure of angle x = ?
2. (2 pts) In the figure, 𝑚 ∥ 𝑛 and 𝑚∠𝑥 = 75°
Find my .
y
m
x
n
3. (2 pts) my  ?
4. (2 pts) In the figure, mQ  90 and 𝑚∠𝑄𝑃𝑅 = 65°. mP  ?
5. (2 pts) If ̅̅̅̅
𝐴𝐵 || ̅̅̅̅
𝐶𝐷, then mx  ?
6. (2 pts) In the plane figure below, MN = NT = TV. Find the measure of TMN .
7. (2 pts) Which of the following reasons can always be used to prove two triangles congruent?
Circle all that apply.
SSS
AAA
SAS
SSA
AAS
8. (3 pts) Given triangle ABC with vertices A(-4, -4), B(2, -2), and C(0, 4), find the location of the
centroid (the intersection of the medians) for this triangle. State the coordinates of the centroid.
9. (1 pt) Two similar triangles have areas in the ratio 16:25. The ratio of a pair of corresponding
sides is
A. 16:25 B. 256:625 C. 25:16 D. 4:5 E. 2: √5
3
10. (3 pts) Reflect the point A(-3, 0) across the line 𝑦 = 2 (𝑥 − 2) + 1. State the coordinates of
the reflection.
11. (1 pt) Given ABC such that 𝑚∠𝐴 = 80° and 𝑚∠𝐵 = 44°. The longest side of the triangle is
A. AB
B. AC
C. BC
D. all sides are equal
E. not enough information is given
12. (2 pts) ABCD is a parallelogram. Solve for x.
13. (2 pts) If the altitude of an equilateral triangle measures 6, then what is the area of the triangle?
14. (2 pts) What is the radius of circle O if PQ = 20.
O
P
Q
15. (2 pts) Find the sum of the interior angles in an octagon.
16. (3 pts) A 12 foot ladder leans against a wall. Its top touches a point on the wall 8 feet above the
floor. How many feet is the bottom of the ladder from the base of the wall? Draw a diagram and put
your answer in simplest radical form.
17. (2 pts) A rhombus has diagonals of 18 and 26. What is the perimeter of the rhombus? Draw a
diagram and put your answer in simplest radical form.
18. (2 pts) In right triangle ABC, BC=15√3. What is the perimeter of the triangle?
A
30
C
B
19. (4 pts) Find the area and perimeter of the triangle below. E = (1, 1), W = (2, 4), and S = (5, 3).
20. (6 pts) Write the following proof either in a two-column format or written in paragraph form.
Given: ABCD is a parallelogram.
Prove: AE ≅ EC
21. (2 pts) What is the area of ∆𝐴𝐷𝐵 in square units?
22. (2 pts) What is the area of parallelogram ABCD in square units?
23. (2 pts) Find the area of a circle whose circumference is 50𝜋.
24. (2 pts) Square ABCD is inscribed in circle with center O. OA = 10. What is the area of the shaded
region in square units?
B
A
O
D
C
25. (1 pt) In the circle below, what is the degree measure of arc AB?
26. (2 pts) The edge of a cube is 3 cm. Find the total surface area of the cube. Include units of
measurements in your answer.
27. (2 pts) The diameter and height of a cone both measures 9 cm. What is the volume of the cone?
28. (2 pts) Find the distance between the points (-5, 7) and (4, -1).
1
29. (3 pts) Write an equation of the line parallel to 𝑦 = 4 𝑥 + 7 going through the point (2, 5). Put
your answer in point-slope form.
30. (2 pts) Given the line segment with endpoints (3, -6) and (5, 4), determine the coordinates (x, y) of
the midpoint.
31. (2 pts) List the letters of the answer(s) that give (s) you enough information to determine that the
quadrilateral is a parallelogram.
A. Both pairs of opposite sides are parallel.
B. Two pairs of consecutive sides are congruent.
C. Diagonals are congruent.
D. Diagonals bisect one another
32. (3 pts) Triangle ABC with A(1, 7), B(2, 3) and C(-3, 4) is transformed according to
T(x, y) = (−2𝑥, −𝑦). Find the coordinates of the transformed triangle.
33. (4 pts) Find 3 points equidistant from the two points E(-2, 1) and W(4, 5).
34. (4 pts) Given below is a circle with two of its tangents drawn in. Find the measure of arc DE and its
arclength.
Level Three Test
In all questions, SHOW YOUR WORK in the space under the question. You may use a
scientific or graphing calculator.
1. (4 pts) Write an equation of a line perpendicular to 8𝑥 − 7𝑦 = 2 and containing
the point (2, -5).
