SATand ACTMath Guide
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SAT and ACT Math Guide
With Cross References to Official SAT and ACT Practice Tests
Matthew E. Boutte
April 2017
SAT and ACT Math Guide page 1
Contents
1 Start Here! 3
2 Basic Algebra 5
2.1 Basic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Solving Basic Algebraic Expressions . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Combining Like Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.4 Fraction Busting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.5 Cross Multiplying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Mid Point and Distance Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Unit Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Exponent and Radical Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.8 Exponential Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.9 Radical Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.10 “In Terms Of” Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.11 Comparison Problems or Effect On Problems . . . . . . . . . . . . . . . . . . . 18
2.12 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.12.1 Solving Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.12.2 Graphing Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.13 Polynomial Equivalence Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.14 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.14.1 Solving Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.14.2 Creating Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Percents 22
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Increasing Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Decreasing Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 “Percent Of” Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 Percent Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Turning other things into percents . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Linear Functions 25
4.1 Finding Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Interpreting Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Graphing Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4 Parallel and Perpendicular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Systems of Equations 28
5.1 Definition and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Methods Solving Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . 28
5.2.1 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2.2 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Types of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3.1 One Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3.2 Zero Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3.3 Infinitely many solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
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5.4 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 Non-Linear Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.6 Turning Word Problems into Systems of Equations . . . . . . . . . . . . . . . 32
6 Factoring and Related Topics 33
6.1 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.3 Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.4 Undefined Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Quadratics 39
7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.3 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.4 Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.5 Vertex Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7.6 Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.7 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.8 Axis of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.8.1 Finding the Axis of Symmetry in Standard Form . . . . . . . . . . . . 43
7.8.2 Finding the Axis of Symmetry in Intercept Form . . . . . . . . . . . . 43
8 Complex Numbers 44
8.1 Definition and Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
8.2 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
8.3 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.4 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9 Geometry 46
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.2 Triangle Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
9.3 Complimentary and Supplementary Angles . . . . . . . . . . . . . . . . . . . . 47
9.4 Vertical Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
9.5 Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.6 Similar Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
9.7 Special Types of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
9.8 Degrees in a Polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
10 Trigonometry 52
10.1 Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
10.2 Complimentary Angle Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
10.3 Law of Sines and Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11 Circles 54
11.1 Basic Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.2 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
11.3 Sector Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
11.4 Central Angles and Inscribed Angles . . . . . . . . . . . . . . . . . . . . . . . . 55
11.5 Central Angle Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
11.6 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
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SAT and ACT Math Guide page 3
12 Statistics and Probability 57
12.1 Counting Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
12.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
12.3 Measures of Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
12.4 Measures of Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
12.5 Regression and Scatterplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
12.6 Basic Statistical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
12.8 Table Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1 Start Here!
How to use this guide
This guide to SAT and ACT math covers all math content tested on the SAT and the ACT.
In addition, each section cross references with problems on official SAT and ACT practice
tests, which are available for free. This provides significant benefit because it means you
will receive instruction on all content areas and only be doing official practice problems and
don’t need to go out and buy any resources. It also means you can do numerous practice
problems from a specific content area and master it before moving on to different kinds of
problems.
The guide is set up somewhat sequentially. However, you will likely have a firm grasp of
some of the content. Therefore, each section is designed to stand alone as much as possible.
If you know specific content areas that you struggle with, you should begin with the content
and practice problems in those sections.
Finally, please feel free to pass this guide on to anyone you know who may need some
guidance studying for the math sections of the SAT and the ACT!
Where to get the tests
The most valuable part of this guide is the references to official SAT and ACT practice tests
for each content area. You should get these practice tests and use them with each section.
The tests are available for free at the links below.
For SAT or ACT tutoring, contact Matt Boutte.
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SAT and ACT Math Guide page 4
SAT 1 www.mattboutte.com/SAT1
www.mattboutte.com/SAT1Answers
www.mattboutte.com/SAT1Explanations
SAT 2 www.mattboutte.com/SAT2
www.mattboutte.com/SAT2Answers
www.mattboutte.com/SAT2Explanations
SAT 3 www.mattboutte.com/SAT3
www.mattboutte.com/SAT3Answers
www.mattboutte.com/SAT3Explanations
SAT 4 www.mattboutte.com/SAT4
www.mattboutte.com/SAT4Answers
www.mattboutte.com/SAT4Explanations
SAT 5 www.mattboutte.com/SAT5
www.mattboutte.com/SAT5Answers
www.mattboutte.com/SAT5Explanations
SAT 6 www.mattboutte.com/SAT6
www.mattboutte.com/SAT6Answers
www.mattboutte.com/SAT6Explanations
SAT 7 www.mattboutte.com/SAT7
www.mattboutte.com/SAT7Answers
www.mattboutte.com/SAT7Explanations
SAT 8 www.mattboutte.com/SAT8
www.mattboutte.com/SAT8Answers
www.mattboutte.com/SAT8Explanations
ACT 2005-06 www.mattboutte.com/ACT2005
ACT 2008-09 www.mattboutte.com/ACT2008
ACT 2011-12 www.mattboutte.com/ACT2011
ACT 2014-15 www.mattboutte.com/ACT2014
ACT 2016-17 www.mattboutte.com/ACT2016
(The ACT frequently puts out the same practice test several years in a row, which is why
some years are skipped.)
How to get a print copy of this guide
If you would prefer to have a print copy of this guide (or to save your printer some ink!),
you can buy a copy of this guide at the link below.
www.mattboutte.com/buy
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SAT and ACT Math Guide page 5
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2 Basic Algebra
2.1 Basic Tools
2.1.1 Keywords
There are lots of keywords that can help with interpreting word problems. Most of these
are pretty intuitive. However, two are worth noting because they occur so frequently and
regularly throw students off.
of – always means multiply
per – always means divide
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SAT and ACT Math Guide page 6
2.1.2 Solving Basic Algebraic Expressions
Solving basic algebraic expressions requires working towards getting a variable alone. To do
this, inverse operations must be undone in reverse order of operations (PEMDAS).
3x2+4=79
Ordinarily, the order of operations would tell us to do the exponent, then the multiplication,
and finally the addition. But because we are solving for x, we do these in reverse order.
3x2=75
x2=25
x=±5
A word about fractions. If you are undoing a fraction that is being multiplied by a variable,
the inverse operation would be to divide by the fraction. This will work, but it is easier to
simply multiply by the reciprocal of the fraction.
By dividing: 3
4x=15
3
4x
3
4=15
3
4
By multiplying by the reciprocal: 3
4x=15
4
3(3
4x)=(15)4
3
2.1.3 Combining Like Terms
Combining like terms is a basic mathematical operation. However, it can be visual confusing.
It is therefore important to create visual cues for yourself, such as crossing out terms once
you have used them.
To combine line terms, simply identify terms that have the same combination of variables
and exponents and then combine their coefficients.
5x2y+12xy2−6xy −7x2y+xy2+5xy
−2x2y+13xy2−xy
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SAT and ACT Math Guide page 7
2.1.4 Fraction Busting
There are numerous types of problems that involve fractions that can be made significantly
easier by eliminating the fractions. This can be done by multiplying all terms by the
common denominator (also known as “fraction busting”).
For example, if we have the following equation
x
2+x+3
3=5
x
we can eliminate all the fractions by multiplying by a term to cancel with all the denomi-
nators. 6 will cancel with the 2 and the 3, and we will need an xas well to cancel with the
xin the denominator.
6xx
2+x+3
3=5
x
6xx
2+6xx+3
3=6x5
x
3x(x)+2x(x+3)=6(5)
3x2+2x2+6x=30
This equation will be significantly easier to solve than the one involving fractions.
As mentioned, fraction busting can be used in several different contexts. For example, it
can be used to make solving systems of equations easier.
x
3=2y−1
2x−y=10
This system would be a lot easier to solve if the first question didn’t involve fractions. We
can make things a lot easier if we multiply it by 3.
3x
3=2y−1
3x
3=3(2y)+3(−1)
x=6y−3
This will make solving the system of equations a lot easier.
2.1.5 Cross Multiplying
Cross multiplying is a powerful tool for eliminating fractions whenever you have a single
fraction equal to another single fraction. (Keep in mind that any whole number is a fraction.)
In fact, cross multiplying should be used whenever you have a single fraction equal to another
single fraction.
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Cross multiplying is done by simply multiplying the numerator (top) of one fraction by
the denominator (bottom) of the other fraction and setting it equal to the denominator
(bottom) of the first fraction times the numerator (top) of the second fraction.
x
4=x+2
6
6x=4(x+2)
6x=4x+8
2x=8
x=4
An important tip to keep in mind is that all numbers are fractions, even if they don’t look
like one. A number (such as 5) or a variable (such as x) or an expression (such as 5y−3) can
be written as a fraction by placing them over one. This can easily turn a fraction problem
into a cross multiplying problem, and thus eliminate the fractions.
7=21
x−4
7
1=21
x−4
7(x−4)=21
7x−28 =21
7x=49
x=7
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SAT and ACT Math Guide page 9
Test Number Test Number Test Number
SAT 1 Sec. 3 1 SAT 1 Sec. 3 5 SAT 1 Sec. 3 8
SAT 1 Sec. 4 2 SAT 1 Sec. 4 10 SAT 1 Sec. 4 35
SAT 2 Sec. 3 1 SAT 2 Sec. 3 8 SAT 2 Sec. 4 3
SAT 2 Sec. 4 3 SAT 2 Sec. 4 6 SAT 2 Sec. 4 32
SAT 2 Sec. 4 33 SAT 2 Sec. 4 37 SAT 2 Sec. 4 38
SAT 3 Sec. 3 1 SAT 3 Sec. 3 2 SAT 3 Sec. 3 5
SAT 3 Sec. 3 17 SAT 3 Sec. 4 6 SAT 3 Sec. 4 7
SAT 3 Sec. 4 10 SAT 3 Sec. 4 25 SAT 3 Sec. 4 33
SAT 3 Sec. 4 37 SAT 4 Sec. 3 5 SAT 4 Sec. 3 6
SAT 4 Sec. 3 10 SAT 4 Sec. 4 5 SAT 4 Sec. 4 31
SAT 5 Sec. 3 6 SAT 5 Sec. 3 17 SAT 5 Sec. 3 19
SAT 5 Sec. 4 4 SAT 5 Sec. 4 5 SAT 5 Sec. 4 8
SAT 5 Sec. 4 33 SAT 6 Sec. 3 15 SAT 6 Sec. 3 17
SAT 6 Sec. 4 1 SAT 6 Sec. 4 9 SAT 6 Sec. 4 32
SAT 6 Sec. 4 33 SAT 7 Sec. 3 2 SAT 7 Sec. 3 16
SAT 7 Sec. 3 20 SAT 7 Sec. 4 2 SAT 7 Sec. 4 23
SAT 7 Sec. 4 23 SAT 8 Sec. 3 1 SAT 8 Sec. 3 12
SAT 8 Sec. 4 1 SAT 8 Sec. 4 9 SAT 8 Sec. 4 32
SAT 8 Sec. 4 38 ACT 2005-06 4 ACT 2005-06 11
ACT 2005-06 15 ACT 2005-06 18 ACT 2005-06 40
ACT 2005-06 41 ACT 2005-06 42 ACT 2005-06 43
ACT 2005-06 44 ACT 2005-06 48 ACT 2008-09 2
ACT 2008-09 6 ACT 2008-09 11 ACT 2008-09 13
ACT 2008-09 21 ACT 2008-09 29 ACT 2008-09 48
ACT 2008-09 52 ACT 2011-12 4 ACT 2011-12 6
ACT 2011-12 8 ACT 2011-12 11 ACT 2011-12 26
ACT 2011-12 31 ACT 2014-15 2 ACT 2014-15 5
ACT 2014-15 7 ACT 2014-15 9 ACT 2014-15 19
ACT 2014-15 21 ACT 2014-15 25 ACT 2016-17 23
ACT 2016-17 48 SAT 5 Sec. 4 21 ACT 2005-06 1
ACT 2005-06 7 ACT 2005-06 39 ACT 2008-09 5
ACT 2008-09 20 ACT 2008-09 34 ACT 2011-12 56
ACT 2014-15 16 ACT 2014-15 53 ACT 2016-17 18
ACT 2016-17 19 ACT 2016-17 45 ACT 2016-17 56
2.2 Mid Point and Distance Formulas
Mid Point Formula The mid-point is simply the middle point between two other points.
