Entering special characters: http://www.alecjacobson.com/weblog/?p=443
Rational number is a fraction of integers. Mnemonic "ratio" + "nal".
Function: Precisely 1 output for each x of the domain.
Rational function is fraction with polonomial numerator and denominator.
Domain is all numbers except where denominator is 0.
This is BEFORE canceling anything from numerator and denominator.
X intercepts: Zeros of the numerator (*,0)
Y intercept: Just solve for x=0 (0,*) (zero or one intercept, depending on domain)
Vertical asymptotes: Check limits for denominator→0 AFTER cancellation.
Horizontal asymptotes: Limit for x→∞, x→-∞
Your line can not cross or touch asymptotes.
GRAPHING FUNCTIONS GENERALLY
g(x) = f(a ∙ x) then (0,y) remains the same but slopes *= a. Horizontal shrink/stretch.
g(x) = f(-x) then f,g mirrors across Y axis. Previous for a=-1
g(x) = f(a + x) then graph is shifted -a.
g(x) = a ∙ f(x) then (*,0)s remain the same but slopes *= a. Vertical stretch/shrink.
g(x) = -f(x) then (*,0) remains the same but slopes inverted. f,g mirrors across X axis. Previous for a=-1
g(x) = a + f(x) then shift graph up a
g(f(x)) = x then g and f are mirrors across x=y line [i.e. inverse functions]
(identity composition)
f(x) = f(-x) Symmetric about Y axis == ODD function
Symmetric (non-function) about X axis: If (x,y) then always (x,-y)
Limits → ∞ and -∞
Ignore lower polynomial terms. Just goes with the highest order x.
There is no limit → ∞ and -∞ for any plain trig function.
Good idea to draw asymptotes as dashed lines since they are not really in fn.
Quadratic Forumula when b=0: ±√(-c/a)
Will be imaginary if signs of a and c the same: 2x² + 3 OR -2x² - 3.
If signs of a and c differ then we have Difference of 2 Squares:
Factors into two () conjugates: (x + √(-c/a)) (x - √(-c/a))
When get 0's for intercept purposes, ignore i results because only real numbers can be plotted on a regular X/Y graph.
Can identify this situation right up front by calculating the quadratic formula discriminant: b² - 4ac
Diff between POINT-SLOPE and POINT-INTERCEPT is
SLOPE-INTERCEPT has mx m x + b (b = y-intercept)
POINT-SLOPE has m(x ± c) yo + m(x - xo)
Graphing lines:
y = yo + m(x - xo) POINT-SLOPE
(don't waste time converting to slope-intercept unless inconvenient
to graph the supplied (xo, yo) point)
Function test
Vertical line test. No multiple Y values for single X value.
[Can have multiple X values for a single Y value.]
Absolute value functions
Solve '|fn| =' poly by doing both
fn = poly
fn = -(poly)
and MUST TEST BOTH ANSWERS because some don't work.
There are no solutions to: |anything| = -(anything)
Perpendicular lines: slopes are negative reciprocals, sum of slopes = -1
Quadratic graphs incorporate a vertical parabola = U shape
To graph, rewrite in perfect square format.
y = +/-(x - h)² + k more generally y = +/-a(x - h)² + k
+/-: opens up/down
a: determines how fast it narrows
h: Vertex moves horizontally
k: Vertex moves vertically
VERTEX = (h,k)
X intercepts of quadratic equation: Factor as much as possible
Radical factoring: Can always factor values under a radical, including to
fractions. E.g.: √(8/3) = √8 / √3.
Can not do this with sums/differences.
Square roots are always positive.
Completing the Square: (If you just need X intercepts, just use quadratic formula!)
x = ±√[(b/2a)² - c/a] - b/2a
0. Start with ax² + bx + c = 0
1. Factor out a to leave x² bare: x² + b/a x + c/a = 0
2. + and - (b/2a)²: [x² + b/a x + (b/2a)²] + c/a - (b/2a)² = 0
3. AUTOMATICALLY: (x + b/2a)² = (b/2a)² - c/a
4. Take √: x + b/2a = ±√[(b/2a)² - c/a]
VERTICAL PARABOLA
VERTEX = (B, C)
d = + opens up; - opens down; value is how fast goes vertical (how thin the U)
N.b. algebra usually concerned with vertex, which is different from classical concentration on directrix and (single) focus.
