1 Section 1.1 - Number Systems

2 Section 1.2 - Coordinates

3 Section 1.3 - Lines and Slopes

4 Section 1.4 - Functions/Graphs

5 Section 1.5 - Combining Functions

6 Section 1.6 - Trigonometric Functions

7 Factoring/Binomial Theorem

8 Solving Equations

The natural numbers, 1, 2, 3, …, are denoted by N

The integers, … ,-3,-2,-1, 0, 1, 2, 3, …, are denoted by

Z. Every natural number is an integer, but not vice versa.

The rational numbers, Q, are defined to be ratios of

integers. Every integer is a rational number, but not vice

versa.

The real numbers, R, can be constructed systematically

from the rational numbers, though this is beyond the scope

of our course. Every rational number is a real number, but

there are real numbers that are not rational (the irrational

numbers).

We will use the common notations for sets. For instance,

the set of all x that satisfy some condition P(x) can be

denoted by {x : P(x)}. If S is a set, and x is an element of

S, we write x ∈ S.

We will use the standard interval notation.

You ought to be aware of what sets of real numbers

inequalities such as etc

The triangle inequality, one of the fundamental
inequalities

in mathematics, is

When we write, for example,sqrt(5), we are dealing with the

exact value of this (irrational) number. The number

2.236067977 is not the same number, but is a (rational)

approximation of sqrt(5).

Every point in the plane (sometimes referred to as R^2),
has

a unique representation as an ordered pair (x, y) of real

numbers.

The distance formula: The (Euclidean) distance between

points (a, b) and (c, d) is

You will need to know how to find the equation of a circle

centered at (h, k) with radius R. Completing the square

plays an essential role here.

You will also need to know the equations of the other

conics - parabolas, ellipses, hyperbolas. (These won’t be

used frequently, but you do need to know them.

Know the definition of slope.

Perpendicular lines have slopes that are negative

reciprocals of each other.

Parallel lines have equal slopes.

You must know how to find an equation of a line through a

point with a give slope - the point–slope formula is

essential.

**Definition**

A function f from a set S to a set T is a rule that assigns at

most one element of T to each element of S. Given x ∈ S, we

denote the element of T that is assigned to x by f (x).

**Definition**

Given a function f : S -> T, the domain of f is the set of all

x ∈ S that are associated to an element y ∈ T (we say

y = f (x)). The range of f is the set of all such y ∈ T

We most often define functions algebraically:

However, we will

also consider piecewise–defined functions (see page 34–35)

**Definition**

The graph of a function f is the set of all points (x, f (x)) such

that x ∈dom(f ). The graph of a function y = f (x) is a curve in

the plane.

**Definition**

A sequence is a function with domain contained in N. We can

write a sequence by listing its values: f (1), f (2), f (3), ….

We’ll often do arithmetic with functions. This should be

routine for everyone.

We’ll also compose functions frequently. Know how to

compute compositions.

We will sometimes use translations of known function

graphs to get graphs of other functions. (see page 49).

Parametric curves - we will address this topic when

needed. Don’t worry about it right now.

You need to know all six trigonometric functions:

sin x, cos x, tan x, sec x, csc x, cot x and their basic

properties (such as period, whether they’re even or odd

functions, their values at the standard angles, etc.)

Take note of the table of standard values of the trig

functions seen on page 63.

A list of basic trigonometric identities are seen on page
62.

You ought to know these identities, although we won’t use

them extensively.

Know how to factor , as
well as

factoring expressions such as Being able to

factor is an essential skill

Be able to factor quadratics, when such a factorization is

possible

You must be able to solve any quadratic equation

You will also, at times have to solve higher degree

polynomial equations. Generally, at our level, this is done

by factoring or by guessing a root and reducing the

polynomial using division.

Recall that a polynomial of degree n has at most n
distinct

real roots.

Let α and β be real numbers and let n be a positive
integer.

Then

It is good to know this expansion, especially for small n
as you

can save a lot of time.