2. (3 pts) A take-out menu offers 7 different beverages, 8 different appetizers, and 13
different main courses. If you want to place a take-out order consisting of 4 different
beverages, 4 different appetizers, and 3 different main courses, how many different
orders are possible from that menu?
3. (2 pts) State the domain & range of the relation graphed below.
Domain:
Range:
4. (2 pts) Is the relation in #3 a function? Explain why or why not.
5. (2 pts) Identify whether the function 𝑔(𝑥) = 𝑥 5 + 𝑥 3 − 𝑥 + 1 is an even function,
odd function or neither. Explain your choice.
6. (2 pts each) Given the point (-3, 4) is on the graph y  f ( x) , what point is on the
graph of:
a. 𝑦 = −2𝑓(𝑥)
b. 𝑦 = 𝑓(𝑥 + 5) + 1
1
c. 𝑦 = − 2 𝑓(𝑥) + 3
7. (3 pts) Let 𝑓(𝑥) = 2𝑥 2 − 3 and 𝑔(𝑥) = 3𝑥 + 4. Find 𝑓(𝑔(𝑥)) and simplify.
8. (2 pts) If 𝑓(𝑥) = 5𝑥 + 2, then f 1 ( x) =_?_
9. (5 pts) Find the EXACT coordinates of the x-intercept(s), y-intercept, and vertex
of the parabola 𝑦 = 6𝑥 2 + 27𝑥 − 15 algebraically. Show your work below.
10. (2 pts) Solve for x: log 3 ( 𝑥 + 8) = 2. Show your work clearly.
11. (2 pts) Simplify:
(𝑥 0 𝑦 −2 )−3 𝑧 4
𝑥 −5 𝑦𝑧 3
. Write your answer with positive exponents.
2
12. (2 pts) Simplify:
8 −3
(27) .
Write your answer as a simplified fraction.
13. (3 pts) Draw a graph of the function 𝑅(𝑥) =
2𝑥+3
𝑥−3
. Be certain to plot at least three
points and include any asymptotes as dashed lines.
14. (1 pts) Write 53 = 125 in logarithmic form.
15. (1 pts) Evaluate: log 3 81.
16. (5 pts) Write down the equation for a polynomial function
consistent with the graph below, which has 𝑥-intercepts at
−2, 0, 3, and 4 and passes through the point (1,12).
17. (3 pts) Solve for x:
1
𝑥+2
=
7
3(𝑥+2)
−7.
18. (2 pts) If the junior class has 95 members, then how many different ways can the
class choose four members to be president, vice president, secretary, and treasurer?
3
19. (2 pts) Solve for x algebraically and show your work: −3 = 4 + √𝑥 + 2.
20. (2 pts) Solve for the missing parts that are labelled with letters:
21. (3 pts) Write an equation of the following graph.
22. (2 pts) Calculate  to the nearest tenth of a degree.
23. (2 pts) Give the exact value (no decimal) of sin 150°.
24. (3 pts) Solve sin 𝑥 = −
√3
2
for x in degrees where 0  x  360 .
25. (2 pts) State the amplitude and period of the graph of
4
𝑦 = −4 sin (5 (𝑥 − 1)) − 3.
26. (3 pts) Graph one full period of 𝑓(𝑥) = 4 sin 𝑥. Be certain to plot at least five
points.
𝜋
27. (3 pts) Graph one full period of 𝑓(𝑥) = − cos (𝑥 + 2 ). Be certain to plot at least
five points.
28. (2 pts) Give the exact solutions (no decimals) of cos 𝑥 = −
0  x  2 .
29. (2 pts) Give the exact solution(s) (in radians) of tan−1 (−
√2
2
1
for x in radians,
).
√3
30. (2 pts) Solve for x algebraically, showing your work: 5𝑥 = 20. Round to the
nearest thousandth.
31. (2 pts) Graph. f ( x)  log 2 x . Be certain to plot at least three points and include
any asymptotes as dashed lines.
y
x
32. (2 pts) Solve for x: ln 𝑥 = 4. Give an exact answer or round to the nearest hundredth.
33. (4 pts) Write an equation of the parabola with vertex (-5, -3) that also contains the point
(1, -2).
Use the following information to answer questions 34 – 36. Round answers to two decimal
places.
An art dealer buys a painting for $5000. Its value increases 3% per year.
34. (2 pts) Write an equation for the value V of the painting as a function of the time t in years
since the dealer bought it.
35. (1 pts) Use your equation to determine the painting value 8 years after the dealer bought it.
36. (2 pts) How many years after the dealer bought it will the painting’s value be $6500? Round
your answer to the nearest tenth of a year.
