ALGEBRA

Basic Identities                                                                                                                                    Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real number

Additive Identity

a + 0 = a

Additive Inverse

a + (-a) = 0

Associative of Addition

(a + b) + c = a + (b + c)

Commutative of Addition

a + b = b + a

Definition of Subtraction

a - b = a + (-b)

Closure Property of Multiplication

Product (or quotient if denominator 0) of 2 reals equals a real number

Multiplicative Identity

a * 1 = a

Multiplicative Inverse

a * (1/a) = 1     (a  0)

(Multiplication times 0)

a * 0 = 0

Associative of Multiplication

(a * b) * c = a * (b * c)

Commutative of Multiplication

a * b = b * a

Distributive Law

a(b + c) = ab + ac

Definition of Division

a / b = a(1/b)                                                                                                                                                             Conic Sections

Conic Sections

Circle	Ellipse (h)	Parabola (h)	Hyperbola (h)
Definition: A conic section is the intersection of a plane and a cone.	Ellipse (v)	Parabola (v)	Hyperbola (v)

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.

Point	Line	Double Line

The General Equation for a Conic Section: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The type of section can be found from the sign of: B² - 4AC

If B2 - 4AC is...	then the curve is a...
< 0	ellipse, circle, point or no curve.
= 0	parabola, 2 parallel lines, 1 line or no curve.
> 0	hyperbola or 2 intersecting lines.

The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).

	Circle	Ellipse	Parabola	Hyperbola
Equation (horiz. vertex):	x2 + y2 = r2	x2 / a2 + y2/ b2 = 1	4px = y2	x2 / a2 - y2 / b2= 1
Equations of Asymptotes:				y = ± (b/a)x
Equation (vert. vertex):	x2 + y2 = r2	y2 / a2 + x2/ b2 = 1	4py = x2	y2 / a2 - x2 / b2= 1
Equations of Asymptotes:				x = ± (b/a)y
Variables:	r = circle radius	a = major radius (= 1/2 length major axis) b = minor radius (= 1/2 length minor axis) c = distance center to focus	p = distance from vertex to focus (or directrix)	a = 1/2 length major axis b = 1/2 length minor axis c = distance center to focus
Eccentricity:	0	c/a	1	c/a
Relation to Focus:	p = 0	a2 - b2 = c2	p = p	a2 + b2 = c2
Definition: is the locus of all points which meet the condition...	distance to the origin is constant	sum of distances to each focus is constant	distance to focus = distance to directrix	difference between distances to each foci is constant
Related Topics:	Geometry section on Circles

Polynomial Identities (a+b) 2 = a 2 + 2ab + b 2 (a+b)(c+d) = ac + ad + bc + bd a 2 - b 2 = (a+b)(a-b) (Difference of squares) a 3  b 3 = (a  b)(a 2 ab + b 2) (Sum and Difference of Cubes) x 2 + (a+b)x + AB = (x + a)(x + b)if ax 2 + bx + c = 0 then x = ( -b (b 2 - 4ac) ) / 2a (Quadratic Formula) Exponential Identities Powers x a x b = x (a + b)x a y a = (xy) a (x a) b = x (ab) x (a/b) = bth root of (x a) = ( bth (x) ) a x (-a) = 1 / x a x (a - b) = x a / x b Logarithms y = logb(x) if and only if x=b ylogb(1) = 0 logb(b) = 1 logb(x*y) = logb(x) + logb(y) logb(x/y) = logb(x) - logb(y) logb(x n) = n logb(x)logb(x) = logb(c) * logc(x) = logc(x) / logc(b) Algebraic Graphs          Conic Sections (see also Conic Sections) Point x^2 + y^2 = 0 Circle x^2 + y^2 = r^2 Ellipse x^2 / a^2 + y^2 / b^2= 1 Ellipse x^2 / b^2 + y^2 / a^2= 1 Hyperbola x^2 / a^2 - y^2 / b^2= 1 Parabola 4px = y^2 Parabola 4py = x^2 Hyperbola y^2 / a^2 - x^2 / b^2= 1 For any of the above with a center at (j, k) instead of (0,0), replace eachx term with (x-j) and each y term with (y-k) to get the desired equation