HORIZONTAL PARABOLA swaps x and y.
Quadratic Formula:
Discriminant = value under the radical
Sign of discriminant determines number of X intercepts.
+: 2 real X intercepts
0: 1 X intercept
-: 2 imaginary X intercepts (no real X intercepts)
Hyperbola (has 2 foci). More like V whereas Parabola like U.
CONSIDERING ONLY THOSE CENTERED ON THE ORIGIN.
For vertical, vertexes at (h, k±√(a)) for vert; (h±√(a), k) for hor.
2 asymptotes with lines of y = k±√(a)/√(b) x for vert;
y = k±√(b)/√(a) x
generally y = k±√(y-denom)/√(x-denom) x
APPLICATION
Lamp reflectors normally hyperbolas. Head lights and flash lights.
Satellite dishes.
Bulbs usually located near that leg's focus
SHAPES NF=Never Functions
Ver. Parabola: ax²... by
Hor. Parabola: ay²... bx
Hor. Hyperbola: (x-h)²/a² - (y-k)²/b² = 1 NF
Ver. Hyperbola: (y-k)²/a² - (x-h)²/b² = 1 NF
Ell.: (x-h)²/a² + (y-k)²/b² = 1 Horizontal NF
(y-k)²/a² + (x-h)²/b² = 1 Vertical NF
Cir. (x-h)² + (y-k)² = r² NF
Ellipse (equation diff from hyperbola only by sign between the 2 terms)
Vertices are the major axis endpoints.
Radial distances are a and b. a is longer radius regardless of x or y.
"semi-major axis" === major-axis-radius, and same for minor
Eccentricity. From 0 to 1: √(1 - b²/a²)
Circle: It is an ellipse with a = b; major axis length = minor axis length
Polynomial: Exponents to variables must be positive integers.
Degree: Highest variable exponent in a polynomial.
Linear, quadratic, cubic, quartic, quintic polynomials
Number of terms: monomials, binomials, trinomials.
Domains always all real numbers.
Graphs always continuous (continuous have no sharp corners)
Number of X intercepts ≤ degree of polynomial.
Ends of graphs always go up or down (never horizontal)
Characteristics determined by degree and sign of leading coefficient:
Even up or down based on ± of leading coefficient
Odd. sign of leading coefficient says whether +x goes up or down; -x goes other way.
TODO??? If your factored =0 (?) equation contains a x^m where m is even, then that x intercept just touches but does not cross the X axis.
Look for difference of 2 squares, like: g² - 81 (where g is an embedded expression)
Can do this recursively on some of results like (g + 9)(g - 9)
Composition: f(g(x)). Just substitute content of g(x) for every x in f(x)'s content.
Solving larger polynomials.
Factor out everything common in all terms FIRST!!! (incl. before any of following tactics).
Similar to difference of 2 squares, look for diff and sum of cubes:
a³ - b³ → (a - b)(a² + ab + b²)
a³ + b³ → (a + b)(a² - ab + b²)
Finally, look for a composition of a quadratic equation: g² + g + c, get x intercepts for g then substitute back to x.
Rational Roots Theorem: (to find all possible real rational roots)
Only possible rational roots are of form [FACTOR-OF-A0]/[FACTOR-OF-An]
Must check all + and - cases.
N.b. DOES NOT TELL US ANYTHING ABOUT NON-RATIONAL ROOTS!
E.g. if either A0 or An is 6, you need to check all (8) of:
6, -6, 3, -3, 2, -2, 1, -1
I.e., minimal checks: 2 if both +/-1, 8 if both non-1 primes.
Factor Theorem: (to find all possible real non-rational roots)
If you find rational root of a, then use a
factor of x-a by dividing original equation by x-a.
Try resulting non-rationals same as you try rationals (from RR Theorem).
(May well result in imaginaries).
When have a fractional rational root a, can just multiply top and bottom to
get rid of fraction, since other side of equation is 0. TODO: RR section?
Descartes Rule of Signs
Number positive and negative real root (rational and non-rational; not imag.).