Thus, it is found simply by averaging the x values of the two points and the y values of the
two points.
x1+x2
2,y1+y2
2
Distance Formula The distance formula is simply the Pythagorean Theorem. If you
can’t remember it, just sketch the two point, form a triangle, and then perform the Pythagorean
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SAT and ACT Math Guide page 10
Theorem.
d=(x2−x1)2+(y2−y1)2
Test Number Test Number Test Number
ACT 2005-06 17 ACT 2008-09 37 ACT 20011-12 9
ACT 2011-12 42 ACT 2011-12 48 ACT 2014-15 39
ACT 2016-17 32
2.3 Proportions
Proportions are two fractions that equal one another. The first fraction represents one
scenario and the second fraction represents a second scenario. The numerators (tops) of each
fraction measure one quantity while the denominators (bottoms of each fraction measure
another quantity.
After setting up the proportions and setting them equal to each other, the unknown value
can easily be solved for by cross multiplying.
If 24 magazines stack to be 5 inches tall, how tall will a stack of 40 magazines be?
5inches
24 magazines =x inches
40 magazines
24x=200
x=81
3inches
Test Number Test Number Test Number
SAT 1 Sec. 4 6 SAT 2 Sec. 4 2 SAT 3 Sec. 4 9
SAT 3 Sec. 4 11 SAT 3 Sec. 4 19 SAT 4 Sec. 3 20
SAT 5 Sec. 3 3 SAT 5 Sec. 4 31 SAT 5 Sec. 4 32
SAT 7 Sec. 4 15 SAT 7 Sec. 4 25 SAT 8 Sec. 4 31
ACT 2005-06 3 ACT 2005-06 32 ACT 2005-06 51
ACT 2011-12 18 ACT 2016-17 3 ACT 2016-17 33
ACT 2016-17 51 ACT 2016-17 54 SAT 6 Sec. 4 26
2.4 Unit Conversion
Unit conversion is the tool for changing from one unit to another. This is done by first
setting up a fraction that will cause the undesired units to cancel out and leave the desired
units. After this is set up, the numbers necessary to make the conversation are inserted into
the fraction.
If we desire to convert 3 years into days, the fraction would be set up with units as follows.
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SAT and ACT Math Guide page 11
3years ⋅days
years
Then we put the approrpiate numbers in the converter fraction.
3years ⋅365 days
1years
Finally, we cancel the units and perform the necessary arithmetic.
3
years ⋅365 days
1
years =3⋅365 days =1095 days
This process can be repeated as many times as necessary to convert from years to seconds
or from miles per hour to feet per second, for example.
Some unit related formulas are important to know.
velocity =change in distance
change in time
acceleration =change in velocity
change in time
density =mass
volume
Test Number Test Number Test Number
SAT 1 Sec. 4 6 SAT 1 Sec. 4 34 SAT 2 Sec. 4 11
SAT 2 Sec. 4 15 SAT 2 Sec. 4 31 SAT 3 Sec. 4 14
SAT 4 Sec. 4 3 SAT 4 Sec. 4 18 SAT 4 Sec. 4 33
SAT 5 Sec. 4 9 SAT 7 Sec. 4 3 SAT 7 Sec. 4 31
SAT 8 Sec. 4 11 SAT 8 Sec. 4 15 SAT 8 Sec. 4 23
SAT 8 Sec. 4 34 SAT 8 Sec. 4 37 ACT 2011-12 40
SAT 5 Sec. 4 37 SAT 6 Sec. 4 29 ACT 2008-09 19
ACT 2016-17 16
2.5 Functions
Functions are simply another way of representing equations such as
y=x2−7x+12
In function notation, this would be
f(x)=x2−7x+12
As such, functions themselves are easy to work with. But people get confused by the
notation. You just need to keep two things straight in your mind.
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SAT and ACT Math Guide page 12
•f(3)means plug 3 in for xbecause the xin f(x)has been replaced by a 3
•f(x)=7 means set the whole equation equal to 7 because f(x)is the “name” of the
function
There is one more piece of information you need to keep in mind with functions. You can
plug anything into a function. For example, using the function f(x)above, we can plug a
heart into it.
f(x)=x2−7x+12
f(♡)=♡2−7♡+12
This may look odd, but there is nothing mathematically wrong with it. The key idea is that
wherever there was an x, you replace it with a ♡.
This is important when dealing with composite functions. A composite function is when one
function is plugged into another function. This is allowed because you can plug anything
into a function. The process is the exact same as plugging the heart in – anywhere there
was an x, you replace it. If we are given the functions
f(x)=−3x+5
g(x)=x2+1
and are asked to find f(g(x)), this simply means that we plug g(x)into f(x). That is, we
replace every xin f(x)with g(x).
f(g(x))=−3(g(x))+5
=−3(x2+1)+5
=−3x2−3+5
=−3x2+2
When graphing functions, if one function is less than the other, this means that its graph
is below the other function’s graph.
Test Number Test Number Test Number
SAT 1 Sec. 3 10 SAT 1 Sec. 4 17 SAT 2 Sec. 4 10
SAT 2 Sec. 4 26 SAT 3 Sec. 4 16 SAT 4 Sec. 3 4
SAT 5 Sec. 4 34 SAT 6 Sec. 3 8 SAT 7 Sec. 3 5
SAT 8 Sec. 3 15 SAT 8 Sec. 4 30 ACT 2005-06 19
ACT 2008-09 33 ACT 2008-09 53 ACT 2008-09 57
ACT 2008-09 59 ACT 2011-12 32 ACT 2011-12 50
ACT 2014-15 14 ACT 2014-15 34 ACT 2016-17 5
ACT 2016-17 42
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SAT and ACT Math Guide page 13
2.6 Complex Fractions
A complex fraction is a fraction made up of two fractions. That is, there is a fraction in the
numerator (top) and a fraction in the denominator (bottom).
5
x
4x
15
There are two steps to handling a complex fraction. Most students remember the second,
but not the first.
•Obtain a single fraction in the numerator (top) and a single fraction in the denominator
(bottom).
•Invert (flip) the fraction in the denominator (bottom) and multiply it by the fraction
in the numerator (top).
2
x+1
3
1
2−x
3
(3
3)2
x+1
3(x
x)
(3
3)1
2−x
3(2
2)
6
3x+x
3x
3
6+2x
6
6+x
3x
3+2x
6
6+x
3x⋅6
3+2x
6(x+6)
3x(2x+3)
2(x+6)
x(2x+3)
Test Number Test Number Test Number
SAT 1 Sec. 3 13
2.7 Exponent and Radical Rules
There are a number of exponent and radical rules that are tested. You should be comfortable
with all of these rules. This means being comfortable using them in situations that you may
not have ever seen before. Each of the rules works in two directions; being comfortable with
the rules requires being able to use the rules in both directions. The rules are always the
same, but they can be tested in novel ways.
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SAT and ACT Math Guide page 14
Product Rule
If the base of two terms is the same and they are being multiplied, the exponents can be
added.
x7⋅x12 =x19
3x⋅35y=3x+5y
Power Rule
If a power is applied to multiple terms being multiplied together, the power can be applied
to each term individually by multiplying the exponents.
(2x3y5z4)6=26x18y30z24
(3a2b3c4)x=3xa2xb3xc4x
Quotient Rule
If the base of two terms are the same and one is in the numerator (top) and the other is
in the denominator (bottom) of a fraction, then they may be combined by subtracting the
exponents.
x9
x4=x9−4=x5
y2
y7=y2−7=y−5
z−8
z−3=z−8−(−3)=z−5
Negative Exponent Rule
If a negative exponent occurs in the numerator (top) of a fraction, it may be removed by
moving the term to the denominator (bottom) of the fraction. If a negative exponent occurs
in the denominator (bottom) of a fraction, it may be removed by moving the term to the
numerator (top) of the fraction.
x−13 =1
x13
1
y−7=y7
Zero Exponent Rule
Anything to the 0 power automatically equals 1.
130=1
x0=1
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SAT and ACT Math Guide page 15
Root of a Fraction
The root of a fraction is simply the same root of the numerator (top) and denominator
(bottom) of the fraction.
3
8
27 =3
√8
3
√27 =2
3
Fractional Exponent and Radical Rule
Fractional exponents can be turned into a radical and a radical can be switched into an
exponent. Both forms have their pros and cons. Generally, whenever you’re stuck in one
form, try switching to the other form.
To switch back and forth:
•The denominator (bottom) of the fractional exponent corresponds with the root of
the radical and vice versa.
•The numerator (top) of the fractional exponent corresponds with the power of the
radical and vice versa.
272
3=3
√272
Note that the 2 can go either inside or outside the radical. Generally, but not always, it is
easier to put it on the outside of the radical.
Getting the same base trick
If two exponentials are equal to each other, then getting them to have the same base makes
them easy to solve.
4x−2=82x+4
(22)x−2=(23)2x+4
22x−4=26x+12
Because the bases are equal, the exponents must now be equal.
2x−4=6x+12
−4x=16
x=−4
Test Number Test Number Test Number
SAT 1 Sec. 3 14 SAT 2 Sec. 3 7 SAT 3 Sec. 3 3
SAT 5 Sec. 3 12 SAT 6 Sec. 3 16 SAT 7 Sec. 3 11
SAT 8 Sec. 4 13 ACT 2005-06 8 ACT 2008-09 4
ACT 2011-12 22 ACT 2011-12 35 ACT 2011-12 49
ACT 2016-17 22 ACT 2011-12 45
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SAT and ACT Math Guide page 16
2.8 Exponential Problems
Exponential growth occurs when a quantity grows at a faster and faster rate. This occurs
because the quantity is repeatedly multiplied by the same value (as opposed to a linear
function, which adds the same amount repeatedly) These equations are of the form
y=a⋅bx
where b>1
x
y
Exponential decay occurs when a quantity shrinks by dividing it by the same amount re-
peatedly. These equations are of the same form, but b<1.
x
y
Test Number Test Number Test Number
SAT 4 Sec. 4 13 SAT 4 Sec. 4 15 SAT 4 Sec. 4 20
SAT 5 Sec. 3 14 SAT 6 Sec. 4 37 ACT 2005-06 49
ACT 2008-09 51 ACT 2008-09 60 ACT 2011-12 30
ACT 2011-12 60 ACT 2014-15 15 ACT 2016-17 7
2.9 Radical Problems
When solving an equation involving a radical, we must first isolate the radical alone on one
side. Then we raise both sides to whatever power is necessary to eliminate the radical.
