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Algebra - Outline

I - Word problems and Logic

(“of” means multiply, “per” means divide, etc)

4 relational properties (commutative, inverse, distributive, and associative)

algebraic order of operations (add, subtract, multiply and divide)

use TI-89 calculator or Wolfram Alpha when necessary

exponent rules: (A^x)*(A^y) = A^(x+y); (A^x)^y = A^(xy), etc

fractional exponents written form of radicands and indices A^(x/y) = yrad(A^x)

some word problems require writing out systems of equations

quadratic physics problems with constant acceleration, initial velocity, and initial position

 

II-Inequalities

solving inequalities:

(works just like solving equations but if multiplying both sides by a negative then the inequality sign flips)

plotting inequalities on number lines:

(blank circle means exclude from domain, filled circle means include in domain)

inequalities of absolute values:

(|X|<A means X is between +/- A, |X|>A means X is split greater than +A and less than -A)

 

III - Functions

vertical line test for linear functions entails all input points on the X-domain mapping to single output points on the Y-range, but not necessarily mapping in the other direction though

continuous functions have no gaps, vertical asymptotes, or undefined points
discontinuous functions which can have X-domain gaps and/or vertical asymptotes and/or undefined points

different ways of plotting a graph: (table method, calculator method)

functions of inequality y > f(x) or y < f(x) ... graph shaded regions greater or less than values with dashed line

 

forms of equations:

multiplicative inverse -> y = 1/f(x)

constant flat liner -> y = k

 

sloped lines -> y = mx+b slope intercept form or ax+by=c for standard form:

slope relation between parallel and perpendicular lines: mperp = -1/mpara

using standard form “ax+by=c” of equations for plotting {x;y}-intercepts = {(c/a,0); (0, c/b)}

using slope-intercept form “y=mx+b” for plotting … slope=m, {x;y}-intercept = {(-b/m,0);(0,b)}

 

monomial expression (ax^2), binomial expression (ax+b)

 

expanded quadratic -> y = ax^2 + bx + c

factoring quadratic equations (if ax^2+bx+c=(rx+t)(sx+v); then a=rs, b=rv+st, c=tv)

finding minimums and maximums of parabolas at the axis of symmetry (x-sym = -b/(2a))
quadratic equation form: y=ax^2+bx+c
quadratic formula: Xint =(-b +/- sqrt(b^2 -4ac))/(2a), for finding zeros of quadratic equations

parabolas that intersect the X-axis 0, 1, or 2 times as determined by the discriminant values (d<0, d=0, or d>0), where the discriminant “d=b^2-4ac” relates to the number of solutions that intercept the X-axis (2 imaginary, 1 real, or 2 real) respectively
 
completing the square: y= a[(x^2+(b/a)x+K)+(c/a-K)], where K=(b/(2a))^2
 
point vertex form:  y = a[x+b/(2a)]^2 + [(4ac-b^2)/(4a)]; where vertex point is {-b/(2a), (4ac-b^2)/(4a)}

 

foiling factored expressions (firsts, outers, inners, lasts)

polynomial expression (…+ax^2+bx+c)

expanded polynomial -> y = An*x^(n) + An-1*x^(n-1) + ... + A1*x + A0 

 

divide polynomials by binomials using long division and synthetic division

factoring the difference of two squares and the sum of two cubes

factored polynomial: y = (B0 X + C0)*(B1X + C1)*...*(BnX+Cn) 
 
 
trigonometric -> y = trig(f(x))
whereby trig can be ... sine, cosine, tangent
or their multiplicative inverses ... cosecant, secant, cotangent
or their angular inverses ... arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent
 
exponential -> y = e^f(x)
hyperbolic -> y = trigh(f(x)) ... whereby trigh can be the hyperbolic functions sinh, cosh, tanh, csch, sech, coth
logarithmic -> y= logA(f(x))
 
absolute value functions -> y=|f(x)| or y = f(|x|)
integer stepping functions -> y = f[x]
 
piece-wise functions -> y = f1(x) on X = {X1,X2}, f2(x) on X= {X3,X4}, ..., etc
 
Graph operations:
shift graphs horizontally along the X-domain by adding or subtracting constants from the x-values
stretch or invert graphs by multiplying the function by positive or negative coefficients

 

IV - Matrices

addition of matrices (Aa,b+Ba,b=Ca,b ... make sure dimensions match, add corresponding matrix elements up) 

multiplication of matrices (Aa,b*Bb,c=Ca,c ... #A rows = #B columns, sum row times column elements)

matrix rules: A(B+C)=AB+A, (AB)C=A(BC)...etc
the identity and zero matrices AI=A, A*0=0
find the determinate of square matrices |Am,m|

use substitution or elimination methods to solve systems of linear equations

write systems of equations in general matrix [An,n][Xn,1]=[Bn,1] and conjoined matrix [A|B] forms

elimination method to achieve reduced-row echelon form rref([An,n|Bn,1]) -> [In,n|Xsolved.n,1]

finding an inverse matrix A*A-1=I by conjoining the [A|I] identity matrix and eliminating to rref -> [I|A-1]

solving 2x2 linear systems of equations (independent systems have multiple lines which can either be inconsistently parallel meaning there are no solutions or consistently intersecting meaning there is only 1 solution that exists at the intersection point, and dependent systems have just one single consistent collinear line with infinitely many solutions)

plot coordinate matrices (similar to bitmaps) and parametric equations on graphs

translate, dilate, and rotate parameterized or coordinate equations with transformation matrices

use the cross product of two displacement vectors V12, V13 V12xV13=det([i,j,k][V12][V13]). which is the vector orthogonal to the plane formed by those displacement vectors, and the magnitude of the cross vector becomes the area of the parallelogram formed by those displacement vectors

 

V - Complex Numbers

the complex plane contains complex numbers or real numbers added to imaginary numbers (z = a +bi)

multiply magic fractions by complex conjugates to get rid of imaginary numbers from denominators 

degrees and radians on the unit circle: (degrees = 180 * radians/pi)

find i^n by rotating n*pi/2 radians times counterclockwise along the unit circle starting with i^0 = 1

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