Lecture 5: More on graphing
[
List of Lectures
|
Math 1100 Index
]
Assignment
These are the problems you should work before January 29.
Review problems, page 75, 4n+1, for n=0..22;
Section 1.1, page 92, 4n+1, for n=0...24;
Section 1.2, page 102, 1-6, and 4n, for n=3...23;
Section 1.3, page 113, 4n+1, for n=0...19.
Brief review of the lecture
We are going to look at how to graph functions using at TI graphing calculator.
The TI-83 and the TI-82 work similarly. The cd rom diskette some of you bought
has a lovely TI-82 or TI-83 simulator.
But first we'll talk about symmetry.
Symmetry means that a figure looks the same from two different perspectives.
We saw in class that the graph of the equation y=|x|. looks the same when the
overhead transparency is flipped around so that the y-axis lands back where it started.
This is an example of a set (or graph) which is symmetric with respect to (wrt)
the y-axis. There are several other kinds of symmetry we need to learn how to diagnose
and use.
Reflection across a line. Symmetry with respect to
1. the x-axis
2. the y-axis
3. the line y=x
4. the origin
We can test these types of symmetry as follows:
1. x-axis: if we get the same set of points when y is replaced by -y.
In other words, if (x,y) belongs to the set just exactly when (x,-y) belongs to the set.
Take (x-1)^2 + y^2=9. You may recognize this as a circle with center at (1,0) and radius 3.
Notice that the very same equations (hence the same set) results if
we replace the y with
-y: (x-1)^2+(-y)^2=9, and the minus sign disappears because of the square.
2. If the same set results when x is replaced by -x.
3. If the same set results when x and y are interchanged.
In other words, (x,y) belongs to the set exactly when (y,x) belongs to the set.
Consider the equation xy=1. This has the same graph as yx=1 (of course),
so it has symmetry wrt the line y=x.
We'll see how to use the graphing calculator to determine
when a curve is symmetric wrt
each of the lines in question.
The material below is left from a previous semester.
Read it if you dare!
- One of the most important types of algebraic expressions is called a
polynomial. The term polynomial (of a single variable)
was defined here as a sum of multiples
of powers of a variable.
- Polynomials may be classified according to
- Number of variables (for our purposes, this will almost always
be 1)
- Degree, e.g., the highest power of all the terms of the polynomial.
Example : x² - xy²z+10 has 3 variables and degree 4
- Multiplication of polynomials was discussed, both from the algebraic and
geometric viewpoints.
Example : (x-2)(x²-3x+5)=x³-5x²+11x-10
- Decimals were defined as a sum of multiples of powers of ten.
Example : 401.03= 4 × 10 ² +
1 × 10 ° +
3 × 10 ¯ ²
- Polynomials were defined as a sum of multiples of powers of a variable.
Example : 4x²+3x+9
- The following table shows how polynomials are ofter
classified by their degree:
| Degree |
Symbolic representation |
Common representation |
Name |
Maximum # of zeros |
| 0 |
a0 |
c |
constant |
none |
| 1 |
a1x+a0 |
mx+b |
linear |
1 |
| 2 |
a2x2+a1x+a0 |
ax2+bx+c |
quadratic |
2 |
| 3 |
a3x3+a2x2+a1x+a0 |
none |
cubic |
3 |
| 4 |
as expected |
none |
quartic |
4 |
[
Next Lecture
|
Math 1100 Index
]