Distance between two points
Midpoint between two points
Slope between two points
Average rate of change of a function
Inverse functions
Transformations of graph of f(x)
f(x − h) + k translates h units ”to the right”,
and k units vertically
−f(x) reflects across x−axis; f(−x) across y−
cf(x) dilates by factor of c vertically
f(cx) dilates by factor of horizontally
Polynomials
General Form
Factored Form
Roots
Rational Functions
Definition where f, g are polynomials with
no common factors
Domain all real numbers except where g(x) = 0
x intercepts at zeros of f
vertical asymptotes at zeros of g
Exponents and Logarithms
Exp onential Function
Definition of Logarithm is equivalent to b ^{y}
= x
Properties of Logs
Trigonometric Functions
Of acute angles:
Of any angle: (circle radius r)
Radians and Degrees 180° =π (radians)
Linear equations
General Form: Ax + By =C
Slopeintercept form y = mx + b
Pointslope form
Doubleintercept form
Quadratic equations
Forms: 
General 


Vertex 


Factored 

Vertex 
General 


Vertex 


Factored 

Roots 
General 


Vertex 


Factored 

Translations, Reflections, Dilations y = Asin(B(x −
C)) + D, y = Acos(B(x − C)) + D
Horizontal Translation (phase shift) C units
Vertical Translation (vertical shift) D units
Horizontal Dilation By a factor of (period
becomes )
Vertical Dilation (amplitude) By a factor of A
Reflections −f(x) reflect across y−axis; f(−x) reflect across x−axis
Trig Identities
Definitions
Pythagorean
Opposite Angle
Periodicity
Complements
Reduction
Addition
DoubleAngle
HalfAngle s
ProducttoSum
SumtoProduct
Inverse Trig Functions
Domain restrictions: sin restricts to ,
cos restricts to [0,π ], tan restricts to
for all θ
only for
only for
only for
Vectors
Position(radius) Given and
Unit Components If =< x, y >, then
Magnitude (length) If =< x, y >, then
Unit vector
Scalar Multiplication If =< x, y >, then k
=< kx, ky >
Addition If =< a, b > and
=< c, d > then +
=< a + c, b + d >
Dot Product If =< a, b > and
=< c, d > then ·
= ac + bd
Angle between and
Polar Coordinates
Trigonometric (polar) form of Complex Numbers
z = r(cosθ + i sinθ ) is the trigonometric form of the complex
number a + bi, where
a = r cosθ , b = r sinθ and
zn = r^{n}(cos nθ + i sin nθ ) with z as above
Matrices
Matrix Multiplication If and
, then
Identity Matrix . For any 2 × n matrix A,
Inverse Matrices and
Matrix equations If AX = B, then
Determinants
Polynomial Functions and Equations
Division Algorithm
Let p(x) and d(x) be polynomials, and assume that d(x) is not the zero
polynomial. Then there
are unique polynomials q(x) and R(x) such that p(x) = d(x) · q(x) + R(x), where
the degree of
R is less than the degree of d. R(x) is called the remainder.
Remainder Theorem
When a polynomial f(x) is divided by x − r, the remainder is f(r).
Factor Theorem
Let f(x) be a polynomial. If f(r) = 0, then x − r is a factor of f(x).
Conversely, if x − r is a
factor of f(x), then f(r) = 0.
Linear Factors Theorem
Any polynomial f(x) of degree n can be be ex pressed as a product of n linear
factors, f(x) =
, where each
is a root of f(x), and may be
a real or complex number.
Some roots may be repeated.
Rational Roots Theorem
If , and all the coefficients are integers .
Then any root of the
equation f(x) = 0 must be of the form , where
p is a factor of (the constant term of f ) and
q is a factor of (the leading coefficient of
f), and p and q have no common factors (so
is in
lowest terms ).
Complex Conjugate Roots Theorem
Let f(x) be a polynomial whose coefficients are real numbers. If a + bi is a
root of f(x) = 0,
and b ≠ 0, then a − bi is also a root of f(x) = 0).
Linear and Quadratic Factors Theorem
Any polynomial with real roots can be factored into linear and quadratic factors
with real coefficients.
Partial Fractions Decompositions
In the following, we always assume the degree of p(x) is smaller than the degree
on the bottom:
,where ax^{2} +
bx + c is irreducible
,where ax^{2} +
bx + c
is irreducible