3√x+5−2=10
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SAT and ACT Math Guide page 17
3√x+5=12
√x+5=4
(√x+5)2=42
x+5=16
x=11
When solving radical problems, you must check your solutions by plugging them back into
the original equation. If they do not work, they are extraneous and must be excluded.
Test Number Test Number Test Number
SAT 1 Sec. 3 20 SAT 2 Sec. 3 5 SAT 4 Sec. 3 9
SAT 5 Sec. 3 5 SAT 6 Sec. 3 9 SAT 8 Sec. 3 7
ACT 2005-06 10 ACT 2011-12 19
2.10 “In Terms Of ” Problems
“In terms of” problems are amongst the easiest problems on the SAT and ACT, but are
frequently overwhelming to students when they first seem them. These problems present
an equation (sometimes an equation that appears to be long and complicated) and then
an explanation of what the equation does and what each of the variables means. Then it
will ask for one of the variable “in terms of” the other variables. This is the only part that
matters (hence the name) – do not spend any time trying to understand the equation or
what the variable are.
There are two varieties of “in terms of problems”: ones where the variable you need to
isolate (get alone) occurs once, and ones where the variable you need to isolate (get alone)
occurs more than once.
When the variable occurs once
To isolate the variable, simply remove items one by one by doing the opposite operation
until the desired variable is alone.
Get min terms of s,t, and u.
t=s+mu
t−s=mu
t−s
u=m
When the variable occurs more than once
To isolate a variable that occurs more than once, follow these steps:
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SAT and ACT Math Guide page 18
•Each term containing the variable must be put on one side of the equation and each
term that doesn’t contain the variable must be put on the other side of the equation.
•Factor the variable out of each term on that side of the equation.
•Divide both sides by what is left over to get the variable alone.
Get pin terms of f,g,h, and k.
p−f
h=p+k
g
g(p−f)=h(p+k)
gp −gf =hp −hk
gp −hp =gf −hk
p(g−h)=gf −hk
p=gf −hk
g−h
Test Number Test Number Test Number
SAT 1 Sec. 3 7 SAT 1 Sec. 4 9 SAT 2 Sec. 3 12
SAT 2 Sec. 4 22 SAT 3 Sec. 4 13 SAT 5 Sec. 4 10
SAT 6 Sec. 3 7 SAT 6 Sec. 4 17 SAT 7 Sec. 4 19
SAT 7 Sec. 4 26 SAT 8 Sec. 3 3 ACT 2005-06 59
ACT 2014-15 54
2.11 Comparison Problems or Effect On Problems
These problems ask students what effect changing the value of a variable has on the total
value of an equation. That is, you compare the value of the original problem to the value
of the problem after the variable is changed.
To do these problems, follow these steps:
•Write the original, unchanged equation.
•Rewrite the original equation but with the new value for the changed formula.
•Do any mathematical operations necessary on the new value, but do not combine it
with any other numbers present. Instead, move the value towards the outside of the
equation.
•Compare the original equation with the new equation.
What is the effect of tripling the radius on the area of a semicircle?
A=1
2πr2
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SAT and ACT Math Guide page 19
A=1
2π(3r)2
A=1
2π9r2
A=9(1
2πr2)
This is 9 times the original equation, therefore the effect is to multiply the area of the
original semicircle by 9.
Test Number Test Number Test Number
SAT 2 Sec. 4 23 SAT 3 Sec. 3 15 SAT 4 Sec. 4 14
SAT 4 Sec. 4 35 SAT 5 Sec. 3 11 SAT 5 Sec. 3 13
ACT 2011-12 29 ACT 2016-17 43 ACT 2016-17 53
2.12 Inequalities
2.12.1 Solving Inequalities
To solve an inequality, simply treat the inequality sign as an equal sign and solve the equation
as you normally would. The only other thing you need to do is turn the inequality around
if you multiply or divide both sides by a negative number.
−2x−7<5
−2x<12
x>−6
If you have an equation with two inequalities in it, you can either break it into two inequal-
ities and treat them as you normally would or you can solve for the variable by doing the
same thing to all three sides of the equation (again remember to turn the inequality around
if you multiply or divide by a negative).
−7≤3x+2≤17
−9≤3x≤15
−3≤x≤5
2.12.2 Graphing Inequalities
To graph an inequality, the ymust be isolated. The line can then be graphed as if it were
an ordinary equal sign, keeping the following points in mind.
•If the equation is less than (<) or greater than (>), the line will be dashed.
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SAT and ACT Math Guide page 20
•If the equation is less than or equal to (≤) or greater than or equal (≥), the line will
be solid.
•If the equation is greater than (>) or greater than or equal to (≥), shade above the
line.
•If the equation is less than (<) or less than or equal to (≤), shade below the line.
If the equations to be graphed are given to you as an equation with two inequalities in it,
simply break the equation into two equations and solve each one for y.
4<2x−y≤7
4<2x−y2x−y≤7
−2x+4<−y−y≤−2x+7
2x−4>y y ≥2x−7
y<2x−4y≥2x−7
If the equation is a circle, the following slightly modified rules must be applied.
•If the equation is less than (<) or less than or equal to (≤), shade inside the circle.
•If the equation is greater than (>) or less than or equal to (≥), shade outside the circle.
Test Number Test Number Test Number
SAT 1 Sec. 4 11 SAT 1 Sec. 4 18 SAT 1 Sec. 4 28
SAT 1 Sec. 4 31 SAT 2 Sec. 4 21 SAT 3 Sec. 4 31
SAT 4 Sec. 4 19 SAT 4 Sec. 4 26 SAT 5 Sec. 4 13
SAT 6 Sec. 4 18 SAT 7 Sec. 3 14 SAT 7 Sec. 4 5
SAT 8 Sec. 4 20 ACT 2005-06 22 ACT 2005-06 47
ACT 2008-09 36 ACT 2008-09 49 ACT 2011-12 36
ACT 2011-12 57 ACT 2014-15 29 ACT 2016-17 36
ACT 2016-17 38 ACT 2016-17 49 ACT 2016-17 52
ACT 2014-15 55
2.13 Polynomial Equivalence Problems
If you are given two polynomials that are equal to each other, then the corresponding
coefficients must be equal. This is a very useful property in numerous problems.
5x2−17x+13 =5x2+(4−b)x+13
17 =4−b
b=−13
Test Number Test Number Test Number
SAT 1 Sec. 3 15 SAT 2 Sec. 3 17 SAT 3 Sec. 3 13
SAT 8 Sec. 3 17
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SAT and ACT Math Guide page 21
2.14 Absolute Values
2.14.1 Solving Absolute Values
There are three steps to solving an absolute value problem. They must be done in order,
and you must be sure not to skip any steps.
Isolate the Absolute Value First, you must isolate the absolute value alone on one side
of the equal sign.
3x−2+5=17
3x−2=12
x−2=4
Now the absolute value is isolated.
Ask Whether the Absolute Value is Possible Once the absolute value is isolated, it
is essential to pause and ask whether the problem is possible. This means asking whether
the absolute value could produce the value on the other side of the equal sign. Specifically,
an absolute value cannot produce a negative value.
x+5=−3
This is not possible because an absolute value can never produce a negative value. Therefore,
there are no solutions.
If you do not pause and ask whether the absolute value is possible and instead immediately
proceed to the next step, you will get the problem wrong. Therefore, you must pause and
ask it after isolating the absolute value.
Break the Absolute Value into Two Problems Once the absolute value has been
isolated and you’ve determined that the problem is possible, you break the problem into
two problems – one positive and one negative.
x−2=4
x−2=4
x−2=−4
Then we simply solve each of these problems.
x−2=4
x=6
x−2=−4
x=−2
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SAT and ACT Math Guide page 22
2.14.2 Creating Absolute Values
Another way of thinking about an absolute value problem is as the two points that are
equidistant from a central point. For example, consider the following absolute value problem.
x−5=2
x−5=2
x=7
x−5=−2
x=3
The two solutions (7 and 3) are each 2 units away from 5 (the central value). Note that
these numbers correspond with the two values in the original absolute value problem.
•The 5 (the central value) is the opposite of the number inside the absolute value.
•The 2 (the distance away from the central value) is the number on the other side of
the equal sign.
Thus, we can answer a question such as this: Find the absolute value that represents the two
points 6 units away from -2. x+2=6
because 2 is the opposite of the central value (-2) and we want two point 6 units away from
this value.
Test Number Test Number Test Number
SAT 1 Sec. 4 8 SAT 4 Sec. 3 1 SAT 6 Sec. 4 28
ACT 2005-06 9 ACT 2005-06 21 ACT 2005-06 53
ACT 2008-09 22 ACT 2008-09 42 ACT 2011-12 1
ACT 2011-12 46 ACT 2014-15 26 ACT 2014-15 60
ACT 2016-17 58
3 Percents
3.1 Introduction
Percent problems are amongst the easiest problems on the SAT and ACT, yet they are
also amongst the most frequently missed problems. The reason for this is that most people
were not correctly taught to use percents at an early age. However, the proper way to do
percent problems can easily be learned. This is good news because percent problems are
very plentiful.
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SAT and ACT Math Guide page 23
We cannot do percent problems with percents. Thus the first step in any percent problem
is to turn the percent into a decimal. This is done by moving the decimal point twice to the
left.
87% =0.87
2.5% =0.025
The second step is to turn this decimal into a multiplier. How the decimal is turned into a
multiplier depends on the type of problem.
3.2 Increasing Problems
Many percent problems involve a quantity that is increasing. Situations that involve an
increasing quantity are frequently in the form of a sales tax, a raise, a tip, growth, interest,
or other increasing quantity.
In these types of problems, the multiplier (m) is formed as follows
m=1+p
where pis the percent turned into decimal form. This multiplier mis then multiplied by
the quantity to find the new quantity.
For example, if someone earns $12.00 per hour and receives a 5% raise, we find the multiplier
as follows
m=1+0.05 =1.05
and multiply it by the old wage in order to find the new wage:
$12.00 ⋅1.05 =$12.60
3.3 Decreasing Problems
Another common category of percent problems involve a quantity that is decreasing. Situa-
tions that involve a decreasing quantity are frequently in the form of a sale, discount, decay,
or other decreasing quantity.
In these types of problems, the multiplier (m) is formed as follows
m=1−p
where pis the percent turned into decimal form. This multiplier mis then multiplied by
the quantity to find the new quantity.
For example, if a $40.00 pair of jeans are 15% off, we find the multiplier as follows
m=1−0.15 =0.85
and multiply it by the original price of the jeans in order to find the new price of the jeans:
$40.00 ⋅0.85 =$34.00
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SAT and ACT Math Guide page 24
3.4 “Percent Of ” Problems
The easiest type of percent problems are “percent of” problems. In these problems we are
looking for some part or portion of the original quantity. In these types of problems, the
multiplier (m) is simply
m=p
where pis the percent turned into decimal form. The multiplier mis then multiplied by the
original quantity to find the new quantity.
For example, if a high school has 1,560 students and 70% of the students are planning on
attending the dance, we simply do
1,560 ⋅0.70 =1,092
to find that 1,092 students will attend the dance.