# Positive roots = How many times does poly in standard form change sign
(+ -2k, down to 0)
# Negative roots = Same thing for f(-x)
Missing terms are fine.
Fundamental Theorem of Algebra consequence.
Polonomial of degree x has x roots, incl. repeats and imaginaries.
Domain of polonomial functions is all numbers.
Can only compute real nth roots if n is odd. TODO: Why?
Can multiply and divide radical expressions as simple as pie.
Can't add radical expressions unless both root-level and discriminant are same.
Rationalize denominator. For single term just multiple top and bottom by it.
If 2 terms then multiply top and bottom by the conjugate.
Mixed powers/roots, like x^(2/3)
Domain allows negatives if DENOMINATOR is odd. TODO: Verify
Range allows negatives if NUMERATOR is odd.
Partial fractions:
Attempt to factor Denom to linear terms.
Say factor to denom terms (DEN_TERM1) and (DEN_TERM2), then
A(TERM2) + B(TERM1) = NUM
regroup so that:
(somethingA)x + (somethingB) = (somethingX)x + (somethingY)
you get: somethingA = something X AND somethingB = something Y
Multiple-occurrence linear terms in denom:
Input denom x(x-1)² → output term denoms: x, (x - 1), (x - 1)²
If can only factor Denom to quadratic terms (incl. just x²), then need to make
one output term of the quadratic form: (Bx + C) / (x²...)
Inverse of polonomial term function is root function. TODO: What??
Exponential and Logarithmic functions/graphs:
Beware radically different for b < 1.
Exponential function ab^x is VERY DIFFERENT from polomonimal term function ax^b.
Property: (x∙y)^b = x^b ∙ y^b (just like multiplicative distribution)
Inverse of exponential function is logarithm X
Domain all numbers except x ≠ 1 ???Verify
(a^b)^c = a^(b∙c). Therefore,
given a^d, can always factor any self-multiple b out of a → (b^c)^d = b^(c∙d)
Range all positive or negative according to sign of a.
Has horizontal asymptote at y=0 (right -1 > b < 1) TODO: WHAT???
(0,a) TODO: WHAT???
f(x) = ab^x = a(1/b)^-x since x^-y = (1/x)^y
Since generally f(-x) is mirror across y axis AND x^-y = (1/x)^y, then for
exponential function (1/x)^y is also mirror across y axis.
(a^b)^c = a^(b∙c) VERY DIFFERENT FROM a^(b^c)
Logarithm logb(x). b ≠ 1, b > 0, x > 0 [Very difficult to introduce a]
Domain is x > 0. [I don't know why odd bases can have x < 0] SIGNIFICANT!
Range is all real numbers.
Has vertical asymptote at x=0.
(1,0)
When given problem: y = logb(x)
translate that to: b^y = x
Crazy property: logb(x^y) = y∙logb(x)
3 Properties of Logarithms.
Basically, if bases are same then can break off factors of the x:
logb(MN) = logb(M) + logb(N)
logb(M/N) = logb(M) - logb(N)
logb(M^c) = c∙logbM
Justification here is that, for x=M^c, M has been root-diminished by c.
If you reduce M to below b, that makes logb(M) < 1 therefore y < c.
GOTCHA: Can do nothing with: logb(x) ∙ logb(y)
Change of Base Forumula (from b to c): logb(M) = logc(M) / logc(b)
Pascal's triangle. 1 at tip, elements in between elements above it.
Each element is sum of higher-left + higher-right.
Elements positioned 0-indexed with tip 1 being (0,0).
C(n,r) = n! / [r!∙(n-r)!]
Binomial theorem: Coefficients of (x+1)^k is kth row of Pascal's triangle.
Therefore, coefficient of power k are C(k,0)... C(k,k). I.e. C(k,a)∙x^a(n-k)
Factorial operation: x!
SUM of all integers up to x.
0! == 1
COMBINATORICS
Number of permutations (order-specific): n! for r=0
If leftover candidates then: P(n,r) = n! / (n - r)!
Number of combinations (order-independent): 1 for r=0
If leftover candidates then: C(n,r) = n! / [r! ∙ (n-r)!]
This is exactly binomial theorem.
Probability where event A occurs m times in n (equally probable) outcomes:
p(A) = m/n