Outline of Math 1650

Math 1650 Outline of the course

(a.k.a. study sheet for final exam)

§ 1:: Graphs of equations

Cartesian (x-y) plane
graph = all points that satisfy the equation
How to graph?
plot points (and fill in gaps)
use x- and y-intercepts
use symmetry
y-axis: (a,b) on graph, so is (-a,b)
x-axis: (a,b) on graph, so is (a,-b)
origin: (a,b) on graph, so is (-a,-b)
Eqn for circle: (x-h)²+(y-k)² = r²

§ 2:: Lines and their slopes

slope= rise over run = (change in y-value)/(corresponding change in x value)
slope-intercept: y = mx+b
point-slope: [(y-y₀)/(x-x₀)] = m
two-point: [(y-y₀)/(x-x₀)] = [(y₁-y₀)/(x₁-x₀)]
same slope: lines are parallel (do not meet)
lines are perpendicular: slopes are negative reciprocals

§ 3:: Functions

function = rule which assigns to each input exactly one output
inputs = domain; outputs = range/image; f:AŽB
y=f(x) : `y equals f of x' : y equals the value assigned to x by the function f
f,x,y, etc. are all placeholders; any other symbols are `just as good'
`implied' domain of f: all numbers for which f(x) makes sense

§ 4:: Graphs of functions

y=f(x) is an equation; graph the equation!
graph = all pairs (x,f(x)) where x is in the domain of f
all functions have graphs, but not all graphs `have' functions
function takes only one value at a point; vertical line test
symmetry (for functions)
y-axis: even function, f(-x) = f(x)
x-axis: XXXXXX
origin: odd function, f(-x) = -f(x)
increasing on an interval: if x > y , then f(x) > f(y)
decreasing on an interval: if x > y , then f(x) < f(y)
constant

§ 5:: Translations and combinations

graph of y=f(x)
shift to right by c; y=f(x-c)
shift to left by c; y=f(x+c)
shift down by c; y=f(x)-c
shift up by c; y=f(x)+c
y=af(x) ; stretch graph by factor of a
reflect graph along axes
y-axis: y=f(-x)
x-axis: y=-f(x)
combining functions: combine the outputs of two functions f,g
f+g, f-g, fg, f/g
composition: output of one function is input of the next
f followed by g = g°f; g°f(x) = g(f(x)) = g of f of x

§ 6:: Inverse functions

Idea: find a function that undoes f
find a function g so that g(f(x)) = x for every x
magic: f undoes g ! Usual notation: g = f^-1
Problem: not every function has an inverse.
need g to be a function; so f cannot take the same value twice.
horizontal line test
Graph of inverse: if (a,b) on graph of f, then (b,a) is on graph of f^-1
graph of f^-1 is graph of f, reflected across line y=x

Chapter 2: Polynomials

§ 1:: Quadratic functions

monomial = axⁿ
polynomial = bunch of monomials = a_nxⁿ+a_n-1x^n-1+ź+a₁x+a₀ = f(x)
a_n š 0, then n=degree of f
deg=0: constant fcn; deg=1: linear fcn; deg=2: quadratic fcn
f(x) = ax²+bx+c ; graph = parabola
Standard form: ax²+bx+c = a(x-h)²+k
complete the square: ax²+bx+c = a(x²+[b/a]x)+c
add half of [b/a], squared, inside parentheses
(and subtract corresponding amount outside!)
standard form Žgraph:
x² to (x-h)² (shift left/right) to
a(x-h)² (stetch/reflect) to a(x-h)²+k (shift up/down)
lowest/highest point of graph = (h,k) = vertex of parabola
axis of symmetry: vertical line x=h