3.5 Percent Word Problems
Many percent problems occur in the form of word problems. The trick to these problems
is to determine whether it is a percent increasing or percent decreasing problem, give any
unknown value a variable, and to carefully walk through the scenario in chronological order.
If you pay $43.20 for a pair of jeans after an 8% sales tax, what was the sticker price of the
jeans?
This is a percent increasing problem and the unknown value is the original price, which we
will call p. Walking through the problem chronologically, the first thing that happened was
that the you picked up the pair of jeans, which had price p. Then an 8% sales tax was
applied, so our multiplier will be 1.08. The result is that the final price became $43.20.
Thus we have the equation that can easily be solved for p.
p⋅1.08 =43.20
3.6 Turning other things into percents
To determine by what percent something has increased or decreased, simply set up an
equation chronologically and use the logic of increasing multipliers and decreasing multipliers
from above.
If the population of a city has increased from 60,000 to 63,000, what percent has it increased
by?
60000 ⋅m=63000
m=1.05
Thus the population has increased 5%.
If a stock was priced at $200 per share but is now priced at $170 per share, what percent
has it decreased by?
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SAT and ACT Math Guide page 25
200 ⋅m=170
m=0.85
Thus the stock has lost 15% of its value.
Test Number Test Number Test Number
SAT 1 Sec. 4 20 SAT 1 Sec. 4 26 SAT 1 Sec. 4 37
SAT 1 Sec. 4 38 SAT 2 Sec. 3 14 SAT 2 Sec. 4 5
SAT 2 Sec. 4 17 SAT 3 Sec. 4 5 SAT 3 Sec. 4 21
SAT 3 Sec. 4 27 SAT 3 Sec. 4 28 SAT 3 Sec. 4 38
SAT 4 Sec. 3 12 SAT 4 Sec. 4 4 SAT 4 Sec. 4 22
SAT 4 Sec. 4 37 SAT 4 Sec. 4 38 SAT 5 Sec. 4 24
SAT 5 Sec. 4 38 SAT 6 Sec. 3 2 SAT 6 Sec. 3 19
SAT 6 Sec. 4 6 SAT 6 Sec. 4 24 SAT 6 Sec. 4 38
SAT 7 Sec. 3 8 SAT 7 Sec.4 10 SAT 7 Sec. 4 12
SAT 7 Sec. 4 27 SAT 7 Sec. 4 30 SAT 8 Sec. 4 22
ACT 2005-06 2 ACT 2008-09 17 ACT 2008-09 47
ACT 2011-12 7 ACT 2011-12 13 ACT 2014-15 46
ACT 2016-17 6 ACT 2016-17 21 ACT 2016-17 25
4 Linear Functions
4.1 Finding Linear Functions
Linear functions are functions that are straight lines when graphed. When put in slope
intercept form
y=mx +b
where mis the slope of the line and bis the y intercept of the line, they are easily graphed.
Finding the Linear Function When the Slope and a Point are Given
If you are given the slope and a point, the equation of the linear function is easily found by
plugging the slope in for mand then plugging the xand yvalues of the point in to solve
for b.
Find the equation of the line with slope −2 and passing through (3,−5).
m=−2
y=mx +b
y=−2x+b
Next we plug the point (3,−5)in for xand yto solve for b.
−5=−2(3)+b
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SAT and ACT Math Guide page 26
−5=−6+b
1=b
y=−2x+1
Finding the Linear Function When Two Points are Given
If two points are given and we are asked to find the equation of the linear function passing
through the two points, we must start by finding the slope.
m=rise
run =∆y
∆x=y2−y1
x2−x1
Once the slope has been obtained, we proceed exactly as we would in a problem in which
the slope and a point are given.
Find the equation of the line passing through (−2,3)and (4,6).
m=6−3
4−(−2)=3
6=1
2
y=1
2x+b
Now we plug in a point for xand yto find b. Either point will work.
6=1
2(4)+b
6=2+b
4=b
y=1
2x+4
4.2 Interpreting Linear Functions
There are several word problems that ask you to interpret the mand bvalues of a linear
function. This is what the slope and y intercept mean in the context of a word problem:
•slope (m): the amount the yvariable changes for an increase of one in the xvalue
•y intercept (b): the initial or starting value or quantity of the problem
4.3 Graphing Linear Functions
To graph a linear function, simply put it in the form y=mx +b. First, plot the y-intercept
(from the bvalue). Second, use the slope (m) to plot a second point, using the rise and run
determined by the slope. Then connect the two points with a line.
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SAT and ACT Math Guide page 27
4.4 Parallel and Perpendicular
If two lines are parallel, this means they have the same slope. Thus, the following two
equations are parallel.
y=−3
5x+1y=−3
5x−4
If two lines are perpendicular, this means that their slopes are negative reciprocals of each
other. Thus, the following two equations are perpendicular.
y=3
4x+5y=−4
3x−1
Test Number Test Number Test Number
SAT 1 Sec. 3 4 SAT 1 Sec. 3 6 SAT 1 Sec. 3 12
SAT 1 Sec. 4 15 SAT 1 Sec. 4 16 SAT 1 Sec. 4 22
SAT 2 Sec. 3 3 SAT 2 Sec. 3 6 SAT 2 Sec. 4 8
SAT 2 Sec. 4 12 SAT 2 Sec. 4 25 SAT Sec. 4. 28
SAT 2 Sec. 4 35 SAT 3 Sec. 3 8 SAT 3 Sec. 4 4
SAT 3 Sec. 4 8 SAT 3 Sec. 4 17 SAT 3 Sec. 4 26
SAT 4 Sec. 3 2 SAT 4 Sec. 3 7 SAT 4 Sec. 3 8
SAT 4 Sec. 4 8 SAT 4 Sec. 4 17 SAT 4 Sec. 4 27
SAT 4 Sec. 4 32 SAT 5 Sec. 3 1 SAT 5 Sec. 3 8
SAT 5 Sec. 3 16 SAT 5 Sec. 4 2 SAT 5 Sec. 4 7
SAT 5 Sec. 4 11 SAT 5 Sec. 4 16 SAT 5 Sec. 4 17
SAT 5 Sec. 4 18 SAT 5 Sec. 4 23 SAT 5 Sec. 4 28
SAT 6 Sec. 3 1 SAT 6 Sec. 3 5 SAT 6 Sec. 4 2
SAT 6 Sec. 4 4 SAT 6 Sec. 4 5 SAT 6 Sec. 4 13
SAT 6 Sec. 4 14 SAT 6 Sec. 4 15 SAT 6 Sec. 4 25
SAT 6 Sec. 4 35 SAT 7 Sec. 3 1 SAT 7 Sec. 3 6
SAT 7 Sec. 3 19 SAT 7 Sec. 4 17 SAT 7 Sec. 4 28
SAT 7 Sec. 4 32 SAT 8 Sec. 3 2 SAT 8 Sec. 3 13
SAT 8 Sec. 3 19 SAT 8 Sec. 4 2 SAT 8 Sec. 4 7
SAT 8 Sec. 4 21 SAT 8 Sec. 4 25 SAT 8 Sec. 4 27
SAT 8 Sec. 4 29 ACT 2005-06 5 ACT 2005-06 16
ACT 2005-06 27 ACT 2005-06 29 ACT 2005-06 35
ACT 2005-06 56 ACT 2008-09 1 ACT 2008-09 3
ACT 2008-09 12 ACT 2008-09 26 ACT 2008-09 31
ACT 2008-09 40 ACT 2008-09 43 ACT2011-12 2
ACT2011-12 3 ACT2011-12 17 ACT2011-12 23
ACT2011-12 24 ACT2011-12 52 ACT 2014-15 24
ACT 2014-15 28 ACT 2014-15 33 ACT 2014-15 35
ACT 2014-15 58 ACT 2016-17 8 ACT 2016-17 9
ACT 2016-17 11 ACT 2016-17 34 ACT 2016-17 35
ACT 2016-17 39 ACT 2016-17 46
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5 Systems of Equations
5.1 Definition and Overview
A system of equations is simply a group of equations. To solve a system of equations, you
need to have the same number of equations as variables. Generally on the SAT and the
ACT, there will be two variables (usually xand y), so you will need two equations to be
able to solve the system.
5.2 Methods Solving Systems of Equations
There are two methods for solving a system of equations with two equations and two un-
knowns. Either will work for any given problem. However, problems will be set up to be
done more quickly and easily in one form or the other. Therefore students should be com-
fortable solving systems of equations using both methods and should solve using whichever
method the system is set up for.
5.2.1 Substitution
Substitution is used when one variable is alone (or nearly alone) on one side of the equal
sign. This variable is then substituted (or “plugged in” to) the second equation.
x=2y−7
−2x−5y=−4
−2(2y−7)−5y=−4
−4y+14 −5y=−4
−9y=−18
y=2
Substituting y=2 into the first equation
x=2(2)−7
x=−3
Thus the solution is x=−3, y=2.
5.2.2 Elimination
The other method for solving systems of equations is elimination (also known as cancel-
lation). Elimination is used when there is a column of xterms, a column of yterms, a
column of the equals signs, and a column of constant terms. One (or both) of the lines is
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multiplied by a constant so that either the xcolumn or the ycolumn will cancel when the
two equations are added (or subtracted).
3x−5y=13
4x+3y=−2
Here the ycolumn will cancel if we multiply the first line by 3 and multiply the second line
by 5.
33x−5y=13
54x+3y=−2
9x−15y=39
20x+15y=−10
The ycolumn will now cancel
9x−15y=39
20x+15y=−10
29x=29
x=1
Substituting x=1 into the first equation
3(1)−5y=13
3−5y=13
−5y=10
y=−2
5.3 Types of Solutions
Systems of equations have three possible sets of solutions.
•one solution
•zero solutions
•infinitely many solutions
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5.3.1 One Solution
Graphically, this occurs when two lines intersect at a single point.
x
y
Algebraically, this occurs when you are able to find numeric values for xand for y.
5.3.2 Zero Solutions
Graphically, this occurs when two lines are parallel and therefore never intersect.
x
y
Algebraically, this occurs when you get an obviously untrue mathematical statement.
3x−4y=10
−6x+8y=−17
Multiplying the first line by 2
23x−4y=10
−6x+8y=−17
6x−8y=20
−6x+8y=−17
0≠3
This is obviously not true, so there are zero solutions.
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5.3.3 Infinitely many solutions
Graphically, this occurs when two lines are exactly the same and lie on top of each other,
thus intersecting at infinitely many points.
x
y
Algebraically, this occurs when you get a mathematical statement that is always true.
5x−2y=4
10x−4y=8
Multiplying the first line by −2−25x−2y=4
10x−4y=8
−10x+4y=−8
10x−4y=8
0=0
This is always true, so there are infinitely many solutions.
5.4 Intersections
If you are asked where two lines intersect, this simply means that there is a system of
equations. You then simply use the tools for solving systems of equations. The xand y
values of the solution (or solutions) are simply the point (or points) of intersection of the
graphs.
5.5 Non-Linear Systems of Equations
Occasionally there are questions with non-linear systems of equations. These are systems
of equations in which there are still two equations and two unknowns (generally xand y)
but one or more of the variables has a power attached to them. These problems are solved
using the exact same methods for solving systems of equations: substitution or elimination.
