Download - Math Level 2
-
8/21/2019 Math Level 2
1/28
SAT* SUBJECT TEST
M TH LEVEL
The subject tests are
designed to test
your knowledge of
a particular
subject
. While
the
Math
Level 2 test will
have some of the same
topics as
the
SAT Reasoning Test,
the word
i
ng of the
questions and
the level
of
math
tested
is different.
You
will find when
ta
king th is test that
you
do
not have to spend as much time interpreting questions. The majority of your t i
me
will be spent
discerning wh ich formula is needed to solve
the
problem .
Keep in mind that most people
take the
Math Level 1 instead of the Math Level 2; the refore, the
curve is much
more
difficult . On
the
Math Level 1, you cannot afford to miss or skip more
than
7
questions and still hope
to
be in
the
700 range. However, on the Level 2, you can miss
up
to 20
questions and
still be in the 700 range. All students
who
have
completed
Algebra
and are
in a
college bound math class should consider taking the Level 2 instead of
the
Level 1 Please check
with
our
office
to
determine which
test
is a
better
fit
for you
.
Remember, a raw score
is
computed by counting the number of questions
correct,
and
subtracting the guessing
penalty
. A
sample
score conversion for Level 1
and
Level 2 is
provided
below
.
Level 1
Level2
Raw
Score Scaled
Score Raw Score
Scaled Score
42 -50
700-800 30-50 700
-800
33 -41
600-690 20-29 600-690
21-32 500-590
8- 19
500-590
10-20
400
-
490
3-7
400
-
490
The
Math
Level 2 test contains
problems
that
test your
knowledge of numbers
operations,
algebra, solid geometry, coordinate geometry, trigonometry,
functions,
probab ility,
and
statistics.
This lesson will provide you with the formulas and
the
practice problems you will need in order to
master
this test .
The
actual test is
SO
questions and 60 minutes long . Please take a pr
act
ice test
or two) at our office to
complete
your
preparation
for this subject test.
Topics
Numbers and Operations
Algebra
Geometry
Coordinate
3-D
Trigonometry
Data
Analysis, Statistics
and
Probability
ercentage
of
Test*
10-
14%
48 - 52%
38-42%
10-14
%
4-6%
12-16
%
6-10%
*
Note:
The values in the table above
are
approx imations.
© 2008 Karen Dillard All rights reserved. - A.5
Number of
Questions
5-7
24-26
13-21
5-7
2
-3
6-8
3-5
No part of this material may be copied
or
used without written permission from Karen Dillard's College Prep
"
3A.T •
1ef, IIS\-ete-d tr• derNtfk of lf•e Collel,l* Boara.
Mncn
were
no
t lf Vol\ltd ltl lhe
ptodur.:;•m
of. ar.d do ntil
endolte,
Uws
p r u J u ~
-
8/21/2019 Math Level 2
2/28
M TH LEVEL 2
GE
C LCUL TORS
The
College Board
calculator
policy is quoted below. Make sure you check
the
website
www.collegeboard.com, to see
if
any updates have been made to the policy.
It s NOT necessary to use a calculator to solve every question on
the
Mathematics
Level
2 Subject Test so it s important to know when and how to use one. For about 35-40
percent of the
questions, there s
no advantage, perhaps even a
disadvantage, to
using
a calculator. For about
55-65
percent
of
the questions a
calculator
may
be
useful
or
necessary.
A scientific or graphing calculator is required
for these
tests. A
graphing
calculator may
provide an advantage over
a scientific
calculator on
some questions. The
tests are de
veloped with the expectation that
most
students are using graphing calculators.
(www. collegeboard. com)
cceptable
Calculators
Calculators permitted during testing are :
• Graphing calculators
• Scientific calculators
•
Four-function
calculators
(not recommended)
Unacceptable Calculators
Unacceptable calculators are those
that:
• Use QW RTY (typewriter-like) keypads
• Require an electrical
outlet
• Talk
or
make
unusual noises
• Use paper tape
• Are
electronic writing
pads, pen
input/stylus
-driven devices, pocket organizers,
cell phones, powerbooks, or handheld or laptop computers
Reminder
f
you
use a
calculator with
a large or raised display
that might
be visible
to other test
takers,
you
will be seated
at the
discretion
of the test
supervisor. You
may not
share
your
ca l
culator
with
another student during
the
test. Any use
of
calculators for sharing or exchanging
or
removing
part
of
a
test book
or any
notes relating to
the
test
from the
test
room may be
grounds
for
dismissal
and/or
cancellation
of
scores. Calculators may
not
be on
your
desk
or
be used on
the verbal sections
of
the test. f
your calculator
malfunctions and
you don t
have a backup
calculator
or
extra batteries,
you may,
if
you wish, cancel scores on
the
Mathematics
Level 1
or
Mathematics Level 2 tests.
© 2008 Karen Dtllard All nghts reserved -A 5
No
part of thts material may be copied or used wtthout written permission from Karen Dtllard s College Prep
-
8/21/2019 Math Level 2
3/28
3
MATH LEVEL 2
P GE
NUMBERS OPER TIONS
Like the Level
1,
Level 2 questions in Numbers Operations can be
drawn
from
operations,
ratio
proportion,
comp
l
ex
numbers, counting, elementary
number
t heory, matrices and sequences.
In addition
,
the
Level 2
may
contain
questions
on series and vectors.