§ 2:: General properties of polynomials

f(x) = a_nxⁿ+a_n-1x^n-1+ź+a₁x+a₀; domain = everything
graph has no gaps, hole, or jumps (f is continuous)
can draw graph without lifting up writing implement
graph has no corners - no sudden turns; graph is smooth
behavior at `ends':
n even, a_n > 0 : high/high
n even, a_n < 0 : low/low
n odd, a_n > 0 : low/high
n odd, a_n < 0 : high/low
root (zero) of f ; f(a) = 0 ; graph of f hits x-axis at a
if f(a) = 0, then f(x) = (x-a)g(x)
nth degree polynomial can have at most n roots
nth degree polynomial can turn around at most (n-1) times
consequence of continuity: intermediate value theorem
if a polynomial takes on two values c and d, then
it also takes on every value in between
application: `finding' roots: if f(a) < 0 and f(b) > 0, then
there is a root of f somewhere between a and b

§ 3:: Polynomial division

root a of f Ť factor (x-a) of f(x)
reason: polynomial (long) division
f(x) = (x-a)g(x) + b ; a=root, then b=0
polynomial = (divisor)(quotient) + remainder
degree of remainder is less than degree of divisor
synthetic division: fast method to divide by (x-a)

§ 4:: Real zeros of polynomial functions

f(x) = a_nxⁿ+ź+a₁x+a₀
`Counting' zeros of f
Descartes' rule of signs
p=number of positive roots of f, q=number of negative roots of f
(number of changes in sign of coeffs of f) - p is ł 0 and even
(number of changes in sign of coeffs of f(-x)) - q is ł 0 and even
Rational roots test
If a_n,ź,a₀ are all integers, a_n š 0, and r = p/q is a rational root of f, then
q divides a_n evenly and p divides a₀ evenly.
backwards: can show roots of a polynomial can't be rational.
Bounding roots: start with a_n > 0.
If c > 0 and the bottom row after synthetic division of f using c are all ł 0,
then no root of f is bigger than c.
If c < 0 and the bottom alternates sign, then no root of f is smaller than c.

§ 5:: Complex numbers

Some polynomials have no roots, e.g., f(x) = x²+1 . Invent some!
i = [Ö(-1)], pretend i behaves like a real number
complex numbers: standard form z = a+bi ; addition, subtraction, multiplication
division: complex conjugate [`z] = a-bi
z·[`z] = a²+b² ( real!) ; z₁/z₂ = (z₁·[`(z₂)])/(z₂·[`(z₂)])
a,b > 0, then [Ö(-a)]·[Ö(-b)] = -[Ö((-a)(-b))] (unfortunately)

§ 6:: The fundamental theorem of algebra

FTA: Every polynomial f(x) (with coefficients in C or R) has a
complex root r ; f(r)=0
Every polynomial factors into linear factors (with coefficients in C)
FTA says it can be done; it doesn't tell you how to do it!
Conjugate pairs; if coeffs of f are real, and r is a root, then so is [`r]
(x-r)(x-[`r]) has real coeffs
every polynomial with real coeffs factors in linear and irreducible quadratic factors.

§ 7:: Rational functions

rational function = quotient of polynomials
p(x) = a_nxⁿ+ź+a₀, q(x) = b_mx^m+ź+b₀ ; f(x) = p(x)/q(x)
domain = where q(x) š 0
vertical asymptote x = a : f(x)ŽąĽ as xŽa
horizontal asymptote: f(x)Ža as xŽąĽ
n < m : horiz. asymp. y = 0
n = m : horiz. asymp. y = a_n/b_m
n > m : no horiz. asymp.
Slant asymptote: n = m+1 . Asymp. = linear part from division of p(x) by q(x)

Chapter 3: Exponential and logarithmic functions

§ 1:: Exponential functions

exponential expressions a^b
Rules: a^b+c = a^b a^c ; a^bc = (a^b)^c ; (ab)^c = a^c b^c
Function f(x) = a^x ; approximate f(x) by f(rational number close to x)
Domain: R ; range: (0,Ľ) ; horiz. asymp. y = 0
Graphs:
a > 10 < a < 1
Most natural base: e = 2.718281829459045.....
Exponential growth: compound interest
P=initial amount, r=interest rate, compounded n times/year
A(t) = P·(1+r/n)^nt
nŽĽ, continuous compounding : A(t) = Pe^rt
Radioactive decay: half-life = k (A(k) = A(0)/2)
A(t) = A(0)(1/2)^t/k