However, these questions are nearly always designed to be solved using substitution.
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5.6 Turning Word Problems into Systems of Equations
Turning word problems into systems of equations is one of the subjects students struggle
with the most on the SAT and the ACT. However, some considerations can be very helpful.
First, clearly define your variables. There will almost always be two variables in these types
of questions (occasionally there will only be one variable). Be sure to state exactly what the
variable is mathematically. The variable will almost always be the quantity of something.
For example, if the problem is about apples and bananas, the variables will not be apples
and bananas; they will be the number of apples and the number of bananas.
Second, if you have two variables, you will need two equations. Generally define what your
two equations will be about. One will be about the quantity of items. The other will be
about some other consideration, such as price, dollars, weight, time, etc.
Third, form the equation about the quantity of items. There are two types of quantity
equations.
•If you are told the total quantity of items, simply add the two variables and set them
equal to this total quantity. For example, if the total number of apples (a) and bananas
(b) is 47
a+b=47
•If the two quantities are compared, you must think much more carefully about this
equation. Students regularly form these equations incorrectly because the English
sentence will look somewhat different than the mathematical statement. For example,
if there are twice as many apples as bananas
a=2b
Student’s natural inclination is to put the 2 with the abecause that is where the 2 is
in the sentence. However, you must ask which quantity is larger (a) and what must
be done to compensate for this to make the quantities equal (multiplying bby 2).
Similarly, if we are told there are 12 more apples than bananas
a=b+12
because there are fewer bananas and we must add 12 to them to make them equal to
the bananas.
Finally, form the equation that isn’t about quantity. This is generally easier than the
quantity equation.
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Test Number Test Number Test Number
SAT 1 Sec. 3 9 SAT 1 Sec. 3 11 SAT 1 Sec. 3 18
SAT 1 Sec. 4 4 SAT 1 Sec. 4 19 SAT 1 Sec. 4 32
SAT 2 Sec. 3 2 SAT 2 Sec. 3 9 SAT 2 Sec. 3 20
SAT 2 Sec. 4 1 SAT 2 Sec. 4 9 SAT 2 Sec. 4 34
SAT 3 Sec. 3 4 SAT 3 Sec. 3 6 SAT 3 Sec. 3 9
SAT 3 Sec. 3 10 SAT 2 Sec. 3 19 SAT 3 Sec. 4 18
SAT 3 Sec. 4 22 SAT 3 Sec. 4 24 SAT 3 Sec. 4 30
SAT 3 Sec. 4 36 SAT 4 Sec. 3 3 SAT 4 Sec. 3 19
SAT 4 Sec. 4 1 SAT 4 Sec. 4 2 SAT 4 Sec. 4 6
SAT 4 Sec. 4 16 SAT 4 Sec. 4 30 SAT 4 Sec. 4 34
SAT 5 Sec. 3 7 SAT 5 Sec. 3 9 SAT 5 Sec. 3 15
SAT 5 Sec. 3 18 SAT 5 Sec. 4 6 SAT 5 Sec. 4 12
SAT 5 Sec. 4 35 SAT 6 Sec. 3 6 SAT 6 Sec. 3 16
SAT 6 Sec. 4 10 SAT 6 Sec. 4 11 SAT 6 Sec. 4 19
SAT 6 Sec. 4 31 SAT 7 Sec. 3 3 SAT 7 Sec. 3 9
SAT 7 Sec. 4 11 SAT 7 Sec. 4 16 SAT 7 Sec. 4 20
SAT 7 Sec. 4 33 SAT 8 Sec. 3 5 SAT 8 Sec. 3 6
SAT 8 Sec. 3 10 SAT 8 Sec. 3 18 SAT 8 Sec. 4 6
SAT 8 Sec. 4 12 SAT 8 Sec. 4 33 ACT 2005-06 57
ACT 2008-09 8 ACT 2008-09 10 ACT 2011-12 27
ACT 2011-12 58 ACT 2014-15 1 ACT 2014-15 3
ACT 2014-15 6 ACT 2014-15 8 ACT 2014-15 10
ACT 2014-15 13 ACT 2014-15 20 ACT 2014-15 41
ACT 2014-15 49 ACT 2014-15 50 ACT 2014-15 51
ACT 2014-15 52 ACT 2016-17 15 SAT 1 Sec. 3 3
SAT 2 Sec. 3 16 SAT 6 Sec. 4 23
6 Factoring and Related Topics
6.1 Factoring
“Factoring” is actually a general terms that describes a number of similar, but distinct,
tools. Therefore, it is important to keep each of the different tools distinct and to know
when to use each one. In addition, any particular factoring problem may require applying
multiple factoring tools or applying the same tool repeatedly.
“Pull Out” Factoring
“Pull out” factoring is the most basic type of factoring and also the most overlooked form of
factoring. As such, you should always ask whether you can perform “pull out” factoring first
when approaching a factoring problem. It will always make the problem easier (sometimes
dramatically so).
“Pull out” factoring is performed by pulling a common factor out of every term in a poly-
nomial. It can easily be done by asking whether a constant can be pulled out of each term
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and then asking whether any variable can be pulled out of each term.
12x3−8x2+14x
Here, a 2 can be pulled out of each coefficient and an xcan be pulled out as well, so we will
pull a 2xout of each term.
2x(6x2−4x+7)
Ordinary Factoring
Ordinary factoring is an essential basic skill for the SAT and ACT, but many students
struggle with it. There are numerous different tools you can use to factor. I will demonstrate
the box and diamond approach. This method only requires the diamond if a=1. If a≠1,
it requires both the diamond and the box.
If a equals 1 If a(the coefficient of the x2term) is equal to 1, you only need to use the
diamond.
x2+4x−12
Simply put the cvalue in the top of the diamond and the bvalue in the bottom of the
diamond.
−12
4
Then list out in order all the factor pairs of the top number (12) (ignoring the sign).
1and 12,2and 6,3and 4
Determine which pair can add or subtract to equal the bottom number (4). Here, it is 2
and 6. Put these numbers on the left and right sides of the diamond.
−12
4
26
Next, determine if either of the left or right numbers need a negative sign so that they
multiply to be the top number (-12) and add to be the bottom number (4).
−12
4
−26
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The left and right numbers tell you what the factors will be.
x2+4x−12 =(x−2)(x+6)
If a does not equal 1 If a(the coefficient of the x2term) does not equal 1, you must
use the diamond and the box.
6x2+5x−4
Multiply the x2(6x2) term and the constant term (-4) and put them in the top of the
diamond. Put the xterm in the bottom of the diamond
−24x2
5x
Then list out in order all the factor pairs of the top number (24) (ignoring the sign).
1and 24,2and 12,3and 8,4and 6
Determine which pair can add or subtract to equal the bottom number (5). Here, it is 3
and 8. Put these numbers with an xon the left and right sides of the diamond.
−24x2
5x
3x8x
Next, determine if either of the left or right terms need a negative sign so that they multiply
to be the top term (−24x2) and add to be the bottom term (5x).
−24x2
5x
−3x8x
Next, we turn to the box. Put the original x2term (6x2) in the top left box and put the
original constant term (-4) in the bottom right box. The terms from the left and right sides
of the diamond go in the remaining two boxes (it doesn’t matter which one goes where).
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8x−4
6x2−3x
Next, pull out factor from each row and column, placing these values along the left and top
sides of the box.
4
3x
8x−4
6x2−3x
2x−1
These terms on the left side and top are the factors.
6x2+5x−4=(3x+4)(2x−1)
Factoring by Substitution
Occasionally an equation can be factored more easily by substituting a new variable for the
previous variable. This is done so that the equation is of the form ax2+bx +cinstead of
having different exponents. The following equation is more easily factored by substituting
u=x3.
x6−7x3+12
u2−7u+12
(u−4)(u−3)
(x3−4)(x3−3)
Factoring by Grouping
If asked to factor a polynomial of degree three with all four terms present, the only tool
available to you for factoring is factoring by grouping. Factoring by grouping is done by
doing “pull out” factoring on the first two terms and then doing “pull out” factoring again
on the last two terms. This will result in a common factor and the process will be completed
by pulling this common factor out. If we are asked to factor the following
x3+5x2−9x−45
we know that we will have to factor by grouping because it is a polynomial of degree three
with all four terms. We begin by pulling x2out of the first two terms and −9 out of the last
two terms.
x3+5x2−9x−45
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x2(x+5)−9(x+5)
Because the (x+5)is common to both terms, we can pull it out.
(x+5)(x2−9)
As mentioned before, the various factoring tools can be applied repeatedly and in various
combinations. Here, (x2−9)can be factored using ordinary factoring to achieve a final
answer. (x+5)(x+3)(x−3)
6.2 Completing the Square
To complete the square, we take a quadratic and put the x2term and the xterm in paren-
theses with a space with a plus in front of it. We place the constant term outside the
parentheses with a space with a negative in front of it.
y=x2−6x+7
y=(x2−6x+)+7−
Then we fill both spaces with the number equal to b
22. Because one is positive and one is
negative, they cancel each other out and we have kept the equation balanced.
−6
22=9
y=(x2−6x+9)+7−9
Finally, we factor the terms in the parentheses (which will always be a perfect square) and
combine the terms outside the parentheses.
y=(x2−6x+9)−2
y=(x−3)2−2
When the avalue (the coefficient of x2) is any number other than 1, completing the square
requires some additional steps.
y=2x2+20x+47
We begin by creating the parentheses and spaces.
y=(2x2+20x+)+47 −
However, we must now pull the 2 out of the parentheses so that the x2is “alone.”
y=2(x2+10x+)+47 −
Then we fill the first space with the number equal to b
22.
10
22=25
y=2(x2+10x+25 )+47 −
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We have just added a 25, but its value is actually 50 since it is being multiplied by the 2 in
front of the parentheses. So we must put 50 in the second blank space.
y=2(x2+10x+25 )+47 −50
Now we can complete the problem by factoring and combining the terms outside the paren-
theses.
y=2(x+5)2−3
6.3 Polynomial Division
Polynomial division problems are presented as follows. The key is to recognize that this is
simply asking for polynomial division to be done.
3x2+2x+4
x−1
The most surefire way to do these problems is with polynomial long division.
x−13x2+2x+4
To do this, we ask what xmust be multiplied by to equal 3x2– namely, 3x.
3x
x−13x2+2x+4
3xis then multiplies by x−1 and subtracted and then the 4 is brought down.
3x
x−13x2+2x+4
−3x2+3x
5x+4
Then this process is repeated.
3x+5
x−13x2+2x+4
−3x2+3x
5x+4
−5x+5
9
There are no more terms to bring down, so the remainder is 9 and the answer is as follows:
3x2+2x+4
x−1=3x+5+9
x−1
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6.4 Undefined Rational Functions
A rational function is undefined if the denominator (bottom) of the fraction is equal to zero.
This frequently requires factoring the denominator (bottom) and setting it equal to zero.
y=1
x2−8x+12
y=1
(x−6)(x−2)
Therefore, the function is undefined if x=6 or x=2.