NUMBER GROUPS
Natural Numbe
rs
Whole Numbe
rs
Integers
Ra tional
Numbers
Irrational
Numbers
Real Numbers
Complex Numbers
COMPLEX NUMBERS
Counting
numbers: 1,
2, 3, . . .
Counting
numbers
with zero: 0,
1,
2, .
Whole numbers with additive inverses: .
2,
-1, 0,
1,
2,
Any number
that
can be
writte
n as a
fraction of
2
integers
,
including integers, repeating decimals, and terminating
decimals.
Numbers
that cannot
be
written
as a fraction
of
2
integers,
such
as
n
e, and roots
that
do
not
simplify
to
integer values.
Numbers that are either rational
or irrational.
See below
A
complex number
is the sum
of
a real number and an imaginary number. An imaginary number
is
defined
as
the square
root
of
-1
and
is denoted
i.
EX MPLE
i
2
=
- 1
·
.
I 1
i
4
=
1
Complex numbers
are written
in the form a +
bi
where a
and
b
are
constants. Treat complex
numbers
just
as
you
would a binomial
that
is, an expression
with two terms)
.
dding
When
adding
complex numbers, be sure
to
add the real parts
to
each
other and the
imaginary
parts to each other. Only combine similar elements:
(a + bi ) + (c + di) = (a + c) + b + d)i.
EX MPLE
(7 - 2i) + (4 + 5i) = (7 + 4) + (- 2i + 5i) = + 3i
Multiplying
Use the
same
method to multiply
complex
numbers as
you
use for
binomials
. Many
mathematicians distribute terms using the FOIL method . However, please note
that the solution
will be a complex number,
not
a trinomial
an
expression with three
term
s
),
as it is
when you
multiply
binomials. The P
term
is equal
to
-
1,
so simplify all P terms.
8 + 2i
) 4
+
3i)
distributes as 32 +
4i
+ 8i + 6P , which simplifies to 26 +
32 i
.
© 2008 Karen Dillard. All rights reserved. - A.5
No part
of
this material may be copied
or
used without written permission from Karen Di llard s College Prep
-
8/21/2019 Math Level 2
4/28
M TH L V L 2
4
GE
bsolute Value
The absolute value of a complex number is its distance
from
the origin. This works just like
finding
the
length of a vector. Applying
the
Pythagorean Theorem, we can find:
VE TORS
Vectors are rays with magnitude and direction. They are represented with ordered pairs in
2-dimensions, or ordered triples in 3-dimensions. Vectors are drawn as an arrow from the origin
to
the ordered
pair or
triple.
To add or
subtract)
vectors, add or
subtract) the
ordered pairs
triples),
component by
component.
Graphically,
the
tail
of
the
vector
being added is placed on
the
head
of
the first
vector. Then, the sum is obtained by drawing an arrow from the origin to the head of the second
vector.
To
subtract, reverse the direction of the second vector.
The magnitude absolute value of a vector is the square root of the sums of the squares of the
components
another
application of
Pythagorean
Theorem).
To obtain a
unit
vector vector of magnitude
1)
in the direction of a given vector, divide each
component by the magnitude of the vector.
The Level 2
test
does not cover
multiplication
of
vectors
.
10
8
6
4
2
10
8 6 4
2
2
4
6
8
10
a b
© 2008 Karen Dillard All rights reserved
-A
5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
5/28
M TH LEVEL 2
5
PAGE
PRIME NUMBERS PRIME FACTORS
Prime numbers
are
defined as positive
integers
with exactly
two
factors, such as
2
3,
5, 7,
11,
13,
17, 19, 23, 29, 31,
etc.
None
of
these
numbers has any factors
other
than 1 and itself. ON is NOT a
prime
number.
The
only
even prime number is 2.
Any positive
integer can be expressed
uniquely
as the product
of primes-called
the prime
factorization.
You probably remember
doing
prime
factorization,
or factor
trees.
SEQUENCES
Some problems
involve
a sequence of numbers. Often you can use a rule to generate successive
numbers in the sequence . f the problem seems to require a
very
high number
of
calculations
(like
asking
for
the
1QQth
term),
do
the
first
several
steps and
look
for
a
trend.
Usually
it s easier
to
find the trend this way than with algebra.
Ar
ithmeti
c
Sequences
are lists of numbers
that
increase or decrease by a constant . All
of
these can be described with a
simple
linear equation . Use the form an + b where n represents
the term number in the sequence, represents the constant difference between consecutive
terms,
and
b indicates how to shift the sequence so
that
it starts on the correct number at n
=
1.
Another
(and often easier)
form for the terms in this sequence is
an
=
a
+ n -
1
d where dis the common difference.
Geometric Sequences are l ists
of
numbers in which each term is multiplied by a constant
to
get
to
the next term. The simplest sequences are the power sequences, such as the powers
of
2 :
1, 2, 4, 8, 16,
...
In general, i f a is the first term , and r is the
constant
multiple, the n th term is r
1
•
Alternatively1
t t
r -
1
where tn is the nth term in the sequence and
r
is the multiplier.
©
2008 Karen
Dillard.
All
rights reserved. -
A.5
No part of this material may be copied or used without written permission from Ka ren Dillard s College Prep
-
8/21/2019 Math Level 2
6/28
MATH LEVEL
2
6
GE
I. If y is a multiple of 3, which of the following must be
a multiple of6?
A) y + 3
B) 6
y 3
2y 6
) y + 3
«.-\: .•r
2.
