Why is WebGraphing.com easier to use than a standard graphing
To graph a function using a standard graphing calculator,
you need to select a viewing window—the plotting
interval and range—guessing
at values you hope will include all the important mathematical
features. This may require several tries, and there is
no guarantee that your
final graph includes everything. WebGraphing.com does
not require trial and error. It uses behind-the-scenes mathematical
analyses to automatically select a viewing window that includes
all the important mathematical features. After seeing the “complete
graph,” you can graph again with a viewing window of
your own choosing or use Smart Zooming to focus on a
particular interval and minimize the guesswork..
How do I become a
Becoming a Member is free and easy: just fill out the registration
form and respond to the confirming email that WebGraphing.com sends
to you. Membership is open to students, teachers, and parents of
A+ membership is by
subscription, but you must first become a Member. There are several
ways to become an A+ Member. Whether you just want to study for
an upcoming test, get material for a math project, or use it for
a whole semester or year, there are subscription periods as short
as 2 days and as long as a year.
What are the
benefits of membership?
Membership entitles you to: (1) complete graphs of all polynomials
of degree less than 4, (2) detailed analyses explaining the mathematics
that produces the complete graphs, (3) access to post and respond
to questions on the forum, and (4) our Tricks of the Trade for
graphing polynomials. In addition, you can sample premium content: complete graphs with analyses for any functions.
A+ membership entitles
you all the benefits of membership plus: (1) complete
graphs of all mathematical functions studied in algebra, precalculus,
and first year calculus, (2) detailed analyses explaining the mathematics
that produces the complete graphs, (3) priority in getting homework
questions answered on the forum by a WebGraphing.com moderator
(mathematician, math teacher, or math professor), and (4) our Tricks
of the Trade for
graphing rational, algebraic, and transcendental (nonalgebraic)
What is the
difference between the basic, intermediate and advanced graphing
The Basic Calculator keypad is designed for algebra and beginning
precalculus students so as not to intimidate budding “apprentice
mathematicians” with unnecessary jargon. It can be used to
enter any arithmetic combination of expressions in x as well
as powers of x, using the power symbol ^ (as in x^2,
read x squared). Each of the basic, intermediate and advanced
keypads can be used interchangeably with your computer keypad to
enter any function, including functions on one of the other keypads.
Note that when you click the π key, the letters “pi” are
entered; alternatively you can type “pi” to enter the
The Intermediate Calculator
keypad additionally has 5 function keys for convenience in entering “named” functions
that students first meet: square root, absolute value, natural
logarithms, logarithms to any base, and exponential functions.
Note that when you click any of these function keys, say log,
the letters with opening left parenthesis “log(“ are
automatically entered for your convenience. After entering the
argument, say 2x+1, you must also enter the closing right parenthesis, “)”,
to keep parentheses paired off, so the final entry reads: log(2x+1).
The Advanced Calculator
keypad additionally has all six trig functions and their inverses
as well as all six hyperbolic trig functions and their inverses.
Note that when you click, say, the inverse sine key, sin-1,
the letters with left parenthesis "arcsin(" are entered.
Inserting the letters “arc” in front of any trig or
hyperbolic trig function is the traditional way of referring to
the corresponding inverse function.
Are there any
special syntax requirements?
When you click, say, the sqrt key, the letters entered are “sqrt(“;
that is, the function name is entered together with the left parentheses. You
need to enter the argument, say, x+1, followed by the right parentheses, “x+1)” so
it finally reads: sqrt(x+1). Entering “sqrtx+1” would
produce the response: “Oops! Syntax Error.” Even if
the argument is simply “x”, our syntax requires that
parentheses be used: sqrt(x).
To enter a root function,
say the cube root of x, you need to write it using the power operator,
^, with a fractional exponent enclosed in parentheses: x^(1/3). As
a consequence, there are two way to enter the square root of x:
sqrt(x) or x^(1/2).
Note that the log key
refers to the logarithm of x to the base e. If you want to refer
to another base, say 5, you would type “log(5,x)”. It
is for this reason that, log(x) can be used interchangeably with
log(e,x) and ln(x).