§ 2:: Logarithmic functions

log_a x = the number you raise a to to get x
log_a x is the inverse of a^x
a = base of the logarithm
log_a (a^x) = x, all x ; a^{log_a x} = x, all x > 0
Domain: all x > 0 ; range: all x
Graph = reflection of graph of a^x across line y = x
vertical asymptote: x = 0
natural logarithm: log_e x = lnx

§ 3:: Properties of logarithms

logarithms undo exponentials; properties are `reverse' of exponentials
log_a (bc) = log_a b + log_a c ; log_a (b^c) = clog_a b
(log_b c)(log_a b) = log_a (b^{log_b c}) = log_a c; so log_b c = [(log_a c)/(log_a b)]
E.g., a = e : log_b c = [lnc/lnb]

§ 4:: Exponential and logarithmic equations

exponential equation: take logs!
a^blah = bleh, then (blah)lna = ln(bleh)
(2^x-3)(2^x-7) = 0, then 2^x=3 or 2^x=7
logarithmic equation: combine into a single log (one on each side?) and
exponentiate both sides
Application: doubling time
time for investment to triple at interest rate of r compounded n times/year:
solve (1+r/n)^nt = 3

Chapter 4: Trigonometry

§ 1:: Degrees and radians

angle: vertex, initial side, terminal side
standard position: vertex=origin, initial side=(positive) x-axis
coterminal angles: same terminal side
measuring size of an angle
one full circle = 360 degrees
one full circle = 2p radians
radian measure = length of arc in circle of radius 1 swept out by the angle
acute, obtuse, reflex angles
A+B = p/2 ; complementary angles (acute)
A+B = p ; supplementary angles (acute,obtuse)

§ 2:: Trigonometric functions

In standard form, terminal side of an angle (t) meets circle of radius 1 in a point (x,y)
x = cost = cosine of t
y = sint = sine of t
[1/x] = [1/cost] = sect = secant of t
[1/y] = [1/sint] = csct = cosecant of t
[y/y] = [sint/cost] = tant = tangent of t
[x/y] = [cost/sint] = cott = cotangent of t
Examples:
sin(p/4) = cos(p/4) = Ö2/2
sin(p/6) = 1/2 ; cos(p/6) = Ö3/2
sin(p/3) = Ö3/2 ; cos(p/3) = 1/2
sin(p/2) = 1 ; cos(p/2) = 0
sin(0) = 0 ; cos(0) = 1
Domain of sint, cost : all t
Range: [-1,1]
point on circle corresp. to t+2p is same as point for t
sin(t+2p) = sint ; cos(t+2p) = cost
sint and cost are periodic
symmetry:
cost , sect are even functions
sint, csct, tant, cott are odd functions

x²+y² = 1 (unit circle): sin² t+cos² t = 1

§ 3:: Right angle trigonometry

Right triangle:
sin(q) = a/c = (opposite)/(hypotenuse)
cos(q) = b/c = (adjacent)/(hypotenuse)
tan(q) = a/b = (opposite)/(adjacent)
``SOHCAHTOA''
Complementary angle = the `other' angle in a right triangle
sin(p/2-q) = cos(q) , cos(p/2-q) = sin(q)
tan(p/2-q) = cot(q) , cot(p/2-q) = tan(q)
sec(p/2-q) = csc(q) , csc(p/2-q) = sec(q)
( i.e., function(``co-angle'') = ``co-function''(angle) )

