SAT MATH SECTION: CHEAT SHEET FORMULAS, RULES, AND DEFINITIONS
Note: important formulas included at beginning of every test not included below

ALGEBRA:
ZERO PRODUCT PROPERTY: If ab = 0, then a = 0, b = 0, or a = b = 0. SYSTEMS OF EQUATIONS:

One solution	Zero solutions	Infinite solutions	
If you get a single x or y value, (i.e. x=5 or y=2} you have one solution.	If you get a statement that is never true (i.e. 5=9 or 0=4}, you have no solutions.	If you get two values that always equal each other (i.e. 3x=3x or 5=5}, you have infinite solutions.	
Comparing equations: You can alternatively put EVERY equation in slope intercept form that
is part of your system of linear equations. Then look at each of the y=mx+b forms and compare the two slopes (i.e. the "m's") and y-intercepts (the "b's").	
One solution	Zero solutions	Infinite solutions	
Different slopes (intercepts don't matter)

On a graph this is two lines intersecting at a single point.	Same slopes (parallel lines), different y-intercepts

On a graph this is two parallel lines.	Same slope, same intercept
i.e. identical equations

On a graph, this is the sa me line (i.e. both equations you have are actually identical but not "simplified" the same).	

When you solve down a system of equations, one of three outcomes is possible as you run the algebra and create a final, simplified equation:

TIP: Look for shortcuts / use elimination on systems of equations questions to get what the question asks for rather than solve down for each individual variable. I.e. if the question asks for x+y, try to use elimination/adding or subtracting the whole equations to end up with x+y straightaway.

FOIL AND FACTORING:
•	A monomial is a single product such as 4x, 7x3,  or 8n2 .
•	A binomial has two elements added together such as 4x + 3 or Sn3 + 3n .
•	A polynomial has multiple elements added together such as Sn3 + 3n2 + 7n + 2 or Sn3 + 3n2 +7n + 2 or 5x2 + 2x + 4  5x 2  + 2x + 4.

•	Difference of Squares: The product of the difference (a - b) and the sum ( a + b )  is equal to a squared minus b squared,   ( a - b)( a + b)  = a2 - b2
•	Square of a Sum: ( a + b) 2 = a2 + 2ab + b 2
•	Square of a Difference: ( a - b)2  = a2 - 2ab + b2

SLOPE FORMULA: For points ( x1,y 1)  and (x2 ,y2 ) , m = Yz -Yi .
X z - x 1
STANDARD FORM SLOPE: If Ax + By = C, slope is equal to -
B

SLOPE-INTERCEPT FORM: y = mx + b
Where m is the slope of the line, and b is the y-intercept of the line at point (0, b).

MIDPOINT FORMULA: The midpoint of two coordinate points ( xv y1) and ( x 2 , y2 ) is:
(x1 + xz )  (yl + Yz ) )

(	2	'	2

( X1 + Xz ) CY1 + Yz ))
(	2	I	2

DISTANCE FORMULA: Given two points, ( xv y1) and ( x 2 , y2 ), the distance between them is:
d =	(X z -x1 ) 2 + (Yz -Y1) 2 d = .J ( x2 - X1 ) 2 + (Yz - Y1)2

SPEED AND RATES DISTANCE FORMULA: d = r t
WORK FORMULA: w = r t
COMBINED WORK FORMULA: w = (r1 + r2 )t
QUADRATICS AND POLYNOMIALS: Important- know how to find the vertex of ANY parabola !
•	VERTEX FORM OF A QUADRATIC: the vertex form of a parabola is: f ( x ) = a ( x -h) 2 +k
o	The vertex of the parabola in this form is ( h, k)
o	When a is positive, the parabola opens upwards, and the minimum is ( h, k).
o	When a is negative, the parabola opens downwards, and the minimum is ( h, k).
o	Alternate form for parabolas with horizontal axis (RARE, unlikely to appear) x =
a( y - k) 2 + h, positive a opens right-ward, negative a opens leftward

•	FACTORED FORM: The factored form of a polynomial usually takes the form:
f ( x)  = a( x - n)( x - m)
o	When a is positive, the parabola opens upwards.
o	When a is negative, the parabola opens downwards.
o	The graph crosses the x-axis at each point when a factored piece equals zero (the zeros). For example, at x=n, x-n equals 0, so the graph crosses through the point (n, O} on the x-axis.

In the case of a quadratic, which will typically have two roots (i.e. two factored pieces that include an x), the midpoint of 2 zeros is the x-value of the vertex. In the example above, the zeros are at x=n and x=m. So the x value of its vertex is the average of n and m.

•	STANDARD FORM: the standard form of the parabola has the general form: f ( x)  =
ax2 + bx + c f (x) = ax2 +bx + c
o	_ _!_ is the x-value of the vertex. The y-value of the vertex can be found by
2a
plugging in this value for x and solving for y (or f (x)). Memorize this!
o	Assuming a vertically oriented parabola, the vertex is always either the maximum or the minimum of the graph. (remember max/min is always the y­ value!)
o	When a is positive, the parabola opens upwards.
o	When a is negative, the parabola opens downwards.
o	The sum of the two roots is _ !:    (not necessary to know, but helpful)
a

o	The product of the two roots is	(not necessary to know, but occasionally
a
helpful)

•	QUADRATIC FORMULA·. x-_ -b ±	b2 -4ac	Memon.ze th'1s. Also, we recommend you
2a
program it into your calculator. (See Tl 84 programs for download in the online course)