Test Number Test Number Test Number
SAT 1 Sec. 4 36 SAT 2 Sec. 3 4 SAT 2 Sec. 3 15
SAT 3 Sec. 3 13 SAT 3 Sec. 3 16 SAT 4 Sec. 3 18
SAT 4 Sec. 4 25 SAT 5 Sec. 3 10 SAT 5 Sec. 4 30
SAT 6 Sec. 3 12 SAT 7 Sec. 3 13 SAT 8 Sec. 3 8
ACT 2008-09 23 ACT 2008-09 38
7 Quadratics
7.1 Definition
A quadratic is simply a polynomial with degree two. That is, it is when the largest exponent
is a 2.
7.2 Terminology
Part of the reason many students find quadratics difficult is because of some of the termi-
nology involved. In particular, there are four words that all mean the exact same thing
and a fifth word with a related meaning. Students have rarely seen these connections and
the designers of the tests know this and design questions based on this misunderstanding.
Several questions become incredibly simple once students understand the connection of all
these words.
The following four words all mean the exact same thing
•x intercept
•root
•solution
•zero
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The word “factor” means one term that is multiplied by other factors to form the polynomial.
The factor is closely related to, but distinct, from the words x intercept, root, solution, and
zero. When a factor is set equal to zero and solved for x, this reveals the x intercept, root,
solution, and zero. Suppose we have the following polynomial
y=x(x+7)(x−3)(2x−9)
The factors are x,(x+7),(x−3), and (2x−9). When we set each of these equal to zero,
we can find each of the x intercepts, roots, solutions, and zeros.
x=0
x+7=0x=−7
x−3=0x=3
2x−9=0x=9
2
Thus the x intercepts, roots, solutions, and zeros are 0, −7, 3, and 9
2.
Working in reverse, if we are told that the x intercepts, roots, solutions, or zeros of a
polynomial are −3, 2, and 5, we know that the factors are (x+3),(x−2), and (x−5).
If a factor is squared, it is a solution with multiplicity 2. This means that the polynomial
“bounces” off the x-axis at the corresponding root, instead of passing through the x-axis.
7.3 Standard Form
The following is the standard form of a quadratic equation.
y=ax2+bx +c
Standard form is the least useful of the three forms. However, there are some useful things
we can do with standard form.
•Standard form is the only form that shows us the values of a,b, and c. These are
necessary to use the quadratic formula or determinant (see below).
•Whether the parabola opens upward (a positive avalue) or opens downward (a nega-
tive avalue).
•The y intercept (the cvalue).
Because standard form is the least useful form, it is important to know how to change from
standard form into the other forms.
7.4 Intercept Form
The following is the intercept form (also called factored form).
y=a(x−m)(x−n)
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As the name implies, we can determine the x intercepts of a quadratic simply by looking at
the intercept form: mand n. Specifically, the x intercepts are found by setting each factor
equal to zero. Therefore, if we have the following quadratic in intercept form
y=(2x+5)(x−6)
we find the x intercepts by setting each factor equal to zero.
2x+5=0
2x=−5
x=−5
2
and
x−6=0
x=6
Thus the x intercepts are −5
2and 6.
The avalue is the same as the avalue in standard form and therefore tells us whether the
parabola opens upward (a positive avalue) or opens downward (a negative avalue).
To get a quadratic from standard form into intercept form, you simply need to factor the
quadratic in standard form.
y=x2−x−12
y=(x−4)(x+3)
Thus the x intercepts would be 4 and −3.
7.5 Vertex Form
The following is the vertex form (also called graphing form).
y=a(x−h)2+k
As the name implies, we can determine the location of the vertex of a parabola simply by
looking at the vertex form. Specifically, the vertex is located at (h, k). Therefore, if we have
the following quadratic in vertex form
y=(x−4)2+1
we find that the vertex is located at (4,1).
The avalue is the same as the avalue in standard form and therefore tells us whether the
parabola opens upward (a positive avalue) or opens downward (a negative avalue).
In order to get a quadratic from standard form into vertex form, you simply complete the
square.
y=x2−6x+7
y=(x2−6x+)+7−
y=(x2−6x+9)+7−9
y=(x2−6x+9)−2
y=(x−3)2−2
Thus the vertex is at (3,−2).
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7.6 Quadratic Formula
The quadratic formula tells us the x intercepts, roots, solutions, or zeros of a quadratic
function in standard form. The a,b, and cvalues from standard form are simply plugged
into the quadratic formula.
x=−b±√b2−4ac
2a
Thus, the quadratic formula is an alternative to factoring a quadratic and putting it into
intercept form to find the x intercepts.
This formula is not given on the SAT or ACT but is extremely useful on the test. Therefore
it is the most important formula to memorize for the test, if you have not memorized it
already.
7.7 Discriminant
The discriminant is the portion of the quadratic formula involving the radical
√b2−4ac
It is a very useful tool when asked how many x intercepts, solutions, roots, or zeros a
quadratic has. The only thing that matters is whether the quantity under the radical is
positive, zero, or negative – not the specific number.
√+– two solutions
√0 – one solution
√−– no real solutions
If we are asked how many solutions the following quadratic in standard form has
y=12x2−7x+5
we simply need to determine whether the discriminant is positive, zero, or negative.
(−7)2−4(12)(5)
49 −4(12)(5)
Even without a calculator, this is obviously negative (the actual number does not matter).
Therefore there are no real solutions.
7.8 Axis of Symmetry
The axis of symmetry is the line across which a parabola is symmetrical (that is, which the
parabola could fold across and “land” back on itself).
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x
y
However, this is not why the axis of symmetry is useful for the SAT and ACT. The reason
it is useful is because the axis of symmetry passes through the vertex of the parabola. That
is, the axis of symmetry quickly and easily gives us the x value of the vertex of a parabola.
7.8.1 Finding the Axis of Symmetry in Standard Form
When in standard form, the axis of symmetry is given by
x=−b
2a
Note that this is simply part of the quadratic formula.
If we have the following quadratic in standard form
y=4x2−2x+3
then the axis of symmetry (and thus the x value of the vertex) is
x=−(−2)
2(4)=2
8=1
4
7.8.2 Finding the Axis of Symmetry in Intercept Form
When in intercept form, the axis of symmetry is found by averaging the two x intercepts.
If we have the following quadratic in intercept form
y=(x+4)(x−6)
then the axis of symmetry (and thus the x value of the vertex) is
x=−4+6
2=2
2=1
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Test Number Test Number Test Number
SAT 1 Sec. 3 16 SAT 1 Sec. 4 25 SAT 1 Sec. 4 29
SAT 1 Sec. 4 30 SAT 2 Sec. 3 13 SAT 2 Sec. 4 7
SAT 2 Sec. 4 29 SAT 3 Sec. 3 7 SAT 3 Sec. 3 12
SAT 3 Sec. 3 14 SAT 3 Sec. 4 12 SAT 4 Sec. 3 11
SAT 4 Sec. 3 13 SAT 4 Sec. 3 15 SAT 4 Sec. 4 12
SAT 4 Sec. 4 28 SAT 5 Sec. 3 3 SAT 5 Sec. 3 4
SAT 6 Sec. 3 4 SAT 6 Sec. 3 11 SAT 6 Sec. 3 14
SAT 6 Sec. 4 20 SAT 6 Sec. 4 30 SAT 6 Sec. 4 34
SAT 7 Sec. 3 7 SAT 7 Sec. 3 10 SAT 7 Sec. 3 12
SAT 7 Sec. 3 15 SAT 7 Sec. 4 6 SAT 7 Sec. 4 24
SAT 7 Sec. 4 35 SAT 8 Sec. 3 11 SAT 8 Sec. 3 14
SAT 8 Sec. 3 16 SAT 8 Sec. 4 8 SAT 8 Sec. 4 14
SAT 8 Sec. 4 19 SAT 8 Sec. 4 35 ACT 2005-06 28
ACT 2005-06 50 ACT 2011-12 21 ACT 2014 59
ACT 2016-17 24 ACT 2016-17 28
8 Complex Numbers
8.1 Definition and Strategy
As you probably learned in one of your high school math classes, i(the imaginary number)
is defined to be √−1. Therefore all complex number problems on the SAT explicitly state
i=√−1
However, this is a distraction. There are no problems in which you will have to turn √−1
into ior vice versa. Therefore cross this out. Instead, you should write
i2=−1
This is the only piece of information about complex numbers that you need to know.
There are only three types of complex number problems that you will encounter; there
will never be any complicated or tricky questions. All complex numbers can be solved by
following these steps.
•If the problem is a division problem, multiply the numerator (top) and the denominator
(bottom) by the conjugate of the denominator (bottom). (See below.)
•Treat ilike any other variable and perform any operations (such as addition, subtrac-
tion, or multiplication) as you normally would.
•Any time an i2appears, replace it with −1.
8.2 Addition and Subtraction
Suppose we have the problem (5−4i)−(7−2i)
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Following the steps above, we simply treat ilike any other variable and perform the indicated
operation (subtraction) by distributing the negative and combining like terms.
(5−4i)−(7−2i)
5−4i−7+2i
−2−2i
Because there is no i2present, we are done.
8.3 Multiplication
Suppose we have the problem (3−2i)(4+i)
Following the steps above, we simply treat ilike any other variable and perform the indicated
operation (multiplication) by FOILing and combining like terms.
(3−2i)(4+i)
12 +3i−8i−2i2
12 −5i−2i2
Next we simply replace any i2with −1.
12 −5i−2i2
12 −5i−2(−1)
12 −5i+2
14 −5i
8.4 Division
Suppose we have the problem 2+3i
1−4i
Following the steps above, we must multiply the numerator (top) and denominator (bottom)
by the conjugate of the denominator (bottom). The conjugate of a complex number is simply
the same complex number, except with the sign in front of the ichanged. Here, the conjugate
of the denominator (bottom) is therefore 1 +4i.
Therefore we multiply as follows: 2+3i
1−4i⋅1+4i
1+4i
Now we simply follow the rest of the steps above for complex number problems. We are
now multiplying two fractions, so we treat ilike any other variable and multiply the two
numerators (tops) and the two denominators (bottoms) by FOILing and combining like
terms. 2+3i
1−4i⋅1+4i
1+4i
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2+8i+3i+12i2
1+4i−4i−16i2
2+11i+12i2
1−16i2
Finally, we simply replace any i2with −1.
2+11i+12i2
1−16i2
2+11i+12(−1)
1−16(−1)
2+11i−12
1+16
−10 +11i
17
Test Number Test Number Test Number
SAT 1 Sec. 3 2 SAT 1 Sec. 3 11 SAT 4 Sec. 3 14
SAT 6 Sec. 3 3 SAT 7 Sec. 3 4 ACT 2005-06 46
ACT 2016-17 29
9 Geometry
9.1 Introduction
There are no pure geometry problems on the SAT or ACT. Instead, all geometry problems
can and should be turned into algebra problems easily and quickly. Only a few basic
geometry concepts are needed to do this. If you find yourself delving into advanced geometry
concepts or doing a proof, you are not approaching the problem correctly. Instead, turn it
into an algebra problem.
Generally this can be done by labeling the unknown quantity that you want to know with
a variable (a new variable if the problem already has other variables), using a geometry
property to turn the problem into an algebra equation, and then solving for the variable
you want to know.
9.2 Triangle Angles
All the angles in a triangle add to be 180 degrees. This is a useful property for turn a
problem into an algebra equation.
The largest angle is always across from the longest side; the smallest angle is always across
from the shortest side.
