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
Vertical Dilation (amplitude) By a factor of A
Reflections −f(x) reflect across y−axis; f(−x) reflect across x−axis
Inverse Trig Functions Domain restrictions: sin restricts to ,
cos restricts to [0,π ], tan restricts to
for all θ
Position(radius) Given and
Unit Components If =< x, y >, then
Magnitude (length) If =< x, y >, then
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
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 = rn(cos nθ + i sin nθ ) with z as above
Matrices Matrix Multiplication If and
Identity Matrix . For any 2 × n matrix A,
Inverse Matrices and
Matrix equations If AX = B, then
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.
When a polynomial f(x) is divided by x − r, the remainder is f(r).
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:
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