All these short descriptions should ring a bell. If you don't know what any line means, go back and figure it out. These are the most important concepts in the course.
1.1 "indirect" (and "direct") p. 6
Algebra is indirect p. 6
operations and order p. 7
1.2 grouping symbols
squaring negatives p. 10, 15
long fraction bar p. 12
extended overhead square root symbol p. 13
superscripts p.13
three significant digits p. 11
1.3 function, argument, image p. 21
placeholder p. 21, 22, 23
(allows operations to be expressed and discussed)
unknown p. 21
"Let f(x) = x2" is about f, a rule, not about x, a number. p. 19, 20, 24
1.4 method p. 30
formula p. 30
identity p. 30, 32
= p. 31
relations between equations p. 32
problem-pattern, solution-pattern p. 33
equivalent
expressions p. 31
equations p. 33
parameters p. 34
1.5 graph, ordered pair p. 43
intersection method p. 44
standard window p. 45
representative graph p. 45
artifact p. 47
pixel p. 46f
direct and indirect (again) p. 48
Guess-and-Check p. 49
1.6 Four Ways to Solve an Equation
1) Inverse-Reverse p. 55
2) Zero Product Rule p. 57
3) Quadratic Formula p. 58
Algebraic (the first three) p. 61
Format requirements p. 56, 57, 59, 60
4) Guess-andCheck (not algebraic, rather "direct") p. 59
2.1 Graph, Window
how changing the widow changes the appearance (all section)
evaluate, solve p. 73
2.2 Composition and Decomposition p.78-80
Notation
adding or multiplying after applying f p. 81
(subtracting and dividing, too) See Table p. 87
adding or multiplying before applying f p. 82-83
(subtracting and dividing, too)
2.3 To find f inverse, f -1, solve "f(x) = y" for x. p. 91
The result is f -1(y). [Switch letters to get f -1(x)] p. 91
one-to-one p. 92
If f is not one-to-one, there are complications. p. 93, 91 (T2-3-1)
Theorems like 2-3-1, 2-3-6, and 2-3-8 tell what to do.
Given (a, b), where is (b, a)? p. 97
3.1 lines in point-slope form (we deemphasize "y = mx + b")
Figure 5 for understanding point-slope form
lines through points on a graph (e.g. Figure 14)
proportional, parameter
linear interpolation
3.2 Quadratics
complete the square and graph location changes
Quadraic Formula, symmetry
QF applies when the letters are different (page 134)
3.3 The distance formula is the Pythagorean Theorem (Figures 1 and 2)
The equation of a circle is the Pythagorean Theorem
completing the square
(Ellipses are less important.)
3.4 Graphical Factoring
The Factor Theorem
the form k(x - b)(x - c) [instead of ax2 + bx + c]
the Factor Theorem does not determine the constant factor
Stating the Factor Theorem
(Non-integer and complex factors are less important.)
3.5 & 3.6 Word problems
cue words, indirect word problems
"Build your own formula" [Repeat this advice!]
Draw and label a picture (if appropriate)
Name the answer (often "x")
Write down relevant basic formulas
Build your own formula
Guess-and-Check for seeing how to build formulas
Start writing! Keep writing!
4.1 Powers and Polynomials
repeated multiplication
basic polynomial shapes of graphs, especially cubics and quartics
local minima, maxima, and extrema
end-behavior model
4.2 Polynomial Equations
monomial equations ("xn = c")
The big difference between n even and n odd
factoring in integers using the Factor Theorem (from 3.4)
Guess-and-Check can solve equations
[Non-integer factors are less important.]
4.3 Fractional Powers
extraneous solutions
squaring both sides is dangerous
The theorem is stated with "if..., then..." and not with "if and only if"
solving "xp = c"
properties of fractional powers
4.4 Percents [One of my favorite sections. Percents are in the news every day!]
think multplication
standard on the bottom
incorporate change
composition (don't add--multiply)
growth of money
averaging percent changes (don't divide--take roots)
(For lack of time, we skip "annunities" on page 243.)
4.5 Rational Functions
solving rational equations
locations of zeros and asymptotes (vertical [p. 254, box] and horizontal) [the most interesting places]
end-behavior model (determines horizontal asymptotes, p. 257, box)
graphing
4.6 Inequalities
Solving equalities is often much different from solving equations
Multiplying or dividing by expressions with variables is not legal unless you keep track of whether the expression is positive or negative.
The Theorem on Zeros and Signs (p. 271)
We recommend you graph inequalities to see the solution.
Absolute value inequalites (Theorems 4-6-5 and 4-6-6)
5.1. Exponential and Logarithmic Functions
exponential functions are like power functions with the letters switched
logs are the inverse of exponentials (exponential and log facts come in pairs)
logs are exponents.
logs are useful for solving for unknown exponents
Theorem 5-1-3L is most important, followed by T5-1-1L
5.2 The doubling-time and half-life models
Four variables, two are parameters.
Given three values, the fourth can be determined.
Compound interest model
Exponential Model (generalizes the others)
5.3 The Richter Scale (for earthquakes) and decibel levels (for sound)
R(a) = log a (base 10)
db(v) = 10 log(v/v0)
log scales are a good way to compare things which change by multiples instead of by addition
(like the growth of investments)
5.4 These application of logs are interesting, but for lack of time we omit this section.
Chapter 6 begins a new subject, trigonometry.
6.1 "Geometry for Trigonometry"
Geometric cases
Angle-Side-Side is the dangerous case and may not determine a triangle (there may be two different triangles with those given parts)
6.2 "Trigonometric Functions"
Definitions of sine, cosine, and tangent
in a right triangle
in a unit-circle (in which case tangent is less important and treated as sine over cosine)
inverse sine is tricky, inverse cosine is not
6.3 "Solving Triangles"
Side-Angle-Side Area Formula
Law of Sines
Law of Cosines
The geometric cases determine which law to use
6.4 "Solving Figures"
Divide multi-sided figures into triangles
Bearing
7.1 "Radians"
definition of radian measure
arc length
sectors, area
deriving results
7.2 "Trigonometric Identities"
Unit-circle figures to illustrate identities with any combination of
θ + - π π/2 2π [for example, sin (θ + π/2)]
Right-triangle pictures [to illustrate and evaluate expressions like sin(tan -1(x/4))]
7.3 "More Identities"
Deriving identities
Figure 7.3.3 yields sine and cosine for the sum of two angles
Other identities follow from them: diference of angles, double angle, squared-function, half angle
7.4 "Waves" This is practical and interesting, but we omit it for lack of time.