Level Four Test
(total points = 75)
In all questions, SHOW YOUR WORK in the space under the question and place your
final answer on the line provided to the right. You may use a scientific or graphing
calculator.
1. Solve for x (no decimals): ln 𝑥 = 9.
1. __________________
(2 pts)
2. To the nearest tenth, solve for x in the triangle shown below.
2. __________________
(3 pts)
3. Use the Law of Cosines to solve for the missing side. Round
to the nearest hundredth.
3. __________________
4. Find the sum of the infinite geometric series. Give an exact
answer.
1 1
1
+
+
+⋯
3 27 243
4. __________________
5.Calculate the 145th term in the arithmetic sequence:
-2, 3, 8,…
5. __________________
(3 pts)
(2 pts)
(2 pts)
6. Find lim
5𝑥 3 −1
𝑥→∞ 4𝑥 2 +3
7. Find lim
𝑎→∞
−10𝑎3 +9𝑎2 −1
4𝑎4 −100𝑎2 +11𝑎
8. Write an equation for the ellipse graphed below.
6. __________________
( 3 pts
7.__________________
(3pts)
8. __________________
(3 pts)
9. Solve showing steps: 2 log 3 𝑥 + log 3 4 = log 3 92. Express
answer as a decimal rounded to the nearest hundredths.
9.__________________
(3 pts)
10. Women’s heights are normally distributed with a mean of
64 inches and a standard deviation of 2.5 inches. Please use the
standard normal distribution table on pages 7 and 8 to answer
the following questions.
a. What is the standard value (z-score) of a women’s height
of 68 inches?
b. Find the women’s height at the 89th percentile.
c. What is the probability that a women has a height of 59
inches or more?
11. The weather forecaster says that there is a 40% chance of
rain for each day. Round each answer to the nearest hundredth.
10.
a.__________________
(3 pts)
b.__________________
(3 pts)
c.__________________
(3 pts)
11.
a. What is the probability that it rains at least one of the next
three days?
a.___________________
b. What is the probability that it will rain exactly 2 days of
the next 7 days?
b.___________________
(3 pts)
(3pts)
12. Given vectors 𝑢 = 〈−2,1〉 and 𝑣 = 〈−3,4〉, make an accurate
sketch on the grid of 2u  v . Include the resultant vector in your
sketch. (1 square = 1 unit)
(3 pts)
13. Prove this trigonometric identity. Show all steps.
cot 𝑥 + tan 𝑥 = sec 𝑥 csc 𝑥
(3 pts)
14. Given the identity:
14. _________________
cos 2 x  cos x  sin x  1  2sin x  2 cos x  1 .
2
2
2
2
Find the EXACT radian solutions for cos 2 x  3sin x  2
where 0  x  2 .
(3 pts)
15. A box with a square base and no top has volume 12 cubic
meters. The material for the base cost $6 per square meter, and
the material for the sides cost $4 per square meter. Express the
cost, C, of the materials to make the box as a function of the
width, w, of the bases. Then, find the dimensions of the box
with the minimum cost. Round your answer to the nearest
hundredth.
15. _________________
16a. Write an equation for a hyperbola with these points:
Center at (0, 5)
One vertex at (0, -1)
One focus at (0, 12)
16a._________________
b. Write the asymptote equations for the hyperbola.
b. __________________
(3 pts)
(3 pts)
(3 pts)
17. Does the graph of 𝑦 2 − 𝑥 2 𝑦 = 5 have x-axis symmetry?
Justify your answer algebraically showing work.
17._________________
18. 9 students are chosen at random for the prom committee
from a group that includes 5 sophomores, 12 juniors, and 10
seniors. What is the probability that the committee will have 0
sophomores, 4 juniors, and 5 seniors on it? Round your answer
to the nearest thousandth.
18.__________________
19. An airplane flying northeast with a speed of 380 miles per
hour encounters a 60 mile per hour wind blowing to the west.
19.
a. What is the new speed of the plane rounded to the hundredths
place?
(3 pts)
(3 pts)
a.___________________
(3 pts)
b. To the nearest whole degree, how many degrees will the wind b.___________________
(3pts)
blow the plane off its northeast course?
20. State the radius and center of the circle with the equation:
(𝑥 − 7)2 + 𝑦 2 = 81
20.
Radius:______________
(1 pts)
Center:______________
(2 pts)
21. Solve 14 − 2(5)𝑥 = 85. Round to the nearest thousandth.
1
22. Write as a single logarithm with base 8: 2 log 8 25 + log 8 4.
21. _________________
(3 pts)
22. _________________
(3 pts)