Percent Change
% Change = (Change)/(Original Value) ?100
Compounded % Change
Original * (1+rate)^(# of changes)
Direct Variation
x?/y?=x?/y?
Indirect Variation
x?y?=x?y?
Law of Sines
Sin(A)/a=Sin(B)/b=Sin(C)/c
Law of Cosines
c�=a�+b�-2abCos(C)
Polar Coordinates
(r,?)
x=rcos(?)
y=rsin(?)
tan(?)=x/y
r�=x�+y�
Even Functions
f(x)=f(-x), reflects over y-axis.
If f and g are even functions...
f x g is even, f+g is even, f-g is even
Odd Functions
f(-x)=-f(x), reflects over origin
Group Problem Formula
Total=group?+group?+neither-both
?
intersection
?
Union
Logarithm Product Rule
Log(xy)=logx +logy
Logarithm Quotient Rule
log(x/y)=logx-logy
Arithmetic Sequence Equation
An=A?+d(n-1)
Arithmetic Sequence Sum
Sum= n�(An+A1)/2
Geometric Sequence Equation
An=A?�r^(n-1)
Geometric Sequence Sum
Sum=A?(1-r?)/(1-r)
Infinite Geometric Sum
Sum=A?/(1-r)
Sum of the Roots of a Polynomial Equation
#NAME?
Product of the roots of a polynomial equation
(constant)/(first coefficient) If first exponent is odd, it's negative, if even its positive
Circles Divided in Sectors, Area and Arc Length
Area= .5?�r�
Arc length = r?
If you divide a polynomial by (x-c)...
the remainder is f(c)
Discriminant
Discriminant= b�-4ac
Positive, 2 real roots
0, 2 equal roots
Negative, 2 imaginary roots
Distance from a line ax+by+c=0 to a point(X,Y)
|aX+bY+c|/?(a�+b�)
Distance from a point (X,Y,Z) to a plane ax+by+cz+d=0
|aX+bY+cZ+d|/?(a�+b�+c�)
To see whether a polynomial rises or falls going to the left and right...
just take the leading polynomial to +/- infinity and multiply by the coefficient.
Rational Zero Test
Assuming integer coefficients, the possible rational zeros of f consist of the combinations of the (factors of constant term)/(factors of leading coefficient)
Descartes's Rule of Signs (for polynomials):
1. The number of positive zeros of f is equal to or less than by an even integer to the number of variations in sign of f(x).
2. The number of negative zeros of f is equal to or less than by an even integer to the number of variations in sign of f(-x).
f(x)=4x^3-6x^2+3x-3
3 or 1 positive roots (4,-6,3,-3)->3 sign changes, 0 negative roots (-4,-6,-3,-3)
Periodic functions:
f(x+p)=f(x)
y=c*f(x) still has period p
y=f(cx) has period p/c
Sinusoidals: y(t)=A�sin(wt+?)+d
y(t)=Tan(bt)
A=amplitude
w=2?/period
Middle line: y=d
For Tan: ?/b=period
Sum of angles of a regular polygon
180 � (n-2)
Triangles
A=�bh=�abSin(?)
Equilateral Area=s��?(3)/4
45/45/90=x/x/x?2
30/60/90=x/x?3/2x
Parallelograms
Area=bh=abSin(?)
Rhombus
A parallelogram with equal length sides. Area=bh=s�Sin(?)
Trapezoids
Area= h�(b?+b?)/2
Cubes
Internal diagonal = s?3
Face diagonal =s?2
Cones
Volume= 1/3�?r�h
Surface Area=?�r�l+?r� l=length from outside base to tip
Spheres
Volume=4/3�?�r�
Surface Area= 4?r�
Pyramid
Volume = 1/3�bh
Parabolas
y=a(x-h)�+k
Axis of symmetry @x=h
Vertex @ (h,k)
a determines amplitude and direction
y=ax�+bx+c
Axis of symmetry @ -b/2a
Sum of roots=-b/a
Circles
(x-h)�+(y-k)�=r�
Center @ (h,k)
Radius=r
Origin: x�+y�=r�
Ellipses
(x-h)�/a�+(y-k)�/b�=1
Center @ (h,k)
Width of ellipse=2a
Height of ellipse=2b
(Distance from center to foci)�=a�-b�; a>b
Origin: x�/a�+y�/b�=1
Hyperbolas
(x-h)�/a�-(y-k)�/b�=1
Center @ (h,k)
If y is second value (-(y-k)�), then it opens left/right
(Distance from center to focus)�=a�+b�
Asymptotes: y=�(b/a)�X
x^2/4-y^2/16=1, asymp@y=4/2=2x,-2x
Finding Asymptotes
Line x=a is a vertical asymptote, if it makes denominator equal to zero.
Line y=b is a horizontal asymptote, if y=b as x goes to +- infinity
If there's a slant asymptote, try long dividing nume by denom to find it.
EX: (2x^2+1)/x ->-> 2x +1/x -> slant asymptote @ y=2x
Permutation
nPr=n!/(n-r)!
Combination
nCr=n!/(n-r)!r! (??) = 4C1
Sin(2x)
2sin(x)cos(x)
Cos(2x)
Cos�(x)-Sin�(x)
Sin(A+B)
SinACosB+CosASinB
Sin(A-B)
SinACosB-CosASinB
Cos(A+B)
CosACosB-SinASinB
Cos(A-B)
CosACosB+SinASinB
Heron's Formula (Triangle Area)
Area=?(s)(s-a)(s-b)(s-c))
s=(a+b+c)/2
Unit Vectors
When finding the unit vector of a vector, divide the x and y values by the magnitude of the vector.
Vectors at the Origin
If a vector has an initial point at origin it's in standard position, it will just give coordinates.
Matrices
Determinant=across diagonal to right - across diagonal to left
If same order, you can add by adding corresponding entries.
If multiplying, just multiply every entry.
Parametric Equations
just substitute/combine equations to solve for y or f(x)
Binomial Probability
If a coin is tossed 10 times, what is the probability of exactly 7 heads.
10 trials, 50% probability, 7 successes binompdf(10,.5,7)=.1172
If a coin is tossed 6 times, what is the probability of up to 3 heads if you are three times more likely to land on heads?
6 trials, 75% probability, up to 3 successes binomCdf(6,.75,3)