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11.  The formal rules of algebra

12.  Rational and irrational numbers

What is a rational number? Which numbers have rational square roots?  The decimal representation of irrationals. What is a real number?


13.  Functions

What is a function? The domain and the range.
Functional notation. The argument.
A function of a function.

14.  Introduction to graphs

The graph of a function. Coördinate pairs of a function. The height of the curve at x.

15.  Basic graphs

The constant function. The identity function.
The absolute value function. The parabola.
The square root function. The cubic function.
The reciprocal function.

16.  The vocabulary of polynomial functions

Variables versus constants.
Definition of a polynomial in x.
The degree of a term and of a polynomial.
The leading coefficient.
The general form of a polynomial.

17.  The roots, or zeros, of a polynomial

A polynomial equation. The roots of a polynomial.
The x- and y-intercepts of a graph.
The relationship between the roots and the x-intercepts.

18.  The slope of a straight line

Definition of the slope. Positive and negative slope. A straight line has only one slope.
"Same slope" and "parallel." Perpendicular lines.
The slope and one point specify a straight line.

19.  Linear functions: The equation of a straight line

The equation of the first degree. The graph of a first degree equation: a straight line.
The slope-intercept form, and its proof.

10.  Quadratics: Polynomials of the second degree

Quadratic equation: Solution by factoring.
A double root. Quadratic inequalities.
The sum and product of the roots.

11.  Completing the square

Solving a quadratic equation by completing the square. The quadratic formula.

12.  Synthetic division by xa

The remainder theorem.

13.  Roots of polynomials of degree greater than 2

The factor theorem. The fundamental theorem of algebra. The integer root theorem. Conjugate pairs.

14.  Multiple roots. Point of inflection.

Concave upward, concave downward.

15.  Reflections of a graph

Reflection about the x-axis. Reflection about the y-axis. Reflection through the origin.

16.  Symmetry of a graph

Symmetry with respect to the y-axis. Symmetry with respect to the origin. Test for symmetry.
Odd and even functions.

17.  Translations of a graph

Definition of a translation.
The equation of a circle.
The vertex of a parabola.
Vertical stretches and shrinks.

18.  Rational functions

Singularities. The reciprocal function.
Horizontal and vertical asymptotes.


19.  Inverse functions

Definition of inverses. Constructing the inverse.
The graph of an inverse function.

20.  Logarithms

The system of common logarithms.
The system of natural logarithms.
The three laws of logarithms.
Change of base.

21.  Logarithmic and exponential functions

Inverse relations.
Exponential and logarithmic equations.
Creating one logarithm from a sum.


22.  Sigma notation for sums

23.  Factorials

24.  Permutations and Combinations

The Fundamental Principle of Counting.
Factorial representations.

25.  The binomial theorem

Pascal's triangle.

26.  Multiplication of sums

A proof of the binomial theorem.


27.  Mathematical induction



Copyright © 2017 Lawrence Spector

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