•	FINDING SOLUTIONS USING THE DISCRIMINANT: Given that j (x) = ax 2 + bx + c , the
discriminant is defined as b2 - 4ac (the piece under the root in the quadratic equation):
o	When this value is positive, there are two real roots o	When this value is 0, there is one real root
o	
EXPONENT  RULES	RADICAL RULES	
Zero Power Rule: a0 = 1	b
Fractional exponent conversion:  Vi =a-;	
Negative exponent rule: a-b = 1
ab	Product of radicals: .ef;efb = ef;;/;	
   Power of a power: ( ir = i c 			

When this value is negative, there are no real roots but two imaginary roots RADICAL AND EXPONENT RULES:

Product of powers: ( abY = ab+c	
Power of a product: ( abY = acbe	
Power of a quotient: a: = (!!....J
c	c	
b
Quotient of powers: !!.c_ = ab-c
a	

CIRCLE THEOREMS:
•	Circle Equation: ( x -h)2 +( y -k )2  = r 2   with center point at ( h, k )

Angle to Arc Length Equations: Central angles are proportional to the arc length they intercept, so you can create a proportion that equates part over whole equal to part over whole:

central angle measure

=	arc  length

total measure of  angle jn a drcle	total  drcumference

DEGREES VERSION OF EQUATION:
•	More specifically, this means if a circle has a central angle a, radius r, and arc length L:

a0	L
=
360°	2Jrr

RADIANS VERSION OF EQUATION:

•	For any arc on a circle with a central angle (in radians) of (}, where r is the radius of the circle, the length L of the arc is given by:
()	L
-= -- which simplifies to a short handy equation that's a good idea to memorize: L = ()r .
2Jr	2Jrr

Angle to Sector Area Equations: Central angles are proportional to the area of the sector they include, so you can create a proportion that equates part over whole equal to part over whole:
central angle measure	sector area
= ------

total measure of  angle jn a drcle

total circle area

DEGREES VERSION OF EQUATION:
•	For any sector on a circle with a central angle (in degrees) of a, where r is the radius of the circle, and the area is A:

RADIANS VERSION OF EQUATION:

•	For any sector in a circle with a central angle (in radians) of 8, where r is the radius of the circle, and the area A of that circle:
()	A

This simplifies to a shorter equation, which may merit memorizing:

1
A = - 8r 2
2

•	Inscribed Angle Theorem: States that an angle (portrayed as b below) inscribed in a circle is equal to half of the central angle (portrayed as a below) that subtends the same arc. Both scenarios below show the central angle theorem:

•	Trigonometry:

op posite	adjacent
SOHCAHTOA: For any angle (} in a right triangle, sin (}  =	cos (}  = ----
hypotenuse '	hypotenuse
and tan (}  =  op posite
adjacent

Passport to Advanced Math:
•	Function Notation: (many of these are unlikely to appear, but you should understand how function notation works)
•	Cf 0 g )C x) means f CgCx))
•	Cf * g )C x) and f gC x) mean f Cx) * g Cx )
•	(L) Cx) means f ( x )
g	g( x )

Problem Solving and Data Analysis: General Statistics:
•	Average: Sum of terms divided by number of terms
•	Median: The numerical middle value of a set of terms
•	Mode: The most common term in a set of terms

Graphs/Plots:
•	Line of Best Fit: A line that best represents the data on a scatterplot. It can be used to estimate the value of points not on the plot itself. TIP: pluck points and use a calculator program to find y=mx+b on scatterplots!

GRAPH BEHAVIOR:
Types of graphs:

Graph of an e.  011emia] ftmorio11:,
Stimda:i:d Form: !( x ) =ax
Example: y = 2..-

Graph of a square woe:
Standard Form: a.{;
Example (Parent Gr:aph) y = ..}x
·l
,I

-6     -!!i       -J	-

-1

Graph of  a oubic funorion :
3
Vertce-.x For:m.: y =a( x -b) +k
Example (Pare,ut Graph): y = x3

Graph of a parabola (quadratic fWJction):
2
Vertex for:m: f ( x ) =a{ x - h) +k
Example (Parent Gr:aph): y = x2

_,

I •

_,	
-··

Graph of a linear equation:
Slope-Inter:cept Rmn: y = mx + b
Example (Parent for:m): y = x

Graph of  a. log:uithnric fimctlon:
Stamhr:d Form: y = log_, x
Example: y = log10 x

-1

-·

=al

Graph of an absolute "-alue fuuction: Vertex F=: y	x -hi + k
Example (Parent Graph): y = lxl

-2

-1

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•	Horizontal Shift: For ALL functions, if you replace all instances of x in a function with
" ( x - h)"  you'll find that the graph moves "h" units to the right.
•	Vertical Shift: If you replace all instances of y in a function with "(y - k)" you'll find that the graph moves "k" units upward.

END BEHAVIOR:

Degree: Even	Degree: Even	
Leading Coefficient: positive	Leading Coefficient: negative	
End Behavior: f (x ) approaches +oo at both
ends of the graph (upward facing)	End Behavior: f (x ) approaches -oo at both
ends of the graph (downward facing)	
Domain: all reals	Domain: all reals	
Range: all reals ;::: maximum	Range: all reals :::; maximum	
Example: y = x 2	Examp Ie: y = -2x6 + 3xs + 4x	

Degree: odd	Degree: odd	
Leading Coefficient: positive	Leading Coefficient: negative	
End Behavior: At graph left, f (x)	-oo.  At graph right, f (x)	+oo  (upward sloping)	End Behavior: At graph left, f (x)	+oo  . At graph right, f (x)	-oo  (downward  sloping)	
Domain: all reals	Domain: all reals	
Range: all reals	Range: all reals	
Example: j (x) = x3	Example: f ( x ) = -3x5 - 2x	

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