# College Algebra Review

The first test (Test #1) will occur on Wednesday, October 1, 2008 and will
cover the following

sections of the text:
**§**1.1-1.6,**§**7.1, and
**§**2.1-
2.6. Below are my comments on
each of these

sections.

It goes without saying—but not without writing—that it is assumed that you have
had a good and

honest attempt at the assigned exercises. Pay attention to these problem types,
especially the ones

that I deemed important enough to grade.

**Chapter 1: Linear Functions, Equations, and Inequalities**

**§1.1 Real Numbers and the Rectangular Coordinate System.** This section
contains some

basic terminology concerning natural number, whole numbers, rational numbers,
irrational

numbers, and real numbers. These are all terms you’ve studied before and should

know. Knowledge of these number systems is assumed, and will not be explicitly
tested

over. Similarly, it is assumed you know the rectangular coordinate system,
plotting and

graphing will occur on this test; know such terms as Quadrant I–Quadrant **IV.
**

There are three important formulas in this section that you will be tested on:

**Pythagorean**

Theorem,the

Theorem,

**Distance Formula,**and the

**Midpoint Formula.**

**§1.2 Introduction to Relations and Functions.**You should know the interval notation, and

the set-builder notation. An example of each follows:

**A relation** is a set of ordered pairs. A relation has a domain and
range. A **function**

is a special kind of relation. You should be able to work with simple discrete
relations

and functions. Relations have graphs, and you should be able to work with the
graph of

a relation, identifying its domain and range. The **Vertical Line Test** is a visual
way of

discerning if a relation is a function.

The function notation is of supreme important. A common with of defining a
function f

is through this notation. For example, f (x) = 2x^{2} − 1 uses an algebraic
expression to

define the function. The variable x is the independent variable (an element in
the domain

of the function). We use this notation as follows: if we wish to find the value
of this

function at x = −2, then f (−2) = 2(−2)^{2} − 1 = 2(4) − 1 = 7.

**§1.3 Linear Functions.** **A linear function** has the form f (x) = ax +
b, where a and b are

numbers. Its graph is a line. As we progressed through this section, we learned
that a

and b had certain (geometric) meanings:

• The number a turned out to be the **slope**, m, of the line.

• The number b is the y-intercept of the line.

Slope of a Line. Given two points (x_{1}, y_{1}) and (x_{2}, y_{2}), the slope of the line
passing

through these two points is given by

(1)

You should know this formula, and how to use it. The sign of m give important
information

about the slant of the line, see page 30 on the Geometric Orientation Based on
Slope.

The slope-intercept form of the equation of a line. Given the slope m and the
y-intercept,

b, of a line, the linear function describing this line is given by

(2)

Know this form, and the interpretation of its parameters (m and b), for
example, the

equation y = 2x − 3 is the equation of a line with slope m = 3 and y-intercept
of

b = −3.

**§1.4 Equations of Lines and Linear Models. **In this section we looked at
several formulas

for developing the equation of a line, notions of parallel and perpendicular
lines were

introduced, and a brief discussion of linear regression was presented.

**Point-Slope Form.** Given a point (x_{1}, y_{1}) and a slope m,
the equation of the line passing

through the point and having the given slope is

(3)

This is a transitional form, normally, we set up the equation using the
point-slope form,

then convert it into one of the others, most notably, the slope-intercept form
(2), or the

standard form (4). You should know how to use this formula, most certainly, and
how to

convert it to the slope-intercept form

**
Standard Form.** The standard form for an equation of a line is of the form

(4)

Be able to convert an equation in standard form to the slope-intercept form;
for example,

2x−3y = −1 converts to . Thus, 2x−3y = −1 is a
line with slope m = 2/3

and crosses the y-axis at y = 1/3.

**Parallel Lines.** Two lines are parallel if they have the same slope.
Example 4, page 39,

is an important type of problem.

**Perpendicular Lines.** Two lines are parallel if their slopes have a
product of −1. Example

5, page 41, is an important type of problem. Note that if the slopes of the
perpendicular

lines are m1 and m2, then m_{1}m_{2} = −1, or m_{2} = −1/m_{1}, this latter formula is
useful if

you wish to compute the slope of any line perpendicular to a given line.

**Linear Models and Regression.** We discussed this topic in class and had a
homework

set over regression. We’ll not cover this on the test itself.

**§1.5 Linear Equations and Inequalities.** A linear equation is one of the
form ax +b = c, or

any equation that reduces to that form. You have been solving such equations
since 8th,

just continue as you did in the past, only without error.