In
the geometric sequence 3, 9.6, 30.72, ... what is the
sixth term?
A) 43.18
8)
98
3
0
C) 191.86
D) 3
14
.57
(@
1006.63
s
"\..
r
3
There are men and
w
women
in
a room. After
15
women leave, there are three times as many men as
women. In terms
of
w how many men are there?
w-15
m =
A)
w+l5
m
3
B)
@m= w
- 15)
m=
3w -
15
(E) m = 3w
+
15
©
2008 Karen Dillard All rights reserved
-A
5
No part of th s material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
7/28
7
M TH LEVEL 2
P GE
LGEBR FUNCTIONS
Like
the
Level
1, the
Level 2
tests items
in expressions, equations, inequalities, representation
modeling, and properties of functions
(linear, pol
ynomia
l, rational,
and
exponential. In addition,
the
Level
2
tests properties
of
logarithmic, trigonometric, inverse trigonometric, periodic, piece
wise, recursive and parametric functions.
LOGS
Do
not let
logarithms
intimidate
you.
They are simp
ly an inverse function
to
exponentiation,
much like division is
to
multiplication, or
subtraction
is
to
addit ion. Here s how
it
works:
ab =
c if and
only
if logac
=
b
Therefore,
it
follows
that
loga ab
=
b
for
all a and
b
f the
base
of the
log is
not given
explicitly, it is assumed
to
be base
10.
Your calculator will
only
compute logs in base
10
or base e in
the
case
of
natural logs.
Ch
f B F I
I
b
_ log b
ange
o
ase ormu a og -
log
This
formula
allows
you to compute
logs
with different
bases on
your
calculator.
The rules
for
exponents have
their
counterparts
as rules
for
logs. Note that
they
correspond
directly
with the
rules
for exponents that
you
already
know.
©
2008 Karen Dillard. All rights reserved. - A.S
No part
of th
is material may be copied
or
used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
8/28
M TH L V L 2
GE
SOLVING EQU TIONS
SYSTEMS OF EQUATIONS
A system of equations
involves
multiple variables in multiple equations. A rule of algebra
states
that you
need at least as
many equations
as variables
to
solve
for
every
variable.
Some
problems on
the
Level 2 Math involve two
variables
and two equations. There are two basic
methods for solving systems of equations. Both methods
involve combining
the equations in
such a way as to get an equation with only one variable which you then solve algebraically using
either
the
substitution method or by
variable
elimination.
Special Cases
There are
two
unusual possibilities that can
occur
in systems of equations. The
first
is
demonstrated by the following:
a
2b
=
4
2a
4b
= 8
f we attempt
to
combine
the
equations
we get a result
like
0 = 0, which provides no
information. As it
turns out
this system cannot be solved: there are an infinite
number
of
solutions. Since
the
second equation is a multiple of
the
first we discover that they are the
SAME equation. Therefore any
solution
to
one
equation will be a
solution to the other
equation.
The other unusual case is this:
a 2b
= 4
2a 4b = 9
f we try to so
lve
this
system
we
get
0
=
1 Since there
are
no values a and b that can satisfy
both equations
there are no solutions. When graphed this
system
looks
like
two parallel lines.
tratqy Hi t
The solution to equations
like
x = 3x 7 can often be
found
by
graphing
both sides of
the
equation with a graphing calculator, and then finding the point of
intersection.
© 2008 Karen Dillard All rights reserved -A 5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
9/28
M TH
LEVEL
2
P GE
DIRECT VARIATION
Direct variation can be identified when the problem states
that
a direct variation exists or
states that a variable is directly proportional to another variable.
In
direct variation, as one
quantity
increases,
the other quantity
increases,
or
i f
one
quantity
decreases,
the other quantity
decreases.
•
x
varies directly as
y
•
x
and
y
change proportionally
•
x
and
y
are in proportion
X
~
All
of
these descriptions describe the same thing:
x
and y increase
or
decrease together.
Therefore Y
will have one value,
y
=
kx, for
some
constant k
X
INVERSE VARIATION
Inverse
variation can be
identified
when
the problem states
an inverse variation exists
or
states
that
one variable is
inversely
proportional to another. Another
way
to recognize inverse variation
is
by observing that,
as one
quantity
increases the other
quantity
decreases.
• x
and y are
in inverse proportion
• x and
y are
inversely
proportional
• x varies inversely
as
y
All
of these descriptions
describe
the
same
thing:
x increases when
y
decreases,
and
x
decreases when
y
increases. Therefore
xy
will have one value so
y = k for
some constant
k X
QUADRATIC EQUATIONS
AND OTHER POLYNOMIALS
A quadratic equation is characterized by having 2 as the highest exponent. In the simplest type
of
quadratic equations, such as x
=
4,
you can solve by taking the square
root of
both sides.
However, you
must
remember that the solution can be positive
or
negative; x
= ± 2. These
solutions
are also called
the
zeros
of
a
function
y = xL 4),
or
roots
of
a polynomial (x
-4) .
In some cases, you may need to factor or use the
quadratic formula.
In general, the solutions
(roots) for
a
quadratic
equation,
ax
+
bx
+
c
= 0
can be found using
the
quadratic formula.
± -
4ac
x
2a
© 2008 Karen Dillard. All rights reserved. - A.5
No part
of
this material may be copied
or
used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
10/28
M TH L V L 2
10
AGE
EXP NDING POLYNOMI LS
Complete binomial expansions are not necessary on the Level 2
test
. Rather, it is more important
to
know
the
number
of
terms an expansion might produce. otherwise, the use
of
Pascal s
Triangle
may
help
you
find
those
middle terms.
