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

"

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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


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


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


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


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


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


8/28

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



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


or



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



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


or

used without written permission from Karen Dtllard s College Prep


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


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


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



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


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

- -


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



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


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



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


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


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


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



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


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.



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


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


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

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Math Level 2

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