§ 4:: Trig functions for any angle

Right angle trig really applies only to acute angles; extend the ideas!
angle q, point (x,y) on terminal side
r = [Ö(x²+y²)]
sin(q) = y/r cos(q) = x/r tan(q) = y/x

reference angle = acute angle that terminal side makes with x-axis
(trig fcn)(q) = (trig fcn)(ref. angle), except possibly for a change in sign:

quadrant I
(x < 0,y > 0)
sin(q) > 0
cos(q) > 0
tan(q) > 0

quadrant II
(x > 0,y > 0)
sin(q) > 0
cos(q) < 0
tan(q) < 0

quadrant III
(x < 0,y < 0)
sin(q) < 0
cos(q) < 0
tan(q) > 0

quadrant IV
(x > 0,y < 0)
sin(q) < 0
cos(q) > 0
tan(q) < 0

§ 5:: Graphs of sine, cosine

sin(q) = y-value of the points (counter-clockwise) on the unit circle, starting with 0
cos(q) = x-value of the points (counter-clockwise) on the unit circle, starting with 1
Graph: note x-intercepts, y-intercept, maximum and minimum; draw a smooth curve
Transformations: y = asin(bx)
vertical stretch by factor of a; amplitude is |a|
amplitude = how far trig function wanders from its `center'
horizontal compression by factor of b; period is 2p/|b|
Translations: just like before
y = cos(x-a) ; translation to right by a
y = cos(x) +a ; translation up by a

§ 6:: Graphs of other trig functions

tanx, cotx, secx cscx
Transformations (same)
Products: sinx, cosx bounce between -1 and 1; so, for example:
y = xsinx bounces between y = x and y = -x
y = e^-xcosx bounces between y = e^-x and y = -e^-x (`damped' trig function)

§ 7:: Inverse trig functions

Inverses of trig functions? No! Not one-to-one. Solution: make them one-to-one!
f(x) = sinx , -p/2 Ł x Ł p/2 , is one-to-one
inverse is called arcsinx = angle (between -p/2 and p/2) whose sine is x
sin(arcsinx) = x ; arcsin(sinx) = x if x is between -p/2 and p/2
f(x) = cosx , 0 Ł x Ł p , is one-to-one
inverse is called arccosx = angle (between 0 and p) whose cosine is x
cos(arccosx) = x ; arccos(cosx) = x if x is between 0 and p
f(x) = tanx , -p/2 < x < p/2 , is one-to-one
inverse is called arctanx = angle (between -p/2 and p/2) whose tangent is x
tan(arctanx) = x ; arctan(tanx) = x if x is between -p/2 and p/2
Graphs: take appropriate piece fo trig function, and flip it across the line y = x
cos(arcsinx) = (cosine of angle whose sine is x) = [Ö(1-x²)] ; etc.

Chapter 5: Analytic trigonometry

§ 1:: Using fundamental identities

Fundamental identities:
Reciprocal: cscx = [1/sinx] secx = [1/cosx] cotx = [1/tanx]

Quotient: tanx = [sinx/cosx] cotx = [cosx/sinx]

Pythagorean: sin² x +cos² x = 1 tan² x +1 = sec² x cot² x+1 = csc² x

Complementarity: sin(p/2 - x) = cos(x) tan(p/2 - x) = cot(x)
sec(p/2 - x) = csc(x) cos(p/2 - x) = sin(x)
cot(p/2 - x) = tan(x) csc(p/2 - x) = sec(x)

Symmetry: cos(-x) = cosx sec(-x) = secx
sin(-x) = -sinx csc(-x) = -cscx tan(-x) = -tanx cot(-x) = -cotx

Trig substitution: rewrite expression in x by `pretending' x=trig function
[Ö(a²-x²)] ; write x = asinq, then [Ö(a²-x²)] = acosq
[Ö(a²+x²)] ; write x = atanq, then [Ö(a²+x²)] = asecq
[Ö(x²-a²)] ; write x = asecq, then [Ö(x²-a²)] = ąatanq