Remember,
the
coefficients
will
change according
to
the leading
coefficient of
the
binomial
being expanded. You can use the
following to
find the
leading
coefficient
for each term:
x+y
n =
t
x n-kyk
O k
Note
that
Z= c k can
be
computed on your calculator.
Pascal s Triangle is
below
and gives the coefficients of each term
of
an
nth
degree
polynomial
in
the
nth
row of the triangle.
Note that
the subsequent row
can be
found
by taking the sum of the
two
numbers above it. Also,
note
that these are the coefficients when the original polynomial has
a 1 as
the
leading coefficien t. Check
with your
teacher
to
figure
out
how
to
use Pascal s
triangle
to
determine
the
coefficient of
other polynomials.
Pascal s Triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1
5
10 10
5
1
SYNTHETIC DIVISION
Synthetic division is much like long division, except you use the coefficients from the polynomial
to divide
one
piece at a time.
x
4
- 3x
3
- 5x
2
+2x-18
x+
2
First, the divisor, x + 2, must be written as a difference, -2),
to
determine that the divider is
-2 . Then, list the coefficients in descending exponent order remember
to
place a 0 wherever a
term
for an ex
ponent
is
missing.)
2
1
-3 -5
2 -18
- 2
10 -10
16
1
- 5
5 - 8 - 2
© 2008 Karen Dtllard All rights reserved - A5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
11/28
M TH LEVEL
2
P GE
Finally,
your solution
becomes:
x
4
-3x
3
-S x
2
+2x-18
3 2 -2
=x -sx 5 x 8 -
x+2
x+2
FUN TIONS
Algebraic functions in the form
of
f x)
= represent
a series
of
operations. When dealing with
functions of this form, apply the
rules relating
to the properties of functions when their
graphs
are required.
Definition F
or
every x-value entered (domain), there is 1, unique
y-value (range)
Domain The set of values t hat may be put into a function.
(The x-values
of
a function)
Range
The set of values that can be produced by a function.
(The
y-values
of
a
function)
Even
Functions A function
for
which f x) = f
- x);
meaning
they
are symmetrical
about the y-axis.
Odd Funct
i
ons A function for which - f x) = f
-
x);
meaning
they are symmetrical
about the origin.
Root
Values in a function s domain
at which the
function equals zero.
OMPOSITE FUN TIONS
A composite function is a combination
of two or
more functions in sequence. Composite
functions
are essentially
functions of
a function -
you
take the
output of the first
function
and put into
the
second function.
f x) =
x
+ lOx + 3
g x) =
+22
What is the value
of
g
( f ( - 4))
?
Step :
f - 4)
=
- 4)
2
+ 10 -4) +
3
Step 2: f (
-
4) = 16
+ -
40
+ 3
=
- 21
Step 3:
g
- 21) = I.J-211+221= 1
Therefore g ( f (
-
4)) =
1
©
2008 Karen Dillard. All rights reserved. - A.S
No part
of
this material may be copied
or
used without written permission from Karen Dtllard s College Prep
-
8/21/2019 Math Level 2
12/28
MATH LEVEL 2
1 2
AGE
The
more complicated
type of
composite
function questions ask you
to
find the algebraic
expression of a
composite function.
Essentially,
this means
you ll be substituting one
function into
another.
f
(
x)
= x
2
+
lOx
+
3
1
9
(x) = .Jx+22
What is
9
f (x )) ?
Step :
9 f x ) )=
1
~ x +10x+3 +22
9 f x ) )
=
1
~ x +10x+25
Step
2:
9 f
x
) )=
1
~ x + 5 )
Step 3:
9 f x ) )=
1
x+S)
Step
4:
INVERSE FUN TIONS
For
the
SAT you may need
to
be able to find
the
inverse of a function, { -
1
(x) as well as
identify
the
graph
of
an
inverse
function.
Not
all inverses are
functions,
only
ones
that
pass
the
horizontal
line test (they are one-to-one).
To find the
inverse
of a function, let
y
=
f
(x), replace all the
x s with y s (and
vice versa) in
the equation, and solve
for y
again. This
is
really solving
for y
1
.
To identify the graph of an
inverse function,
reflect
the function s graph across the line y
=
x
Two functions
f
and
9
are inverses i f
f
(9
(x))
=
9
f
(x ) )
=
x. The domain of a function is the
range of
its
inverse, and vice versa.
©
2008 Karen
D llard
All rights reserved. -A 5
No
part
of
th s
material may be copied or used without written permission from Karen Dillard s
Co
lege Prep
-
8/21/2019 Math Level 2
13/28
EX MPLE
Let y f
x)
x-2)
2
y-2)
2
I f
y
.:.__ .:.._
.
switch
variables)
---
x
-
4 4
Solve
for dependent
variable:
y-2)2
X= : : ______:_
4
4x y-2)
2
2-JX=y-2
y=2.JX+2
Therefore
f -
1
x) 2 £
+2
TRANSFORMATIONS AND TRANSLATIONS
For
c > 0,
in the coordinate plane,
let y f x)
• y f
x)
+ c
•
y
f x)
- c
•
y f x)
• y = f - x )
is shifted upwards c units
is shifted downwards c units
is reflected over the x-axis
is reflected
over
the
y-axis
13
Constants a, b, c, and d
affect
the position of a function; in f x) f
bx +
c
d
• a
affects
the
shrink/stretch on the
y-axis for
f
x)
and vertical reflection
M TH LEVEL 2
P GE
•
b
affects
the shrink/stretch on the x -axis
for f x) and horizontal
reflection
• c translates the function along the x-axis
to
the right or left £units)
b
• d
translates
the
function along the
y-axis
Students who are successful on the Subject Tests understand how
to
use function notation by
substituting both numbers and expressions for the independent variable x) in a function like
f x).
