Algebra II: Unit 13: Review
The area of a circle
(A = π r²) π(d²/4)
Binomial Coefficients
(n, r) or nCr or C(n, r) or C^nr nCr = n!/r!(n - r)! = n(n - 1)(n - 2)...(n - r + 1)/r! permutations of n objects taken r at a time; (n, r) = nPr/r!
Circle conic
(x - h)^2 + (y - k)^2 = r^2
The center-radius form of the circle equation
(x - h)^2 + (y - k)^2 = r^2 center; the point (h, k) radius; "r"
Ellipse conic
(x - h)^2/a^2 + (y - k)^2/b^2 = 1 (y - k)^2/a^2 + (x - h)^2/b^2 = 1
Hyperbola conic
(x - h)^2/a^2 - (y - k)^2/b^2 = 1 (y - k)^2/a^2 - (x - h)^2/b^2 = 1
the equation for a parabola
(y - mx - b)^2 / (m^2 +1) = (x - h)^2 + (y - k)^2 focus; (h, k) directrix; y = mx + b
Conjugates
* a + b ➡ a - b * x^2 - 3 ➡ x^2 + 3
Radians
1 radian = 180°/π π radians = 180° To go from radians to degrees: > multiply by 180, divide by π To go from degrees to radians: > multiply by π, divide by 180
Simplified form of radicals requires that:
1. No perfect n^th power factors of a radicand may exist in an n^th degree radical. 2. No fractions may exist in the radicand. 3. No radical may exist in the denominator of a fraction.
The process for determining the sine/cosine of any angleθ
1. Starting from (1,0)(1,0), move along the unit circle in the counterclockwise direction until the angle that is formed between your position, the origin, and the positive x-axis is equal to θ. 2. sin(θ) is equal to the y-coordinate of your point, and cos(θ) is equal to the x-coordinate.
Anti-log
10^n = x n: common log (base 10 log) x: inverse log 10^x; > the base for your antilog is always 10 > x is the number with which you are working example; 2.6452 > 2 = characteristic > 6452 = mantissa > 10^2.6452 = 441.7
The general equation for any conic section
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 A,B,C,D,E and F are constants
combinations
C(n, r) = n!/r!(n - r)!
Radicals
I. (xy)^a = x^ay^a II. (x/y)^a = x^a/y^a III. a√x = x^1/a IV. √x = x^1/2 V. √xy = √x*√y VI. √x/y = √x/√y VII. √x^2 = x
The perimeter of a circle
P = 2πr πd
The Basic Trigonometric ratios
Sine = opp/hyp = a/c Cos = adj/hyp = b/c Tan = opp/adj = a/b
Imaginary number
a complex number that can be written as a real number multiplied by the imaginary unit i, which is defined by its property i2 = −1
Rational Powers
a^(1/n) = n√a a^(m/n) = (n√a)^m or n√(a^m)
The Discriminant
b^2 - 4ac of the quadratic formula b^2 - 4ac > 0; positive | 2 real roots b^2 - 4ac = 0; 1 real root b^2 - 4ac < 0; negative | no real roots
Natural Logarithms
base that is often used is e (Euler's Number) which is about 2.71828 generally written as ln x, loge x > base e is implicit InN = x ↔ N = e^x
Distance Formula
d = √(x2 - x1)^2 + (y2 - y1)^2
Arithmetic Sequence
example; -6, -1, 8, 15, 22 diff = 7 nth term; an = a1 + (n - 1) d
Geometric Sequence
example; 2, 4, 8, 16, 32 diff = *2 nth term; an = ar^n-1 > an = nth term of the sequence > a = first term of the sequence > r = common ratio > n = term number
The Reciprocal functions
general form of a reciprocal function; r(x) = a / (x - h) + k The vertical asymptote of r(x) is x = h The horizontal asymptote of r(x) is y = k csc = hyp/opp = c/a sec = hyp/adj = c/b cot = adj/opp = b/a
Common Logs
log x ↔ log10 x a logarithm is written without a base ; usually means that the base is really 10 logN = x ↔ N = 10^x
Series Notation
n ∑ an k = 1 n = last value an = formula for the terms k = index of summation 1 = first value of n
Permutations formula
n things taken r at a time: P(n, r) =n!/(n-r)! n things taken n at a time: P(n, n) = n! Circular permutation of n things taken r at a time: P^c(n, r)/r = n!/(n - r)!r Circular permutations of n things taken n at a time: P^c(n, n) = (n - 1)!
Factorials
n! = n*(n - 1)*(n - 2) n is a positive integer n! is the product of positive integers ≤ n
Rate
r = d/t > d = rt
The Unit Circle
raduis of 1 x^2 + y^2 = 1
conditional probability
represented by P(A/B) occurs when an event A occurs after event B has occurred
The Pythagorean trigonometric identity
sin2θ + cos2θ = 1 tan2θ + 1 = sec2θ 1 + cot2θ = csc2θ csc(x) = 1/sin(x) sec(x) = 1/cos(x) cot(x) = 1/tan(x) = cos(x)/sin(x) sin(x) = 1/csc(x) cos(x) = 1/sec(x) tan(x) = 1/cot(xx) = sin(x)/cos(x) sin(-t) = -sin(t) cos(-t) = cos(t) tan(-t) = -tan(t)
The Quadratic Formula
x = -b ± √b^2 - 4ac/2a
negative exponents
x^-n = 1/x^n x^n/x^m = x^n-m
Parabola conic
y = a(x - h)^2 + k x = a(y - k)^2 + h
Inverse Variation Equations
y = k/x k = xy
Direct Variation Equations
y = kx k = y/x
Logarithms
y = logb(x) ➡ b^y = x logb(a) = c ↔ b^c = a - b = base - c = exponent - a = argument logaN = x ↔ N = a^x
the "slope-intercept" form:
y=mx+b m = slope b = y-intercept