There is some discussion about the graphical interpretation of intersecting
graphs; I doubt

if that will be explicitly on the exam, but you should be able to analytically
find the

intersection point; for example, where do the two lines y = 4x − 1 and y = 2x +
1

intersect? We begin by setting the y-values equal:

4x − 1 = 2x + 1 | Set the functions equal |

2x = 2 | Subtract 2x and add 1 to both sides |

x = 1 | divide ! |

However, when x = 1, y = 4(1) − 1 = 3. The two lines intersect at the point
(1, 3)

Linear inequalities are also covered. Their solution is typically in interval.
Standard

algebraic tricks work except for dividing or multiplying both sides by a
negative number,

in this case the inequality is reversed.

The solution set is then (−∞,−1 ] . Actual inequalities seen on the test my
be more line

Example 6, p. 58, where some additional routine algebra is needed.

**§1.6 Applications to Linear Functions.** We covered **Applications to
Linear Equations,**

(simple geometric problems, mixture problems) and **Break Even Analysis.**
Expect problems

of the type that were on the homework.

## Chapter 7: Systems of Equations

**§7.1 Systems of Equations.** We popped into this section to pick up the
techniques for solving

a system of linear equations:

ax + by = c

dx + ey = f

Such a system has one of three possible solutions: (1) a unique solution; (2) no
solution,

or the empty solution ; or (3) infinitely many solutions.

There are two techniques for solving such a system: (1) the **substitution
method;** (2) the

**elimination method.**

**The Substitution Method** is the one you should be most familiar with. Here
you solve

for one variable in one of the two equations, and substitute it into the other
equation.

**The Elimination Method.** The idea behind the elimination method is to
eliminate one

of the two variables, and solve for the remaining variables. We do this by
matching the

coefficients of the variable we want to eliminate. For example:

5x + 7y = 6 | (1) | |

10x − 3y = 46 | (2) | |

−10x − 14y = −12 | (3) | multiply (1) by −2 |

−17y = 34 | add (2) and (3) | |

y = −2 | (4) | solve |

5x + 7(−2) = 6 | Subst. (4) into (1) | |

x = 4 | (5) | …and solve for x |

From (4) and (5), the solution set is { (4,−2) } .

We studied only linear systems—not the nonlinear systems—and the exercises
reflected

that.

Chapter 2: Linear Functions, Equations, and Inequalities

**§2.1 Graphs of Basic Functions, Symmetry.** You need to have a basic
understanding of

continuity (at least, be able to pick one of the crowd). By looking at a graph,
you should

be able to determine where it increases and decreases.

The following functions are basic, you should know there graphs, and able to
answer

questions on continuity, increasing, decreasing, domain, and range. You need to
know

what the graphs of these basic functions look like:

• The identify function f (x) = x

• The squaring function f (x) = x^{2}. Its graph is a parabola, and is symmetric
with
respect to the y-axis:

f (−x) = f (x) Criterion for y-Axis Symmetry

• The cubing function f (x) = x^{3}. Its graph is symmetric with respect to the
origin:

f (−x) = −f (x) Criterion for Origin Symmetry

• The square root and cube root functions f (x) =and
f (x) =.

• The absolute value function f (x) = |x|

• The x = y^{2} relation and x-Axis symmetry

An **even function** is just one that has y-axis
symmetry, while an **odd function** is one

with origin symmetry.

**§2.2 Vertical and Horizontal Shifting of Graphs.** Geometrically easy to
understand. Know

this material.

**
§2.3 Stretching, Shrinking, and Reflecting **Stretching and shrinking are a
little trickier to

understand graphically. Not much emphasis will be placed on this section except for

reflecting. We reflect a graph y = f (x) across the x-axis by graphing the function

y = −f (x), and we reflect a graph with respect to the y axis by graphing the function

y = f (−x).

**§2.4 Absolute Value Functions: Graphs, Equations, Inequalities.**Consider the function

y = f (x), and define y = |f (x)|. There is a definite relation between these two graphs.

This is an important point.

The other content of this section concerns solving equations and inequalities that involve

absolute values. Some simple exercises will appear on the exam.

**§2.5 Piecewise-Defined Functions.**Be ready to graph and calculate the values of a piecewise defined

function.

**The greatest integer function,**p. 141, will not be covered.

**§2.6 Operations and Compositions.**The simple arithmetic operations were briefly discussed

(p. 149), a greater emphasis was placed on

**composition of functions,**p. 152. Examples 5,

6, and 7 are of particular importance.

That was a lot of (review) material!! Hope we can slow the
pace as we progress through the book.

The test, no doubt, will have some short answers (true/false, fill in the blank)
and some questions

involving computation, algebra and graphing. Good luck, but more importantly,
good knowledge.

I shall attempt to construct a fair test over these topics.