©
2008 Karen Dillard. All rights reserved.
-AS
No part
of
this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
14/28
M TH
LEVEL 2
14
GE
DOMAIN AND RANGE
o
figure out
the limits
of
a function s domain, use the following laws:
• A
fraction
with
a
denominator
of
zero is undefined.
Any
values
that
would
make
the
denominator
of
a fraction equal to zero
must
be excluded from the
domain
of that
function.
• Square roots of
negative numbers result
in imaginary numbers. Thus any values
that
would
make a
number
under the square root sign negative must be excluded
from the domain of that function.
o figure out the
limits
of
a function s range you need to be familiar with the following rules:
• An even exponent produces only non-negative numbers. Any
term
raised to an even
exponent must be positive or zero.
• Like even powers a square
root cannot
result
in
a negative number.
• Absolute values
produce
only non-negative
numbers.
ASYMPTOTES OF RATIONAL POLYNOMIALS)
Asymptotes are lines that the graph of a function gets closer
to
as
the variable
or
the
function
gets indefinitely large
or small.
Vertical asymptotes act much like points
of
discontinuity since the graph will never cross that
particular line. Vertical asymptotes are found in functions where
certain
values
of
x make the
denominator
of
the
function
equal
to
zero.
Horizontal asymptotes are found in functions that have
polynomials
in both the numerator and
the denominator with the same
degree.
y 0 is a horizontal asymptote if the exponent in the
denominator
is
greater than the exponent in the
numerator.
Specifically let
f
x) A@-a x )
n b x)
f m > n there are no horizontal asymptotes.
A
f m n the horizontal asymptote is y
8
.
f m < n the
horizontal
asymptote is y = 0.
Oblique asymptotes are not covered on SAT Subject Tests.
© 2008 Karen Dillard. All rights reserved
-A
5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
15/28
4.
If 4 x
+
3 = 0, what is x x - 2)(x -
3 ?
(A) - 270
®
-
90
(C) 0
(D)
30
E) 90
4lt
• - \
~ - J
- \
l -?
-7 - \-3)
- 3
- ~ > < - - b
1 t
-
b
~ .
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~
5. Which of he following is
NOT
equivalent to
3x - 2
= 95
x ?
6.
(A) x - 2 = 2(5 -
x)
~ x
= 3
I
(C) - x - 2) = 5- x
2
(D) log Y -
2
)
= log (9
5
-
x)
d - e 5 7d - 7e
f = - then
x + y 8
5x
+ 5y
7
(A) 40
1r-
2
4
B)
-
3
9
7
-
8
D)
7
-
5
(E)
13
5
7
-
© 2008 Karen Dillard All rights reserved. - A.5
15
M TH LEVEL
2
PAGE
7.
If
x
1
- y = 96 and
y - x = 8, what is y
+
x ?
fA) - 12
(If) - 8
(C)
- 2
(D) 2
(E)
12
_, 6 - )I
. ~ - \
8. What are the zeros of x
4
- 16 ?
(A) x = - 2
~
x = {-
2,2}
I
y = {- 2, 2}
(E)
y = 2
9. If h
x)
=I
x -
2 1 which of the following must be true?
h (0) = h (2)
)B)
h (2) =h (- 2)
h O) = O
(p ) h (4) = h (- 2)
@h 4 = h O
No part of this material may be copied or used without written permissi
on
from Karen Dtllard s College Prep
-
8/21/2019 Math Level 2
16/28
MATH LEVEL
2
16
GE
x - 3
10
If
f x)
= there is a vertical asymptote at x
=?
2x+5
-
2.5
1.2
- 0.6
0.4
3.2
II
Given h, 4) on the graph
of
f x)
=
log
2
(x
2
+
6x ,
which
of
the following is a solution for
h?
A) 36
B) 3.8
C) 1.6
D)
3
§)
- 8
~
12
If
h g x - 3x - I and g (x) = 2x, h (x) could equal
which
ofthe
following?
X
. )-
2
fRr3x + I
D) 6x
E) I
13
The inverse
of
a function f is defined by f
1
x).
What is F
1
(x)
if
f x)
=
4x
-
9?
@ x+9
x 9
B)
4
9
C)
x
4
4
D)
9
E)
4x - 9
- -
-
8/21/2019 Math Level 2
17/28
17
M TH LEVEL
2
PAGE
GEOMETRY
About
10
questions
on the
test
will be geometry.
The
College Board arranges
the geometry
by
coordinate
and
3-D there
is no
plane
geometry
on
the
Level
2).
There
are 5
geometry formulas
given to
you
on the test, all
are for
three-dimensional objects.
COORDINATE GEOMETRY
For problems in
which no diagram
is provided,
you may want to
sketch one.
MIDPOINT COLLINEAR PROPERTIES
The coordinates of the midpoint of
two
points can be found using the following formula :
Midpoint Formula
This is
the
average
of the
x-coordinates and
the
y-coordinates, which gives
the
numbers
halfway
between
the
x-coordinates and halfway between
they-coordinates
.
