Functions, Graphing, and X(PLORE)
D. P. Morstad, University of North Dakota
Objectives of Assignment
1.
2.
3.
4.
Learning to use X(PLORE) on an IBM type personal computer.
Determining the domains and ranges of functions in order to graph them.
Categorizing functions by viewing their graphs.
Interpreting graphs of functions.
IMPORTANT NOTE: if you are not doing this assignment in the Math Computer Lab,
you must copy Assign1.xpl, Assign2.xpl, Assign3.xpl, and Slowpara.xpl
from a Math Lab computer to your X(PLORE) diskette. They are also
available by anonymous FTP at ftp.und.nodak.edu/pub/math.
I.
Three Steps to Start X(PLORE)
1)
Turn on the computer , and turn on the monitor.
2)
Wait. The computer takes some time to load the Windows program.
3)
Double click on the X(PLORE) icon. This means that you use the mouse
to move the cursor over to the X(PLORE) icon. Then press the left
mouse button twice in quick succession.
After completing these three steps, there should be stuff typed in white at the top
of a black screen and two big, nearly empty, outlined rectangles that take up most of the
rest of the screen — welcome to X(PLORE).
(If you are running X(PLORE) from a diskette, get to the DOS prompt, change to
the diskette drive, type “xpl”, and press E. Do not type the quotation marks.)
II.
How to Make X(PLORE) Graph Your Functions
To make X(PLORE) graph a function, you only need to tell it three things:
1) tell X(PLORE) what function you want graphed,
2) tell X(PLORE) where on the x-y plane you want it graphed, and
3) tell X(PLORE) to draw the graph.
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III.
Example One: Plotting a Graph
Since you've taken a trigonometry class and already know everything there is to
know about graphing trig functions, let's start by graphing f ( x ) = 2 sin( 3x ) :
Î
Type " f ( x ) = 2 sin( 3x ) " and press E.
(don't type either " mark, but do type all parentheses)
Î
Type "window(-500,500,-500,500)" and press E.
Î
Type "graph(f(x),x)" and press E. Ta-dah!! A perfectly lousy graph.
This graph is lousy due to the 500's in the window command. If you had to graph
" f ( x ) = 2 sin( 3x ) " on a trig test, you'd probably mark the x-axis from −2 π to 2 π and the
y-axis from –4 to 4. To make X(PLORE) do the same, use "window( −2 π ,2 π, −4,4) ".
To reset the window, first press X or E to return to the input screen. Next,
change the window command that's already there. Use the arrow keys to get the cursor on
one of the 500's and then use = or the backspace Q to get rid of all the 500's. Replace
them with –2 π ,2 π ,–4,4. Type Ap to get π . Press E while the cursor is on the
window line, and then press E again when the cursor is on the graph line. There! A
much better graph.
Determining a good window setting requires a little trial and error and a lot of
thinking. The more thinking you do, the less trial and error it will take. The computer
does the busy work of drawing graphs, but you still have to provide the brain power.
It's simple to graph more functions on the same axis system. For example, go
back to the input screen and change the function line to f ( x ) = 4 sin( 6 x ) . Just change the
numbers, don't retype the whole thing. Press E and then move to the graph line and
press E. Don't press E while on the window line this time. If you want to erase
everything that's been graphed, type "erase" on its own line and then press E.
IV.
Example Two: Interpreting a Graph
Suppose a particle is moving along a number line and that at time x seconds its
position along the number line is given by the function
− ( x 3 − 11x 2 + 23x + 33)
g( x ) =
+ 18.
x+2
2
Is this particle moving faster between the first and second second or between the fifth and
sixth second?
To use X(PLORE)'s graphing abilities to help solve this, first erase the trig
functions and reset the window to (0,6,0,20) (you can thank me later for figuring out the
best window for you), and then have X(PLORE) graph g(x). You will have to be careful
typing in this function. Type it in as
g(x) = –(x^3 – 11x^2 + 23x + 35)/(x + 2) + 18.
The "^" symbol is the upper character on 6. Notice the importance of the
parentheses. Before you enter "graph(g(x),x)", type and enter "color(lightgreen)" on its
own line. A list of all color choices is on page 53 of the X(PLORE) manual.
The graph you should be looking at represents the function which describes the
motion of the particle. The x-axis represents time and the y-axis represents the particle's
position along the aforementioned number line. To figure out how fast the particle is
moving, you need to find out how far it went between the first and second second and
between the fifth and sixth second.
Use the arrow keys to move the crosshairs down onto the graph. The ordered pair
in the lower right of the screen reports the location of the crosshairs. Position the
crosshairs as close to x = 1 as possible while keeping it on the curve of the graph. It
should be about (1.0634, 2.370) - if your computer screen has a different resoultion, you
will get numbers that are just slightly different.
These coordinates mean that at time one second, the particle is at 2.370 on the
number line. Now do the same thing at x = 2 to find out where it is at time two seconds.