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9.3 Complimentary and Supplementary Angles
If two angles are complimentary, they add to be 90 degrees. If two angles are supplementary,
they add to be 180 degrees. In either situation, an algebraic equation should be set up
immediately.
xy
Here, the angles are supplementary. Therefore angles should immediately be labeled and
the algebraic equation x+y=180 can be used.
x
y
Here, the angles are complimentary. Therefore, angles should immediately labeled and the
algebraic equation x+y=90 can be used.
9.4 Vertical Angles
When two (or more) lines intersect, the angles across the intersection point from each other
are equal (congruent). Any vertical angles should immediately be marked as equal.
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9.5 Parallel Lines
Parallel lines are not of any use in and of themselves on the SAT or ACT. The only reason
parallel lines are useful is because alternate interior angles are congruent. Therefore, when-
ever you are told that lines are parallel (whether you are told directly, it is indicated in a
diagram, or implied by a particular shape such as a trapezoid or parallelogram), you should
immediately think of alternate interior angles.
Alternate interior angles are the angles that occur on opposite (alternate) sides of the
transversal (the line that cuts through the two parallel lines) but inside (interior ) two
parallel lines.
Using alternate interior angles and vertical angles, several angles can be related to each
other.
Sometimes alternate interior angles are hidden. However, they are still there. To uncover
them, the parallel lines and the transversal should be extended.
9.6 Similar Triangles
If two triangles are similar, it means two things.
•All the corresponding angles are congruent.
•All the corresponding sides are proportional.
A8B
4
C
6
50○
v
60○
D
y
E
2
F
z
60○
70○
x
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First we solve for vand x:
50 +60 +v=180
v=70
60 +70 +x=180
x=50
This means the triangles are similar because all the angles are the same. Specifically,
△ABC ∼△DEF .
Therefore, the sides are proportional and we can solve for yand z.
4
2=8
y
4y=16
y=4
4
2=6
z
4z=12
z=3
9.7 Special Types of Triangles
Equilateral Triangles
Equilateral triangles have the following properties:
•All three side lengths are the same.
•All three angle measures are the same (60○).
The formula for the area of an equilateral triangle is useful to know.
A=√3
4s2
where sis the length of a side of the equilateral triangle. If you don’t remember the formula
for the area of an equilateral triangle, it can be derived easily using the 30-60-90 special
triangle (see below).
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Isosceles Triangles
Isosceles triangles have the following properties:
•Two side lengths are the same.
•Two angle measures are the same.
It is important to remember that if you know one of these items is true, you know that the
other is true as well.
30-69-90 Triangles
A 30-60-90 triangle is a right triangle in which the other two angles are 30○and 60○. 30-60-
90 triangles are particularly useful because they result from splitting an equilateral triangle
in half. The formula for 30-60-90 triangles is given to you on the SAT.
To use the 30-60-90 formula, side lengths should only be compared to the side across from
30○.
To get the hypotenuse, multiply the side across from 30○by 2. To get the side across from
60○, multiply the side across from 30○by √3.
If you have the hypotenuse, divide it by 2 to get the side across from 30○. If you have the
side across from 60○, divide it by √3 to get the side across from 30○.
3√36
3
30○
60○
45-45-90 Triangles
A 45-45-90 triangle is a right triangle in which the other two angles are equal. Thus, it is an
isosceles triangle. Therefore, the other two angles are each 45○. The formula for 45-45-90
triangles is given to you on the SAT.
Each of the two legs will be equal in length. If you have the length of a leg, simply multiply
it by √2 to get the length of the hypotenuse. If you have the length of the hypotenuse,
simply divide it by √2 to get the length of the legs.
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5
5√2
5
45○
45○
Pythagorean Theorem and Pythagorean Triples
You must know the Pythagorean theorem.
a2+b2=c2
where aand bare the legs of a right triangle and cis the hypotenuse of the right triangle.
Pythagorean triples are three whole numbers that work together in the Pythagorean the-
orem. The most common Pythagorean triple is 3 −4−5. The second most common
Pythagorean triple is 5 −12 −13. Any multiple of a Pythagorean triples also works. There-
fore, 9 −12 −15 is also a Pythagorean triple. Recognizing Pythagorean triples on the test
can save a lot of time.
9.8 Degrees in a Polygon
The number of degrees in a polygon with nsides is given by
(n−2)180
If the polygon is regular this means
•All side lengths are the same.
•All angle measures are the same.
Therefore, in a regular polygon, each interior angle is
(n−2)180
n
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Test Number Test Number Test Number
SAT 1 Sec. 3 17 SAT 1 Sec. 4 3 SAT 2 Sec. 3 18
SAT 2 Sec. 3 19 SAT 2 Sec. 4 30 SAT 3 Sec. 3 11
SAT 3 Sec. 3 18 SAT 4 Sec. 3 16 SAT 5 Sec. 3 20
SAT 5 Sec. 4 19 SAT 5 Sec. 4 26 SAT 6 Sec. 3 18
SAT 6 Sec. 4 16 SAT 7 Sec. 3 17 SAT 7 Sec. 4 14
SAT 8 Sec. 3 4 SAT 8 Sec. 4 5 ACT 2005-06 6
ACT 2005-06 12 ACT 2005-06 13 ACT 2005-06 20
ACT 2005-06 23 ACT 2005-06 24 ACT 2005-06 25
ACT 2005-06 33 ACT 2005-06 36 ACT 2005-06 37
ACT 2005-06 58 ACT 2008-09 9 ACT 2008-09 14
ACT 2008-09 15 ACT 2008-09 16 ACT 2008-09 18
ACT 2008-09 35 ACT 2008-09 39 ACT 2008-09 44
ACT 2008-09 45 ACT 2008-09 46 ACT 2008-09 58
ACT 2011-12 5 ACT 2011-12 10 ACT 2011-12 12
ACT 2011-12 16 ACT 2011-12 20 ACT 2011-12 25
ACT 2011-12 34 ACT 2011-12 39 ACT 2011-12 43
ACT 2011-12 44 ACT 2011-12 59 ACT 2014-15 11
ACT 2014-15 17 ACT 2014-15 27 ACT 2014-15 30
ACT 2014-15 31 ACT 2014-15 38 ACT 2014-15 40
ACT 2014-15 47 ACT 2016-17 12 ACT 2016-17 13
ACT 2016-17 17 ACT 2016-17 20 ACT 2016-17 26
ACT 2016-17 27 ACT 2016-17 40 ACT 2016-17 41
ACT 2016-17 44 ACT 2016-17 50 ACT 2005-06 54
SAT 8 Sec. 4 10 ACT 2008-09 27 ACT 2014-15 23
ACT 2014-15 44
10 Trigonometry
10.1 Trigonometric Ratios
You probably remember the term “SOH-CAH-TOA” for remembering trigonometric ratios
in a right triangle. It is important to be able to set up these ratios.
adjacent
opposite
hypotenuse
θ
sin(θ)=opposite
hypotenuse cos(θ)=adjacent
hypotenuse tan(θ)=opposite
adjacent
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10.2 Complimentary Angle Theorem
If two angles are complimentary, then the sine of the first angle and the cosine of the second
angle are equal. For example:
sin(37)=cos(53)
This is easily shown in a right triangle.
a
b
c
θ
90 −θ
If we compare sine of θand cosine of θ−90, we will see that they are equal.
sin(θ)=b
ccos(90 −θ)=b
c
10.3 Law of Sines and Law of Cosines
On the ACT, the Law of Sines and Law of Cosines are occasionally tested. Theses laws are
used to solve non-right triangles. The formulas will be given to you and you only have to
set them up – not solve them. Therefore, you only need to know which law to choose.
•If you know a side across from an angle, use Law of Sines.
•If not, use Law of Cosines.
Test Number Test Number Test Number
SAT 1 Sec. 3 19 SAT 3 Sec. 3 20 SAT 3 Sec. 4 23
SAT 4 Sec. 3 17 SAT 7 Sec. 3 18 SAT 7 Sec. 4 36
SAT 8 Sec. 4 36 ACT 2005-06 34 ACT 2005-06 45
ACT 2005-06 50 ACT 2005-06 52 ACT 2008-09 25
ACT 2008-09 32 ACT 2008-09 55 ACT 2008-09 56
ACT 2011-12 38 ACT 2011-12 41 ACT 2011-12 53
ACT 2014-15 22 ACT 2014-15 42 ACT 2014-15 56
ACT 2014-15 57 ACT 2016-17 30 ACT 2016-17 57
ACT 2016-17 60 SAT 5 Sec. 4 20
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11 Circles
11.1 Basic Formulas
On the ACT, you need to know the formulas for the circumferences (perimeter) of a circle
and for the area of a circle. These formulas are given to you on the SAT.
C=2πr
A=πr2
If you are given a point on a circle, the best way to approach the problem is to draw a line
from the point to the x axis to create a triangle.
You will also need to know the general graphing form of a circle:
(x−h)2+(y−k)2=r2
Where (h, k)is the center of the circle and r(not r2) is the radius of the circle. Thus the
following circle would have its center at (−5,1)and would have a radius of 9.
(x+5)2+(y−1)2=81
If you are given an equation with both an x2term and a y2term, this is a circle. To put
it in graphing form, you need to complete the square twice: once for xand once for y.
Completing the square twice below would result in the same equation of the circle above in
graphing form.
x2+y2+10x−2y=55
11.2 Arc Length
Arc length is simply a portion of the circumference (perimeter) of a circle. It is determined
by setting up a ratio between the arc length and the total circumference (perimeter) on the
one hand, and the central angle and the total number of degrees (360○) on the other hand.
arc length
circumference =central angle
360○
If we want to find the length of arc
>
AB:
O
A
B
12
135○
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x
2π12 =135○
360○
360x=3240π
x=9π
11.3 Sector Area
Sector area is simply a portion of the area of a circle. To determine the sector area, we
simply set up a ratio between the sector area and the total area of the circle on the one
hand, and the central angle and the total number of degrees (360○) on the other hand.
sector area
area =central angle
360○
If we want to find the area of ÂAOB:
O
A
B
6
30○
x
π62=30○
360○
360x=1080π
x=3π
11.4 Central Angles and Inscribed Angles
A central angle is an angle with its vertex at the center of a circle. An inscribed angle is
an angle with its vertex on the edge of the circle. If a central angle and an inscribed angle
point to the same arc, then the central angle will be twice the inscribed angle.
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O
A
B
C
80○
40○
Here, ∠BOA is a central angle and ∠BCA is an inscribed angle. Therefore, ∠BOA is twice
∠BCA.
11.5 Central Angle Measure
If you are asked to find the central angle measure, you simply need to use the ratios described
above for arc length or sector area (depending on what you are given) and then solve for
the central angle measure.
11.6 Tangent Lines
A line that is tangent to a circle is a line that touches the circle at exactly one point. In
addition, The tangent line will be perpendicular to the radius from the center of the circle
to the point of tangency. Therefore, the angle should immediately be marked as being 90○.