Di
stance
For
mula
EQUA ONS OF LINES
Slope-Intercept Form
where m is
the
slope
and
b is the
y-intercept
y mx
b
Standard
Form
where
A, B, and C are integers, and A is positive
Ax C
Po int-Slope Form
where m is
the
slope and
x
1
,
y
1
)
is a point on the
line
Y
1
= m
x
- x
1
)
2008 Karen Dillard. All rights reserved. - A.5
No part of th is material may be copied
or
used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
18/28
M TH
LEVEL 2
1
8
GE
CONICS
The most common conics on the Level are circles and parabolas; Level 2 may include ellipses
and hyperbolas.
ircle
x h)2 +
y
k)
2
= r2
where r is the radius of the circle, and h, k) is the center.
Parabola
Ellipse
Vertex
Form
y =
a x h)2 +
k
or
x =
a y
- k 2 + h
where
h, k) is the
vertex
Standard Form
y
=
ax
2
+
bx
+ c
a, b, and c are constants
x-h)2
+
y
- k)2 = 1
a2 b2
a and
bare
constants and h, k) is the center
Hyperbola
x-h)2- y-k)2 =1
a2
b2
a and b are constants and h,
k)
is
the
center
THREE DIMENSIONAL GEOMETRY
The test provides you
with
the following formulas. In addition
to
solids, surface area and volume
cylinders, cones, pyramids, spheres, and prisms) tested on
the
Level 1, the Level 2 could include
coordinates i ree dimensions.
Volume
of a right circular cone
with
radius r and
height
h
2
n
r
3
Lateral Area of a right circular cone
with
circumference of the base
l
and slant height
1
5 = c
£
2
© 2008 Karen D1llard. All rights reserved -A 5
No part
of th1s
material may be copied or used Without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
19/28
Volume
of
a sphere with radius r
V = i
3
3
Surface Area
of
a sphere
with
radius
r
S = 4
7t
r
2
Volume
of
a pyramid with base area B and height h
V Bh
3
In addition
you
may
also need
to
use
the
following :
Volume of a Prism h
9
M TH
LEV
EL
2
P GE
B is the area of the base, and h is the
height
of the prism . Note
that
this is not different from
a rectangular solid
whose
area is length
times width times
height
the
area
of the
base is a
rectangle
with length
and
width.
s o n
of
Pythagorean Theorem in 3
Dimensions
:
b
+ c2 ;}..
and
c are
sides
of
a
rectangular
solid, and d is
the
diagonal.
Volume
of
a Cylinder
= 1tr
h
r is the rad ius and h is the height
©
2008 Karen Dillard All rights reserved A.5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
20/28
M TH LEVEL
2
2 0
GE
TRIGONOM TRY
Be prepared
to
use sines, cosines, and
tangents on
more than one problem. Occasionally,
secant, cosecant, and cotangent are also used. For
many
trig
questions
you
will only
need to
know which function
to
use, not necessarily how
to
use it. Many people use
the mnemonic
SOH -
CAH
- TOA
to
remember the three main functions.
SOH
- CAH - TOA stands for
the following
parts of a right triangle.
S =sine
0 =
opposite
side
H = hypotenuse
C =cosine
A =
adjacent
side
H = hypotenuse
T =tangent
= opposite side
A = adjacent side
For
angle e
n
the right
triangle below,
the following
relationships can be observed.
Q
Vl
a.
a.
adjacent
sin = opposite
hypotenuse
e
hypotenuse
esc
=
opposite
e
adjacent
cos = . =
hypotenuse
e
hypotenuse
sec =
~
adjacent
tan = sin e = opposite
cose
adjacent
cot
8
= cos
e
= adjacent
sin e opposite
u will be given 2 of the 3 elements of one of the equations above, and you will be asked
to
lve for the third element. Be sure
to
use
the
correct equation.
To
solve for
the
angle, use
the
propriate inverse function: cos-
1
, sin-
1
, or tan-
1
•
© 2008 Karen D llard All nghts reserved -A 5
No part of this material may be copied or used Without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
21/28
2
MATH LEVEL
2
PAGE
Note that these are
not
the same as secant, cosecant and cotangent. The differences are
outlined below.
INVERSE FUNCTIONS
Remember
that
arc
is
the same
as inverse
arc
sin = sin ·
1
[sin·l (sin 8)] =
[cos·l
(cos
8)] =
[tan·l (tan 8)]
=
MULTIPLICATIVE INVERSES
(sin
8)(csc 8)
= 1
(cos
8) sec
8) = 1
(tan
8)(cot 8)
= 1
When
solving
trig
functions with
a
calculator
make
sure the calculator
is
in radians
or
degrees
depending
on
what
you
are
trying to
find. Also,
note
that
the
calculator only
gives
solutions in
the
principle ranges as follows: '
7r 7r
sin·
1
and tan·
1
--+
- -
to
-
2 2
cos·
1
--+ 0 to r
f your range is outside these principle ranges, you will need
to
adjust the
answer
given by the
calculator.
