ELEMENTARY & INTERMEDIATE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Elementary Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/431034. See "Tell me more about Keely's GM book".
Lines & Intercept Points
GOLDen Mathematics - Elementary Algebra: Section 4.1
Supplemental Sites: MathOL Links - Alg 4.1
Topics of Importance
Definition of linear equation in two variables
Standard form vs. general form of a linear equation in two variables
Find intercept points algebraically and graphically
Graphing lines by the intercept method
Introduction to graphing horizontal and vertical lines
Comments and Cautions
Hopefully you are beginning to
feel more comfortable with functions, graphs, and your electronic grapher as
this chapter is graphing intensive. Over the next few days we are going to learn
all about graphing lines and analyzing linear functions. Most of our work will be
algebraically "by hand", but we will also verify things electronically and use
the technology to expand basic problems into more complicated and realistic
applications. The main topic that we study today is finding x and y-intercept
points of a line and graphing the line using those points. You should be able to
do this algebraically by:
 | Find the x-intercept point by letting y=0 and solving for x. The x-intercept point will be of the form (#,0). |
| Find the y-intercept point by letting x=0 and solving for y. The y-intercept point will be of the form (0,#). |
You should also be able to find the intercept points graphically by:
| Use the ZERO feature of your grapher to find the exact x-intercept point. See my online Calculator Guide: x-Intercept Points for TI-calculator steps. |
| Use the VALUE feature of your grapher to find the exact y-intercept point. See my online Calculator Guide: Finding Points for TI-calculator steps. <<Link to be posted soon.>> |
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 3.2.)
| It is a good idea to know ch 1.3 and 3.1 well before beginning to study the remainder of chapter 3. |
| ch 3.2 pg 203 example 8 - Notice that you cannot graph the line x=5 on a calculator. You can graph example 7, y=-4 on a calculator, but not example 8. Why is this??? (Discuss in class.) |
Lines & Slopes
GOLDen Mathematics - Elementary Algebra: Section 4.2
Supplemental Sites: MathOL Links - Alg 4.2
Topics of Importance
Find slope of a line from a graph
Use slope to find third point on line
Find slope of a line from two points
Slopes that are positive, negative, zero, undefined
Slope of parallel and perpendicular lines
Horizontal and vertical lines: graphs, intercepts, slopes, equations
Applications of slopes and rates
Comments and Cautions
Today we will focus on the
concept of "slope" which measures the steepness of a line. Slope is an important
aspect of linear functions and has applications to real-life situations in the
form of "rate" of growth. Some cautions to note:
| Watch your sign when you are finding slope! Anytime you move from point to point and you go down or left the sign will be negative on that side of the slope triangle. |
| As you move from point to point, be sure to do just that in one smooth motion. Don't start at the right angle and move to one point, then start back at the right angle and move to the other point. If you do the signs may get messed up. |
| When using the formula to find slope, it doesn't matter which point is (x1,y1) and which point is (x2,y2). But you do have to be careful that you don't mix the order up and use, say, y2-y1 on the top and x1-x2 on the bottom. If you do your signs will get messed up. |
| Some textbooks say that vertical lines have "no slope". Don't confuse that with a "zero" slope! Horizontal lines have a slope of zero; vertical lines have "no slope" or better yet "undefined slope". |
Be sure to read Ask Dr. Math's "Why is the letter m used for slope?" and PurpleMath's "The Meaning of Slope and Y-Intercept" linked from the supplemental sites. Highly recommended!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 3.3.)
| ch 3.3 pg 208 It is strongly recommended that you memorize the formulas given in the "definition of slope" box. |
| ch 3.3 pg 209 Take note of the caution/common error in the "study tip" box. |
Equations of Lines
GOLDen Mathematics - Elementary Algebra: Section 4.3
Supplemental Sites: MathOL Links - Alg 4.3
Topics of Importance
Slope-intercept equation of the line
Graphing lines by the slope-intercept method
Given various information about the line, find its equation
Point-slope formula
Applications of the slope-intercept equation of the line
Comments and Cautions
The slope-intercept method is the most useful method for graphing a linear
function. Be sure to memorize the formula y=mx+b and learn to use it well! Be sure that you are able to graph a line from its equation manually by each
of the following methods:
| Plug-n-chug method |
| Intercept method |
| Slope-intercept method |
So far we have primarily been starting with a linear equation and then producing the graph. But you should also be able to work that process backwards, beginning with the graph (or information about the graph) and finding the equation of the line. This will allow us to take real-life data and find a representative linear equation thus opening up even more avenues for solving practical applications.
By the end of this chapter you should be able to accomplish the following tasks:
| Given equation, find m and b |
| Given graph, find m, b, and equation |
| Given equation, graph line by the slope-intercept method |
| Given m and a point (not y-intercept), find equation (including by using the point-slope formula) |
| Given 2 points, find equation |
| Given various information about the line, find equation (including info about parallel, perpendicular, horizontal, or vertical lines) |
| Describe what slope and y-intercept value mean in practical terms |
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 3.4-3.5.)
| ch 3.4-3.5 are super important sections filled with formulas, terminology, and processes. Spend significant time learning them well. The next chapter deals with "systems of linear equations" where the graphs will have more than one linear equation and we will be analyzing how they connect including finding intersection points and interpreting this information in terms of fairly advanced applications. I cannot stress enough how important it is to have the processes of "equation -> graph" and "graph -> equation" down well before we start adding the additional lines to form multi-equation multi-variable systems! |
Graphing Linear Inequalities
GOLDen Mathematics - Elementary Algebra: Section 4.4
Supplemental Sites: MathOL Links - Alg 4.4
Topics of Importance
Definition of linear inequality in two variables
What it means to be a solution to a linear inequality in two variables
Graphing the solution to a linear inequality in two variables including:
dashed vs solid line, shading the solution set
Applications of linear inequalities
Comments and Cautions
Now that you are well-versed in
graphing and analyzing linear equations, we're going to throw some inequality
signs into the mix. This will have a couple of effects on the graphs including
dashed vs. solid lines and shading the solution set. This graphical
interpretation of a solution to an application problem will be particularly
useful if you are a business or social science major where
you are likely to run into graphs of linear inequalities frequently. The shaded
solution set could represent, for instance, corporate profits and losses, a
geographic region where a flu epidemic has broken out, or the limits of the
human body when placed under physical stress. One of the goals will be to take
all the possible solution points and then maximize some function. For instance
you may need to find the size package that meets the criteria to be checked on
an airline and shipped by USPS, but that also has the maximum possible volume
inside.
If you plan to take a finite math class to satisfy a college quantitative skill requirement, you will find yourself solving a variety of applications by graphing linear (and non-linear) inequalities. Often you will need to graph several at once each representing different criteria and then put the solutions all together coherently. That involves solving a system of linear inequalities ... coming soon to a lesson near you!
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections N/A.)
| THIS TOPIC IS NOT COVERED IN THIS COURSE AND THUS WILL BE OMITTED AT THIS TIME. |
| ch 8.9 pg 656-657 The final answer is shown as the region shaded in dark pink or gray. The light pink and light blue shadings are just the intermediate shadings. |
Originally written: 2006-006-15
Last revision:
2008-01-04 10:22 PM