General NOTES for Math 169
Fifth Edition
Chapter 3

Introduction to Detailed Notes

This is a set of notes that have been made on reading the textbook. There is no real attempt to have comments on absolutely everything in the book noted here. At the same time, there is supplementary material here that is not in the book.

After writing out the notes for the first few sections, it has become clear that there is a tendency to make this a "teaching" document. As much as possible, efforts will be made to not do this. Rather, if there is teaching material to be presented then that will be done in separate pages, with pointers inserted here.

Chapter 3: Linear Functions

3.1 Constant and Linear Functions: Slope

The material on the calculator and the calculator display is important. The text on page 126 contains the Technology box. However, there is not much in the way of explanation for how one gets the displays on the page 126. Nor, is there much information about how one turns off the plot of points shown on that page. The web page Chapter 3 Plot gives a detailed explanation of this. In addition, that web page takes you through the steps needed to move from the graph on page 126 to the graphs in the Technology box on page 128.

When we plot points on the calcualtor, the values for X and Y are set into lists. The calculator uses those pairs of points to select the appropriate pixels on the screen to identify with the points. When we graph a function on the calculator, values for X are chosen by the calculator and subsequent values of Y are determined, based on the fuinction. For any particular X and Y pair, the calculator determines the pixel on the screen that represents that pair, and then the calculator turns on that pixel. Note that the pixel is not "illuminated", rather it is turned opaque, i.e., it appears black. To really understand the graph it is important to realize that any one pixel on the screen represents much more than just a single ordered pair. The page pixels has been developed to give futher explanation to this last statement, tuned to the TI-83.

The idea of the Technology box at the top of page 128 is that changing the viewing window settings (Xmin, Xmax, Ymin, and Ymax) changes the way a function looks. This is an important concept. A great deal of the material presented in the next few pages of the text references "how the graph of a function appears." Usually, when we talk about "how the graph of a function appears" we assume equal spacing on the X and Y axes. Before we had graphing calculators, when we asked students to graph functions on standard graph paper we could expect that students would use each pre-drawn line as a unit, and that students would end up with a graph having equal X and Y spacing. This is not always the case on graphing calculators. Unequal X and Y spacing will affect the appearance of the graph of a function.

Page 128 is probably as good a place as any to remind us that we are using two different abbreviations for the same concept. In particular, we use

f(x)=5x-2 or y=5x-2 as shortcuts for defining a function as f={(x,y)|y=5x-2} After all, a function is a special type of relation, and a relation is merely a set of ordered pairs. Therefore, a function is a set of ordered pairs that has an additional restriction, namely, that each first coordinate is matched with exactly one second coordinate. f={(x,y)|y=5x-2} defines f to be a function. However, at times we drop most of this and simply write y=5x-2, while at other times we emphasize the function name and the idea that for any given x value we need to produce a single function value by writing f(x)=5x-2. It is perhaps unfortunate that we slip back and forth between these different representations for the same function. And then, to complicate matters, the calculator insists on a slightly different form. Where we want to write y=5x-2, to put this into the calcualtor we need to identify that this is one of possibly many functions to be graphed at the same time. We have the practice of identifying different functions by giving them different alphabetic names. For example, in a problem we may talk of f(x), g(x), and h(x). The calculator uses a form closer to the y=5x-2 example, but the calculator distinguishes different functions by using Y with a subscript. Thus, to graph the function f(x)=2x-1 on page 128, we first change the function to y=2x-1, and then we enter the function into the TI-83 calculator as Y1=2X-1. (Note the change to capital letters.)

Again, in the text on page 129 we see a function f(x)=2 and then we talk about the "y-intercept". The "y" comes from

f(x)=2 being shorthand for f={(x,y)|y=2} and the fact that we can graph f on the xy-plane.

3.2 Mathematical Models of linear Functions

This section is an attempt to make linear functions more user friendly. It tries to demonstrate real world examples of various linear functions. The really important concepts here are

recognizing linear functions, including constant functions
recognizing when a relation is not a function
giving an interpretation to the intercepts
Giving an interpretation of slope.

At the bottom of page 145 the text presents a calculator screen and challenges the reader to "find a viewing window which allows you to see the graph f(x)=7500". It would be nice if the text gave a solution as it does for every problem example. In this case, we will need to move to the WINDOW screen on the calculator and chage the YMIN and YMAX values to include the value 7500. For example, we could set Xmin to be -10 and Ymax to be 8000.

Also, the graph at the bottom of page 145 uses the continuous function y=7500 to generate a graph for all values of X. Is it possible to generat a graph for just those points where the function is defined (in this example, just for whole number values of X between 1 and 25)? The answer is yes, however the process is a bit ugly. There is a demonstration page that shows a relatively "clean" method for "plotting" the discontinuous function. That page goes on to show the "ugly" discontinuous function. And, finally, the page concludes with the development of the TI-83 screen at the bottom of page 147.

It is a bit danerous to employ the term "discontinuous" to these examples. For example, on page 148 Q4 we have a problem that gives

s(x) = 6.5x + 100 where x is the number of hours that Joe works
and s(x) is his weekly salary in dollars. The answer to part b) of the question states that the function is discontinuous, and it probably is given its real life meaning. However, in the abstract world of mathematics, Joe could work any number of hours, inclusing fractional hours, between 0 and 7*24 hours in a week. And, his resulting pay would be any value between 100 and 6.5*24*7+100 dollars. The book does not state why this is discontinuous. The earlier example of the Beanie Baby sales is discontinuous because the Domain values necessarily jump from 1 to 2 to 3 and so on. We do not sell 3/4 of a stuffed animal! However, in this case, Joe can work 3/4 or an hour, or the square root of 2 hours, or "pi" hours in a week. Even if we say that we will round off time to the nearest minute, then Joe can still work the square root of 2 hours, but he will be paid the same as if he worked that amount of time rounded to the nearest minute. This becomes the "step function" introduced later. Alternatively, we could insist that Joe is paid his weekly salary rounded to the nearest penny. Again, this produces a "step function". The fact that a "step function" jumps from level to level is what makes it discontinuous.

On page 149, we need to change Q5 a) to read "The estimated height..." The function is defined to give an "estimated" height. Given any reasonable limit to the precision of our measurements, and given the billions of women in the world, it is certain that there are at least two women who really have identical lengths for their tibia and yet those two women will be of different actual heights. Thus, a woman's height can not be a function of the length of her tibia. By the way, the problem states that the formula is valid for a person, and yet the questions relate to "woman". Is there a different formula for a man? Why the distinction in one place but not the other?

3.3 Equations of Lines & Graphing

The first paragraph in the box on page 165 presents the various ways that we represent functions. Remember that a function is a special kind of relation and that a relation is a set of ordered pairs. Thus, we should define a linear function as

f = { (x,y) | y = mx + b but, at times, we merely write y = mx + b or, we use the function notation f(x) = mx + b and, this leads us to say y = f(x)

Note that the graph at the top of page 172 should be a step function, and it should be discontinous. See the demonstration page for this section to see how this should appear. In addition, that page illustrates the use of the SHADE command that is pointed out at the bottom of page 181.

3.4 Writing Equations of Lines

to be done, but there is a demonstration page for section 4. That page points out the steps needed to use the Linear Regression feature illustrated on page 196 of the text. That same page goes on to illustrate both the SLOPE and the SLOPE1 programs for the TI-83.