One
way
to
remember this is by using
the
mnemonic All Sad Tigers Cry, in which
the
first
quadrant shows
ALL
of the functions
as positive,
the
second quadrant
shows
only Sin
(Sad)
as
positive,
the third
quadrant shows
only
Tan (Tigers) as positive, and
the
fourth
quadrant
shows
only
Cosine
(Cry)
as positive.
y
Quadrant
I
Quadrant Quadrant Quadrant
V
o o
rr
7r 3 7r
3
7r
-
to
rr rr
to
- -
to
2
rr
2 2 4 4
Sine
+ +
- -
Cosine
+
-
+
\
Tangent
+
+
-
© 2008 Karen Dillard All rights reserved. - A.S
No part of this material may be copied or used without written permission from Karen Dillard 's College Prep
-
8/21/2019 Math Level 2
22/28
M TH LEVEL 2
2
2
GE
Another helpful
visual tool is
the unit
circle.
The
circle uses
the trig
identity cos2 + sin 2
= 1. I t
s
analogous
to
x
2
+ y
2
=
1, where the cos has its values on t
he x-axis
and sin has its values on
they
-axi
s.
/
Quadra nt
I I
(cos, s
in)
( I +
(cos, s
in)
( - , -
Quad
rant
I I I
Quad rant
I
(co s, s
in
)
{+, + )
(cos, sin)
+ , -
Quadra
nt
V
Common Identit ies
You might want
to
keep in mind any
algebraic
variations of the
following
as well as the
additional
trig identities for
some
of the
more difficult
problems
on
the test.
sin
2
x ) + cos
2
x ) = 1
tan
2
x )
+
1
=
sec2
x )
c
ot
2
(x ) +
1
=
csc
2
x)
The
followi
ng are only
found
on the SAT Level
2 tests,
and are
always given
in th e problem-so
make s
ure you
know how to use
them:
sin x + y) = sin x cosy cos x sin y
cos x + y) = cos x cos y - sin x sin y
GRAPHS OF TRIG FUNCTIONS
Standard form for a trig function is
y
=
a sin (bx
+
c)
+
d
sin x - y) = sin x cos y cos x sin y
cos
x
- y)
=
cos x cos y + sin x sin y
a is
the
amplitude that wi ll determine
the maximum and minimum
values
b affects the period
and the
width of the curve
c is the phase
shift which
t ranslates
the curve
left and right
d is t he
vertical
shift
which
t rans l
ates
t he
curve
up and down
©
2008 Karen D llard All nghts reserved
-A
5
No part of th s material may be copied or used Without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
23/28
3
M TH LE
VEL 2
P GE
The period
of
a
trig
function is
the amount of
space on
the x-axis that
is required
to
go
throug
h
one complete cycle . Sin and
Cos
have a
standard
period
of
21i
and
Tan has a standard period
of
1i.
To
find the period
of
a function
divide
the standard period
of the
function by the coeffeicient
of x
For triangles where
no right
angle has been defined,
the
laws
of
si
ne and
cosine are required.
LAW OF SINES
sinA sinB sine
=
a
b
c
POLAR COORDINATES
/
LAW OF COSINE
For a given r e e t coordinate x , y) a polar coordinate r,
8
can be found using
the
followin ·
/
__
,.
© 2008 Karen-Dillard....A11 rights reserved. - A.5
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
24/28
PAGLE 4
- - - - - - - - - - - - - - - - - - - - - - ~ ~ /
15
Where
will
the
cen t
er of
cir
cle x
+
3)
2
+
y- 2f =
9
be if it is moved up 2 units and to the right I unit?
A) -3, 2)
B)
-3,
4)
C) -
5,
3)
@ (- 2, 4)
E) - 2, I)
16 If y
=
5 - 3 and x
=3t
+ I, what is the y-intercept
of
the line defined by x and
y?
@-
;
D: J.., _.,
1 '- •
- \
7
· = - ~
3
.J
~
4
3
1;..
s
{ - ~
J
4
?
(9f
- j:. -
3
-
- . .
J
~
19
-s-
•C
5
. . , ~
-
. . . /
- \ 1
~
17 Wh
i
ch
of
the
following is
the
equation for
the
line
containing point - 3,
2)
and having an x-intercept
greater than 2 ?
Jrlf)y
=
2x
+
8
~ = x 5
(12 r y = 3x + I I
@ y
= -
2f
y=
x
I
?
18
Which
of
the following is equal to sin
y?
(tR (tan
x
. :8)
cosy
~
cosx
cot y
~
f
19.
If
the
graph above is shifted
2
to
the
left, which
of
the
following would be the resulting equation?
(B ) cos 2 x
+
r
)
, .D)
~
c o s 2 x + ~
2
c o s 2 x - ~
2
© 2008 Karen D1llard All rights reserved -A 5
No part
of thiS material may be copied
or
used Without written permiss1on from Karen
D1llard
s College Prep
-
8/21/2019 Math Level 2
25/28
20. Which of the following is N Ta polar coordinate of
3, 3J3) ?
.{1 5 (-6 4
(-
6-
;
)
-5
6 2
- I
e
. . . ~ . . :1
Q + ~ ~ - l h
1
~ ~ . ; . . . fi
©
2008 Karen Dillard All rights reserved. - A.5
5
MATH
LEVEL
2
P GE
21. What is the length
of AC
on
M BC if LB
=
45
°,
L
=25°, and side
BC
= 11.4 ?
A)
6.71
7.43
fO 8.58
D) 9.14
E)
10 .6
:.
g r { \1 1 )
•, f I< )
t? ) 9
Remember to check
the
mode on
your
calculator for degrees or rad ians.