O
A
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Test Number Test Number Test Number
SAT 1 Sec. 4 24 SAT 2 Sec. 4 24 SAT 2 Sec. 4 36
SAT 3 Sec. 4 34 SAT 4 Sec. 4 24 SAT 4 Sec. 4 36
SAT 5 Sec. 3 2 SAT 5 Sec. 4 29 SAT 5 Sec. 4 36
SAT 6 Sec. 3 20 SAT 6 Sec. 4 27 SAT 7 Sec. 4 29
SAT 7 Sec. 4 34 SAT 8 Sec. 3 9 SAT 8 Sec. 3 20
ACT 2005-06 26 ACT 2005-06 31 ACT 2005-06 38
ACT 2005-06 60 ACT 2008-09 30 ACT 2008-09 54
ACT 2011-12 15 ACT 2011-12 37 ACT 2011-12 54
ACT 2011-12 55 ACT 2014-15 36 ACT 2014-15 37
ACT 2014-15 45 ACT 2014-15 48 ACT 2016-17 14
ACT 2014-15 18
12 Statistics and Probability
Many students have not had any exposure to statistics and feel overwhelmed by the statistics
questions on the SAT and the ACT. While there are statistical concepts on the SAT and
ACT, they are all cursory and easy to master.
12.1 Counting Problems
If asked to determine how many possibilities there are in a scenario, simply draw a line
for each decision that must be made. Then write the number of options there are for each
slot/decision. Finally, multiply all of these together.
If someone must choose an outfit from 4 pairs of shoes, 5 pairs of pants, and 8 shirts, how
many possibilities are there?
4⋅5⋅8
160
12.2 Distributions
A distribution is simply a visual representation of how data is spread out, such as with a
histogram (bar chart).
Normal distribution A normal distribution is one that looks like this. It is symmetrical
across its center. The ends that trail off on either side are known as “tails.”
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Skew Skew occurs when one of the tails of a distribution is pulled out or elongated. If
the right tail is pulled out or elongated, it is “right skew.” If the left tail is pulled out or
elongated, it is “left skew.” Skew is about an overall trend in the data, not individual data
points.
Left skew
Right skew
Outliers Outliers are similar to skew in that they represent the data stretching out in one
direction. But it is about individual data points lying outside the general trend of the data.
In the following graph, 9 is an outlier.
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12345678910
0
1
2
3
4
5
12.3 Measures of Center
Measures of center are various ways to determining what the center or middle of a group of
data is.
Mean The mean of data is just the average of all the data points. That is, it can be
calculated by adding all the data points together and dividing by the number of data points.
The key thing to remember about the mean is that it is not resistant to outliers or skew.
That is, the mean will move towards any outliers or towards the skewed tail of a distribution.
Therefore if there is skew or if there are outliers, the mean is a bad measure of center to
use.
Median The median is the middle data point when the data points have been arranged
from least to greatest. Students frequently forget to arrange the data points from least to
greatest first. If there is an even number of data points, the median is the average of the
middle two data points. The key thing to know about the median is that it is resistant to
outliers and skew. Therefore, if there is skew or if there are outliers, it is a better measure
of center than the mean.
Mode The mode is simply the most frequently occurring data point. If there are multiple
data points that are repeated the same number of times, then each of them is the mode.
The mode is not a particularly useful measure of center and should only be used if you are
specifically asked for it.
Expected Value Expected value is computed by multiplying the value of each outcome
by its probability and then adding all of these numbers.
Prize Probability
$0 0.3
$1 0.4
$5 0.2
$10 0.1
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Here the expected value would be $0(0.3)+$1(.4)+$5(.2)+$10(.1)=$0.00+$0.40+$1.00+$1.00=$2.40
Getting the Mean and Median from a Table or Graph If you are presented with a
table or chart with data, finding the mean and the median is easy if you know what you’re
looking for.
•To find the median, you just need to find the middle item in the already organized
data.
•To find the mean, you need to determine what would be in the numerator (top) and
denominator (bottom) of your fraction and then use the table to get this data.
Number of Pets Students
0 12
1 10
2 7
3 3
4 2
5 1
Total 35
There are 35 students, therefore the 18th student will be the middle student. There are
12 students with 0 pets and then there are 10 students with 1 pet, which means we have
passed over 22 students. Therefore, the 18th student will be in the group with 1 pet and
the median number of pets will be 1.
To determine the mean, we need to figure out what we’re looking for. If we’re looking for
the mean number of pets per student, the fraction will look like this
total number of pets
number students
We know that there are 35 students. To determine the total number of pets, we simply
multiply each number of pets by the number of students in that category.
0(12)+1(10)+2(7)+3(3)+4(2)+5(1)
35 =46
35 =1.314
12.4 Measures of Spread
Measures of spread are various ways of determining how spread out or how clumped together
data is.
Range Range is simply the largest data point (the max) minus the smallest data point
(the min). Thus it is easy to calculate. However, it is not particularly useful since it only
compares two of the data points.
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Inter Quartile Range (IQR) The inter quartile range (IQR) is subtracting the first
quartile (Q1) from the third quartile (Q3)
Q3−Q1
The first quartile (Q1) is the median of the minimum and the median. That is, it is the
middle value between the minimum and the median.
The third quartile (Q3) is the median of the median and the maximum. That is, it is the
middle value between the median and the maximum.
Standard Deviation or Variance Standard deviation and variance are more technical
measures of the spread of data. You will not have to calculate standard deviation or variance
on the SAT or the ACT. However, you can think of the standard deviation and variance
as being the average distance of the data points from the center of the data. Thus a large
standard deviation or variance means the data is more spread out; a small standard deviation
or variance means the data is more closely clumped around the center of the data.
12.5 Regression and Scatterplots
Scatterplots are plots of data that has two variables, such as hours spent studying for the
SAT and score on the SAT.
01234
0
1
2
3
4
5
6
7
8
9
Each point in the scatterplot is an actual data point that represents an observation.
The line passing through the scatterplot is known as the line of best fit. It describes the
data and can be used to predict what one variable will be if the other variable is known.
The residual is the distance between an actual data point and what the line of best fit
predicts for the yvariable.
There are several terms that can be used to describe a scatterplot.
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•Data is linear if it generally fits into a straight line. Data is nonlinear if it is generally
curved.
•A scatterplot has strong association or correlation if it falls close to the line of best
fit. It has weak association or correlation if it is more spread out from the line of best
fit.
•A scatterplot has a positive association if the slop of the line of best fit is positive. It
has a negative association if the slope of the line of best fit is negative.
12.6 Basic Statistical Concepts
There are a few statistical concepts that you need to have a basic understanding of.
Sampling Bias Sampling bias occurs when the people you choose to question are not
truly random. That is, if the people you choose to question are not truly random, then
there is bias.
For example, if you want to gather statistics about a city and stand on a street corner and
ask questions of everyone who walks by, this may appear random at first. But it is not
actually random because only certain segments of the city population may pass by that
street corner.
Survey Bias Survey bias occurs when the way that you ask questions in your survey
influences how people respond to the questions.
Inference Inference is the concept that the statistics you gather cannot apply beyond the
group that you drew your sample from. For example, if you do a random survey of people
at your school, you cannot use that data to draw inferences about everyone in your city
because the two populations (people at your school and people in your city) are different.
12.7 Probability
Probability of an event Ais defined as follows:
P(A)=number of ways to get what you want
total number of possibilities
“And” Probability The probability of two independent events both occurring is as fol-
lows:
P(A and B)=P(A)⋅P(B)
The probability of flipping a Heads on a coin and rolling a 6 on a die is as follows:
P(H and 6)=P(H)⋅P(6)=1
2⋅1
6=1
12
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“Or” Probability The probability of either one event or another event occurring is as
follows:
P(A or B)=P(A)+P(B)−P(A and B)
The probability of flipping a Heads on a coin or rolling a 6 on a die is as follows:
P(H or 6)=P(H)+P(6)−P(H and 6)=1
2+1
6−1
12 =7
12
12.8 Table Data
When data is presented in a table and the question asks the probability of choosing a
certain item from the table, careful attention must be given to the numerator (top) and the
denominator (bottom) of the fraction.
Has pet No pet Total
Boy 24 16 40
Girl 12 18 30
Total 36 34 70
These questions must be read carefully because subtle wording can change the meaning of
the question significantly. The trick is to focus carefully on what the denominator (bottom)
and numerator (top) will be.
What is the probability that someone chosen at random has a pet? 36
70
What is the probability that someone chosen at random is a girl with no pet? 18
30
What is the probability that a boy chosen has a pet? 24
40
Given that someone chosen has a pet, what is the probability that they are a girl? 12
36
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Test Number Test Number Test Number
SAT 1 Sec. 4 5 SAT 1 Sec. 4 12 SAT 1 Sec. 4 13
SAT 1 Sec. 4 14 SAT 1 Sec. 4 21 SAT 1 Sec. 4 27
SAT 2 Sec. 4 13 SAT 2 Sec. 4 14 SAT 2 Sec. 4 16
SAT 2 Sec. 4 18 SAT 2 Sec. 4 19 SAT 2 Sec. 4 20
SAT 3 Sec. 4 2 SAT 3 Sec. 4 15 SAT 3 Sec. 4 20
SAT 3 Sec. 4 29 SAT 3 Sec. 4 32 SAT 4 Sec. 4 7
SAT 4 Sec. 4 9 SAT 4 Sec. 4 10 SAT 4 Sec. 4 21
SAT 4 Sec. 4 23 SAT 4 Sec. 4 29 SAT 5 Sec. 4 14
SAT 5 Sec. 4 15 SAT 5 Sec. 4 22 SAT 5 Sec. 4 25
SAT 5 Sec. 4 27 SAT 6 Sec. 3 10 SAT 6 Sec. 4 7
SAT 6 Sec. 4 8 SAT 6 Sec. 4 12 SAT 6 Sec. 4 21
SAT 6 Sec. 4 22 SAT 6 Sec. 4 36 SAT 7 Sec. 4 1
SAT 7 Sec. 4 4 SAT 7 Sec. 4 7 SAT 7 Sec. 4 8
SAT 7 Sec. 4 18 SAT 7 Sec. 4 21 SAT 7 Sec. 4 22
SAT 7 Sec. 4 37 SAT 7 Sec. 4 38 SAT 8 Sec. 4 3
SAT 8 Sec. 4 4 SAT 8 Sec. 4 16 SAT 8 Sec. 4 17
SAT 8 Sec. 4 18 SAT 8 Sec. 4 24 SAT 8 Sec. 4 26
SAT 8 Sec. 4 28 ACT 2005-06 14 ACT 2005-06 30
ACT 2008-09 7 ACT 2008-09 24 ACT 2008-09 28
ACT 2008-09 41 ACT 2011-12 14 ACT 2011-12 28
ACT 2011-12 33 ACT 2011-12 47 ACT 2014-15 4
ACT 2014-15 12 ACT 2014-15 32 ACT 2014-15 43
ACT 2016-17 1 ACT 2016-17 2 ACT 2016-17 10
ACT 2016-17 31 ACT 2016-17 37 ACT 2016-17 47
ACT 2016-17 55 ACT 2016-17 59 SAT 1 Sec. 4 1
SAT 1 Sec. 4 7 SAT 1 Sec. 4 33 SAT 2 Sec. 4 27
SAT 3 Sec. 4 1 SAT 3 Sec. 4 3 SAT 3 Sec. 4 35
SAT 4 Sec. 4 11 SAT 5 Sec. 4 1 SAT 6 Sec. 4 3
SAT 7 Sec. 4 13 ACT 2011-12 51
For SAT or ACT tutoring, contact Matt Boutte.
meb@mattboutte.com — www.mattboutte.com/tutor