§ 2:: Checking trig identities

Basic differences: an identity is supposed to be true for every value of x;
an equation is solved for the correct values of x
Basic idea: use identities that we already know (like the list above)
convert things to sines and cosines
play with the two sides of the identity
add 0 ! multply and divide by the same expression!
Examples: cscx-sinx = [1/secxtanx]
[(tanx+tany)/(1-tanx tany)] = [(cotx+coty)/(cotxcoty-1)]

§ 3:: Solving trig equations

Idea: just like exponential and logarithmic equations; try to rewrite as
(single trig function) = (single value)
Wrinkles:
Polynomials: 2cos²x+3cosx+1 = 0 ; (2cosx+1)(cosx+1) = 0
2cosx+1 = 0 or cosx+1 = 0
Trig identities: tanx+secx=4 ; tanx = 4-secx ; square both sides
tan² x (= sec² x-1) = 16-8secx+sec² x = ....

Problem: `ghost solutions' = solutions which `appear' only after manipulating
equation; (stupid) Ex: sinx = 1 and (sinx)² = 1 have different sets of solutions!

§ 4:: Angle sum and difference formulas

sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB-cosAsinB
cos(A+B) = cosAcosB-sinAsinB
cos(A-B) = cosAcosB+sinAsinB

Note: it is easy to derive any three formulas from the remaining one, using even/odd
and complementarity formulas.

tan(A+B)= [(sin(A+B))/(cos(A+B))] = [(tanA+tanB)/(1-tanAtanB)]
tan(A-B)= [(sin(A-B))/(cos(A-B))] = [(tanA-tanB)/(1+tanAtanB)]

Some uses: complex multiplication! (side trip to part of Section 6.5)

(a+bi)(c+di) = (ac-bd)+(ad+bc)i
pretend z=a+bi=cosA+isinA, z^˘=c+di=cosB+isinB, then this reads
z·z^˘=(cosAcosB-sinAsinB)+(sinAcosB+cosAsinB)i
=cos(A+B) = isin(A+B)
Problem: z=a+bi=cosA+isinA. then a²+b²=sin² A+cos² A=1 (every time)
Solution: think z=a+bi=r(cosA+isinA), where
r²=a²+b²;. i.e, think zŤ (a,b) (in plane) = point in plane at distance r
from origin, making angle A with (positive) x-axis
i.e., think z=a+bi Ť (a,b) Ť (distance,angle) ; polar coordinates
then complex multiplication multiplies distance and adds angles:
(r(cosA+isinA))(r^˘(cosB+isinB)) = (rr^˘)(cos(A+B)+isin(A+B))

Another use: find values of trig functions at new angles:
Example: 105^° = 60^° +45^° (i.e. 7p/12 = p/3+p/4), so
cos(7p/12) = cos(p/3+p/4) = cos(p/3)cos(p/4)-sin(p/3)sin(p/4) =
(1/2)(Ö2/2)-(Ö3/2)(Ö2/2) = (Ö2-Ö6)/4

§ 5:: Multiple angle, product-to-sum formulas

Double angle formulas: set A = B in formulas above!
sin(2A) = sin(A+A) = 2sinAcosA
cos(2A) = cos(A+A) = cos² A-sin² a = 2cos² a-1 = 1-2sin² A
Triple angle? sin(3A)=sin(2A+A)=....

sin² x=(1-cos(2x))/2 , cos² x=(1+cos(2x))/2 ; these give

Half-angle formulas:
sin(x/2) = Ö{(1-cosx)/2} ; cos(x/2) = Ö{(1+cosx)/2}
tan(x/2) = [sinx/(1+cosx)] = [(1-cosx)/sinx]
Product-to-sum formulas:
sin(A+B)+sin(A-B) = 2sinAcosB, so
sinAcosB= [1/2](sin(A+B)+sin(A-B)) Simlarly,
cosAcosB = [1/2](cos(A+B)+cos(A-B)), and
sinAsinB = [1/2](cos(A-B)-cos(A+B))
Sum-to-product formulas:
set A+B = x, A-B = y (solve: A=[(x+y)/2], B=[(x-y)/2]), plug in above!
sinx+siny = 2sin[(x+y)/2]cos[(x-y)/2]
cosx+cosy = 2cos[(x+y)/2]cos[(x-y)/2]
cosx-cosy = 2sin[(x+y)/2]sin[(x-y)/2]