No part of this material may be copied or used without written permission from Karen Dillard s College Prep
-
8/21/2019 Math Level 2
26/28
MATH LEVEL 2
2
6
GE
DATA ANALYSIS PROBABILITY
STATISTICS
Often probability
questions
are asked where
you
need
to count the number of
possible events
to
determine the denominator for your answer. Counting techniques are usually helpful.
In
addition
to what you learned
for the
SAT, it may be helpful to know
the
following for the Level 2 test.
Conditional dependent): This
probability
depends upon an outcome of a separate event
that
occurred earl ier so
that the
probability
of the
second
event
changes, depending on
the outcome
of the first.
For example, let s say we are pulling
different
colored socks
from
a bag. Once
the first
sock is pulled, the probability
of the next
sock being pulled is changed, because the
total number
of
socks in
the
bag and
the
colors available have also changed.
Multiple events
independent): In some
problems, the probability
of
each
event
is
independent
of the
previous event. Examples include flipping a coin and rolling a die. The
probability
of
getting
a head
on the
second
flip of
a coin is still 0 .5,
no
matter what the
results
of
the first
flip.
COMBINATIONS PERMUTATIONS
When deciding
whether
a problem involves computing a
combination
or a permutation, it is
important to determine whether or
not
putting
the
items in a different order will change a
grouping.
In combinations, order does NOT matter. There will be fewer groups to count because group AB
is considered
the same as group BA The
formula
that
is in
your graphing calculator under math
functions for combinations is:
C
nl
n r rl n - r ~
In
permutations, order is important. There are more permutations than combinations because
the number 23
is
different than the number
32.
The
formula
for permutations
in
your graphing
calculator is:
P
= _
n r (n-r)
© 2008 Karen Dtllard All nghts reserved - A 5
No part of thts material may be copied or used wtthout written perm ss
on
from Karen Dtllard s College Prep
-
8/21/2019 Math Level 2
27/28
22. An urn contains
18
white marbles and 5 black marble
s.
•
If
you select 4 white marbles without replacement, what
is
the probability
of
selecting 2 black marbles on your
next two selections?
10
171
tl
I
-
(B)
9
\\..1-l--
r : 1
35
(C)
10
p
-
91
-
-
-
5
•'1
1 0
(D)
18
\ 1
'
(E)
10
-
253
•
• • • •
c
D
E
F
23. How
many line segments are there with endpoints that
are two
of
the six points?
(A) 8
-
DF..
~ 1 2
~ l l
~ .
(.J>
-
5
A'--
c€..
t>F
~ j )
-
D) 20
-
-
~ t ; c
u
(E) 24 AD
-
---..
B ~
~
-
F
r ~ ~ ~ 1 ~ t •
VI
l..+'
rO-t-5 : :
to
{ or
1...
©
2008 Karen Dillard. All rights reserved . - A.5
7
M TH
LEVEL 2
P GE
24.
How
many 7-digit telephone numbers can
be found
if
the first digit cannot
be
zero?
(A) 15 ,000
(B) 70,000
(C) 200,000
(D) 4,000,000
®
9,000,000
)( 10
(0
25. An urn contains
I I
red marbles and 7 blue marbles.
Two red marbles are selected without replacement.
What is the probability that a t least 3 of the next 4
marbles selected are blue?
3
(A)
-
26
4
(B)
-
26
9;6
(D)
9
26
(E)
2
< - (
+
B .Z..B
r-
~
tLB13
fJ-{176
t ~
,
·-
IJ
No part of this material may be copied or used without written permission from Karen Dillard's College Prep
-
8/21/2019 Math Level 2
28/28
MATH LEVEL
2
2 8
GE DDENDUM
1. If 1x - 4 1= 9, what could be the solution for Ix + 2 1? 5. Which
of
the following has the same symmetrical area
~
3
I.-, \ s
0
->
4
(C)
8
\ : - ~ - t t -
-
(D)
II
(E)
16
\11
+
.
\
-
-
2.
If x = ~ , what
is (4 -
. ;
r
A)
0.52
B)
0.77
3.18
6.
69
7.43
I
3. If f x )
=
x
7
-
3x + 6, what is the minimum value
of
f
within - I
::0 x :::;
I ?
(A) 2.
98
(B) 3.23
fa 3.77
{6)
4.6 2
E) 6.10
7 ~ b - :s
7 v f
, . c . ~
7
) , . 8 ,
6 1-
0
4.
What
is
the
length
of the
longest
side of
a
tr
iangle th
at
ha
s points at 0, 0), (7,
3)
, and
f 3 :: s
?
A) 9.76
CD
B)
10.85 \ p\
(5=i
©
11.18
D)
12.2 1
{o ,o
(E)
13.0 ) r -
- 1,
11
about f x)
and
1
x)?
(A) or i
gin
(B) x-ax
is
J
y-axis
(lj)
y = x
(E) It
is not symmetrical
6.
If
the two
triangles
above
are
similar
,
what
is sin
B
5
~ 3
12
r;I .3
(B)
13
10
(C)
29
(D)
13
5
(E)
24
-
5
7.
Wh
i
ch of th
e
o w i n
a v e
the
s
ame
period
as y 2 s i n ( 3 x ~
1 )):, = 2 s in 3x + 8) + 2
('13} y = 2 sin 3x + 6) + 4
t }
y
= 2 sin (3x)
y
= 2 sin 4x + 6) + 2
~ y
= sin
3x
+
6)
+ 2
1n
3
©
2008 Karen Dillard. All rights reserved - A.5
J