This week we have 3 basic topics:
More on the calculator
A review of lines and their graphs on the calculator
A reintroduction to quadratic functions.
Hmm, exactly what are we doing on the calculator? A careful reading of the syllabus might leave the attentive student a bit confused as the topics don’t necessarily match up with the assignments. Maybe it is best to summarize.
We want to have these skills on the calculator:
Being able to use the calculator as a scientific calculator. This means understanding the symbols + * - / ^ and being able to read scientific notation. In addition, there are some notes on how to edit your entries to fix mistakes.
Being able to graph on your calculator. This requires a few things:
entering a function (for example $f(x) = x^2)$
Viewing the graph of the function (ZOOM 6)
Manipulating the graphic screen to see the important part of the function. (Setting the x and y windows. This is done in the WINDOW screen)
Graphical solutions of equations. We want to be able to use the graph of a function to solve equations. This boils down to two basic problems:
Finding the intersection of a graph with the x-axis. To do this, you would want to zero in on the value of the graph as it crosses the x-axis. (The TRACE command)
Finding the intersection of two different graphs. Again , we do this with the trace commands. This skill is also used to find solutions to inequalities.
Tables of values. Sometimes, we want numbers and not graphs. Like most graphing programs, the caclulator typically creates a table of numbers for x and y, then it plots these points and then connects the points with lines. The TABLE command allow you to see the numbers it creates, as well, you can manipulate the numbers. For example you can specify where to start the table and how big each increment should be.
Now, these are the skills, they can be done on the TI-82. As well they can be done on most graphing calculators. Even my free calculator that runs on my expensive PALM pilot. Do you need a TI-82? No, Do you need these skills? Yes.
The line is one of the basic objects of mathematics. Part of being a student of math is learning about lines. A big part. In this class we do a review of some basics we hope you have already. Here are some highlights.
A line is the graph of an equation that typically looks like
$$y = mx + b \quad\quad \text{point-slope form}$$
The $m$ is the slope and is given by the equation
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
better known as the rise over the run formula. This tells us how fast the line is increasing. If the line is straight up and down the slope is $\infty$, if it is flat it is 0. If it points up the slope is positive, if it points down it is negative.
The $b$ is the intercept on the $y$ axis. Most line will intersect the $y$ axis (which don’t?). It happens just once (except in odd occasions. Again which ones?) and this is $b$.
There are other forms for the line depending on your taste
$y-y_0=m(x-x_0)$
$y=mx+b$
$Ax + By + C = 0$
For this section, the homework covers all of the concepts. If shouldn’t be hard, but there are quite a few.
If lines are important then the next slightly harder but still very important thing are quadratic functions – parabolas. First we learn about their graphs.
Why are parabolas important? Let’s see can we say satellite dishes, the flight of a ball or aerodynamics. These are things that are modelled using curves or pieces of curves that are parabolas.
First we learn about the graphs of paraboloas in lesson 13. We cover the following:
graphs related to the graph of $f(x) = x^2$. That is graphs like $f(x) - 2$, $f(x-2)$ etc.
Using your calculator to draw the graph of a parabola. We should fix the window size to be able to identify the vertex (the high or low point) and any intercepts (x and y)
Find the equation of a parabola from the vertex and a point then graphing to verify. This is a little tricky. You will need to now that a parabola can always be written in the form
$$y = a(x-h)^2 + k$$
and if this is done then the vertex is at $(h,k)$. I would consider this a less important skill for this class. I would focus on the first two things.
After graphing, there is stuff on applications. This is covered in homeworks 53 to 59 in lesson 14. Essentially, you need to graph a function and find its maximum. The trickiest part is setting the graphing window to be an appropriate size. Keep in mind the default window for the TI-82 is from -10 to 10 on both the $x$ and $y$ axis. This might be awful if you are talking about your yearly salary. If you have difficulties let me know.