OK, so what's the point? It's alot easier to remember what these formulas (in the previous two sections) say if you remember where they come from. We built all of these formulas up from one formula; cos(A-B) = ..... . If you remember how each follows one from the other, then you don't `have to' remember (most of) the formulas!

Chapter 6: Additional Topics in Trigonometry

§ 5:: DeMoivre's formula

The translation:
z = a+bi (complex number) = (a,b) (point in plane) =
(r,q) (distance from origin and angle with (positive) x-axis)
yields an interpretation of complex multiplication:
z=a+bi = r(cosq+isinq) , w=c+di = s(cosf+isinf) , then
zw = rs(cos(q+f)+isin(q+f) . setting z = w yields
zⁿ=rⁿ(cos(nq)+isin(nq) (DeMoivre's formula)
setting r=1: (cosq+isinq)ⁿ = cos(nq)+isin(nq)
Similarly, z/w = (r/s)(cos((q-f)+isin(q-f)

Think backwards; solve zⁿ = w
Need: rⁿ = s , cos(nq) = cosf , sin(nq) = sinf ; i.e.
r = s^1/n , nq = f+2kp, i.e., q = f/n = 2kp/n
So zⁿ = w has n distinct solutions, coming from k = 0,1,ź,n-1

§ 1:: The Law of Sines

Idea: triangle congruence theorems (SSS,SAS,SSA,ASA,AAS) tell us that three pieces of information about a triangle (except the 3 angles) essentially are enough to determine the triangle (hence determine all 6 pieces of information). But how to determine those extra 3 pieces of info?
Law of Sines: In a triangle with side lengths a,b,c , angles A,B,C
(A opposite a, etc.),

[sinA/a] = [sinB/b] = [sinC/c]

So if you know the sizes of an angle and the
side opposite, and any other piece
of info, you (almost) can calculate everything else.
(AAS,SSA,ASA)
Glitch: knowing sinB (for example) doesn't tell you B
there is an acute and an obtuse angle with that value of sine
SSA: Knowing A (an acute angle) and b, if:
a < bsinA , there is no triangle with that data
bsinA < a < b , there are two triangles; B is acute/obtuse
a = bsinA or b < a , there is one triangle; B is right/acute
SSA: If A is obtuse, then if:
a Ł b , there is no triangle
a > b , there is one triangle

§ 2:: The Law of Cosines

As above, but use for SSS,SAS:
c² = a²+b²-2abcosC (to calculate c from a,b,C)

Rewrite: cosC = [((a²+b²)-c²)/2ab] (to get C from a,b,c)
Note: cosC does determine C, since 0 Ł C Ł p

Heron's formula: calculate area of triangle from length of the three sides

Area = [1/2]bh = [1/2]absinC
= [1/4][Ö((2ab)²-((a²+b²)-c²)²)]
= [Ö(s(s-a)(s-b)(s-c))] , where s = [(a+b+c)/2]

Chapter 10: Topics in Analytic Geometry

§ 1:: Lines

Line L: ax+by+c = 0 , slope m=-a/b
Angle of inclination = angle q that upward side of L makes with the rightward
part of the x-axis
tanq = m , but since q lies between 0 and p,
it is NOT always the same as arctanm !
Angle between two lines: L₁, L₂, with angles of incl. q₁, q₂
angle q between line is one of q₁-q₂ , p-(q₁-q₂) , q₂-q₁ , p-(q₂-q₁)
(whichever is between 0 and p/2 !)
Better approach: L₁, L₂ , with slopes m₁, m₂

tanq = |[(m₁-m₂)/(1+m₁ m₂)]| = M

(from the angle difference formula for tangent!)
So q IS equal to arctan(M)
(Note: this says lines are perpendicular (q = p/2) if 1+m₁ m₂ = 0)
Distance from a point P = (x₀,y₀) to a line L = (ax+by+c = 0):
idea: find line through P perpendicular to L, see where they hit, measure distance

distance = [(|ax₀+by₀+c|)/([Ö(a²+b²)])]