It should be about (2.0033, 6.739). The particle's speed over this period is
Speed =
distance
6. 739 − 2.370
=
= 4. 65 units sec
time
2. 0033 − 1. 0634
Doing the same to find the speed between the fifth and sixth second, you should
find
Speed =
18.839 − 18.167
= 0.715 units sec
5.9798 − 5.0399
It's going much faster during the period between the first and second second.
When a graph represents the movement of a particle and time is plotted on the
horizontal axis, the steeper the graph, the faster that particle is moving.
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V.
Important Practice Problems (hand-in problems will rely on what you
learned doing these practice problems)
Problems 1 – 4 display basic similarities and differences between types of
polynomial functions. You will need to be able to recognize these similarities and
differences.
1.
2.
3.
4.
Graph f ( x ) = x 2 − 1, g ( x ) = − x 2 + 3x + 2, and h( x ) = 6 x 2 + 2 x + 3 on the same
axis system. These are all quadratic, or second degree polynomial functions.
Erase previous graphs. Graph f ( x ) = x 3 − 1, g ( x ) = x 3 − 9 x , and
h( x ) = x 3 + x 2 − x − 1 on the same axis system. These are all cubic, or third
degree polynomial functions. Notice the difference between quadratic and cubic
polynomial functions.
Erase. Graph f ( x ) = x 4 − 1, g( x ) = x 4 − 5x 2 + 4, and h( x ) = x 4 + x 3 − x 2 − x on
the same axis system. These are all fourth degree polynomial functions. These
are similar to quadratic polynomial functions or else have an extra "hill or valley"
in them.
Erase. Graph f ( x ) = x 5 − 1, g( x ) = x 5 − 5x 3 + 4x, and h( x ) = x 5 + x 4 − x 3 − x 2
on the same axis system. These are all fifth degree polynomial functions.
Compare the possible number of "hills and valleys" in these to the cubic
polynomials.
Problems 5 – 7 demonstrate some subtle differences between polynomial
functions which at first appear similar.
5.
Erase. Graph f ( x ) = x 2 , g ( x ) = x 4 , and h( x ) = x 6 on the same axis. Note the
increasing "bluntness" of the vertex as the degree increases.
6.
7.
Erase. Graph f ( x ) = x 3 , g ( x ) = x 5 , and h( x ) = x 7 on the same axis.
Erase. Graph f ( x )= x 2 , g( x ) = x 4 − x 2 , and h( x ) = x 6 − x 4 − x 2 on the same
axis.
Problems 8 – 10 demonstrate how critical it is to know how to use parentheses
when entering the function.
8.
Erase. Graph f ( x ) = sin 2 x 2 , g ( x ) = sin 2 x , and h( x ) = sin x 2 on the same axis.
Hint: f ( x ) = sin 2 x 2 should be entered as f(x) = (sin(x^2))^2.
9.
32
Erase. Graph f ( x ) = 3 , and g ( x ) =
on the same axis.
x
10.
Erase. Graph f ( x ) =
2
x
x+1
x+1
, and g( x )=
+ 1 on the same axis.
2x + 1
2x
4
VI.
Further Investigations
The graphing you have done so far is based on the rectangular coordinate system.
Another useful coordinate system is the polar coordinate system. Ordered pairs in the
polar system represent a distance from the origin and an angle (counterclockwise) from
the positive x-axis. For example, the polar point (1, π/2) is on the terminal side of the
angle π/2, which means it is on the positive y-axis, and it is a distance of 1 away from the
origin. In other words, it has rectangular coordinates (0, 1).
X(PLORE) can graph in polar coordinates. First type and enter
window(–8, 8, –6, 6)
grid(1, 1)
to help you interpret the following polar curves. To make X(PLORE) graph a polar
curve, use the command polarg. For example, type and enter
polarg(3, t = 0 to 2π) .
This graphs all points that are a distance of 3 away from the origin for all angles from 0 to
2π radians. It is simply a circle of radius 3 centered at the origin.
Try some more interesting polar curves:
1)
2)
3)
4)
5)
polarg(3sin(t), t = 0 to 2π)
polarg(3cos(t), t = 0 to 2π)
polarg(3sin(4t), t = 0 to 2π)
polarg(3cos(4t), t = 0 to 2π)
polarg(tan(t), t = 0 to 2π)
6)
7)
8)
9)
10)
polarg(3.5sec(t), t = 0 to 2π)
polarg(3.5sec(2t), t = 0 to 2π)
polarg(3.5csc(t), t = 0 to 2π)
polarg(3.5csc(2t), t = 0 to 2π)
polarg(3.5sec(2t)^2, t = 0 to 2π)
11)
12)
13)
14)
15)
16)
polarg(2 + 2sin(6t), t = 0 to 2π)
polarg(2 + 2sin(6t)^23, t = 0 to 2π)
polarg(t, t = 0 to 2π)
polarg(0.01t, t = 0 to 198π)
polarg(0.01t, t = 0 to 225π)
polarg(0.005t, t = 0 to 299π)
(erase)
(erase)
Make up some of your own polar graphs – try adding, multiplying, or dividing
trigonometric functions, or whatever other variations you can imagine.
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