Application: area of a triangle =
(1/2)·(length of a side)·(distance from third corner to the side)

§ 2:: Parabolas

Old view: graph of y = ax²+bx+c
standard form: (y-k) = a(x-h)²
vertex: (h,k) ; axis of symmetry: x = h

New view: the collection of points which are the same distance from a
certain point (focus)
and a certain line (directrix)

Basic idea: vertex and focus lie of axis of symmetry, directrix is perpendicular to it
New standard form(s)
(directrix horizontal): (x-h)² = 4p(y-k) , where
(h,k) = vertex, (h,k+p) = focus (y = k-p) = directrix
(directrix vertical): (y-k)² = 4p(x-h) , where
(h,k) = vertex, (h+p,k) = focus (x = h-p) = directrix
Rule of thumb: the bigger |p| is, the further apart vertex and focus are,
the fatter the parabola is

Important geometric property of parabola:
shine a light from focus, bounce off parabola, light ends up traveling parallel to
the axis of symmetry.
`bounce off': incoming angle equals outgoing angle
wait: angle of a line with a curve?
Use: line tangent to the parabola for measuring angles
How to find it? Use this geometric property!
point of tangency; second point on line lies the same distance away from focus,
down the axis of symmetry
OR... tangent line = points same distance from focus and the
point (A) on the directrix
slope: perpendicular to line from focus to A

§ 3:: Ellipses

Geometric description: two foci P,Q (assume on horiz. or vert. line (for simplicity)
Ellipse = points with (distance to P)+(distance to Q) = certain constant

midpoint between foci = center = (h.k)
distance center to focus = c
line through foci = major axis ;
line perpend. to major axis, thru center = minor axis
distance center to ellipse along horiz. axis = a (a > c)
distance center to ellipse along vertical axis = b
Standard form: [((x-h)²)/(a²)]+[((y-k)²)/(b²)] = 1
b²+c² = a² (major axis horiz.) or a²+c² = b² (major axis vert.)
Measure of roundness: eccentricity e = c/a
c < a, so 0 Ł e < 1 e = 0, then c = 0 so foci=center; ellipse=circle
small e, round ellipse; large e, flat ellipse

§ 4:: Hyperbolas

Geometric description: two foci P,Q (same assumptions, for simplicity)
Hyperbola = point with |(distance to P) - (distance to Q)| = constant

(note: constant must be less than distance P to Q)
midpoint between foci = center = (h.k)
distance center to focus = c
line through foci = transverse axis ;
line perpend. to transv. axis, thru center = conjugate axis
distance center to hyperbola along transverse axis = a (a < c)
points of hyperbola along transverse axis = vertices
Standard form: [((x-h)²)/(a²)]-[((y-k)²)/(b²)] = 1 (tr. axis horiz)
or [((y-k)²)/(a²)]-[((x-h)²)/(b²)] = 1 (tr. axis vert.)

with a²+b² = c²
Hyperbolas have two branches (= pieces);
the four ends become asymptotic to two lines = asymptotes
(tr. axis horiz. (other case is similar)):
[((x-h)²)/(a²)] = [((y-k)²)/(b²)], i.e. (y-k) = ą[b/a](x-h)

Eccentricity: e = c/a = [Ö(1+(b/a)²)] (e > 1)
e close to 1, then b small, so hyperbola closer to transv. axis
e big, then b big, so hyperbola closer to conj. axis

File translated from T_EX by T_TH, version 0.9.