To minimize syntax errors,
you can use any of the onscreen keypads . For example, to graph
the inverse tangent of x, if you click the inverse tangent
key, tan-1, you will see “arctan(“ entered
for you in the entry field after “y=”. You can complete
the entry by typing “x)” so the final entry reads:
arctan(x). Also, when you click the key for the exponential function, ex,
you will see “e^(“ in the entry field. Again, you
need to enter the argument, say x, followed by the closing right
parenthesis so it reads: e^(x).
Why would I generate
a random graph?
This is a quick way for a visitor to get a live sample of what the
web site has to offer. When a visitor clicks the generate a random
graph button, WebGraphing.com returns a complete
graph of a randomly generated polynomial function up to degree 3
along with descriptive answers in The Analyzer that
explain the mathematics needed to produce the graph by hand. A visitor
will see that by rolling over the graph, it turns from red and blue,
which highlight segments where the graph is increasing or decreasing,
to green and purple, which highlight segments where the graph is
concave up or concave down. In addition, a visitor can explore the
breadth of mathematical coverage and the details that the descriptive
answers provide in The Analyzer.
When a Member clicks
the generate a random graph button, WebGraphing.com returns
a complete graph of a randomly generated polynomial function up
to degree 5 along with descriptive answers in The Analyzer that
explain the mathematics needed to produce the graph by hand. This
is a quick way to explore a number of live higher degree polynomials—what
they look like and their various properties.
When an A+ Member clicks
the generate a random graph button, depending on the level
of graphing calculator selected, WebGraphing.com returns
a complete graph of a randomly generated elementary, intermediate
or advanced function with descriptive answers in The Analyzer that
explain the mathematics needed to produce the graph by hand. This
is a convenient way to explore various live intermediate-level
and advanced mathematical functions—what they look like and
their various properties.
Why does the graph
change color when I roll over it?
To aide visualization, when you move your cursor over your graph,
it changes from red and blue to purple and green, where purple highlights
the concave up curve segments and green highlights the concave down
the benefits of seeing color-coded graphs?
In addition to aiding your visualization of the increasing/decreasing
segments and concave up/concave down segments, the color coding serves
the purpose of identifying turning points and points of inflection
even when a curve appears flat in a certain region. You can be sure
that if a curve is flat but has both red and blue meeting at a juncture,
there is a turning point at the juncture. In our Rational Examples,
see the case of the “hidden” turning point. This “hidden” turning
point can be revealed on resizing in the vicinity of the juncture
of the two colors, most conveniently by using Smart Zooming. The
same can be said for “hidden” changes in concavity. If
a curve appears flat in a certain region that includes the juncture
of green and purple colors, despite the appearance of being flat,
the shape of that curve is changing concavity at the juncture and
it may be possible to confirm this visually by resizing the graph
in the vicinity of the juncture.
What happens to the color coding for straight
Straight lines are colored red when they are increasing, blue when they are decreasing,
and black when they are horizontal since a horizontal line is neither increasing
nor decreasing. Also, since straight lines are neither concave up nor concave
down, when you move your cursor over the graph, the line is colored black to
signify the absence of concavity.
Aside from color-coding, what other “important
mathematical features” are included in the graph?
Certain functions have breaks, discontinuities, and asymptotic behaviors, which
all good math textbooks illustrate by using dashed lines and small empty or filled
circles. The illumination of these important mathematical features is an invaluable
addition to a graph, required to visually identify the special attributes of
a function. WebGraphing.com automatically includes these features in its
graphs, while standard graphing calculators do not.
What is Smart
After seeing the complete graph, if you enter only the lower (Xmin)
and upper (Xmax) bounds on the x-values and click the Graph
Again button, Smart Zooming takes over
and WebGraphing.com automatically determines
optimal lower (Ymin) and upper (Ymax) bounds for the y-values;
the resulting viewing window includes all the important mathematical
features on your selected interval (Xmin,Xmax). This is convenient
and more powerful than ordinary proportional zooming since you don’t
have to go through trial and error to get an optimal graph. Of course,
you can enter your own lower (Ymin) and upper (Ymax) bounds on the y-values
and click the Graph Again button to resize
the graph to your own liking. In fact, you can do this repeatedly.
any other unique features on WebGraphing.com?
Whenever feasible, in our descriptive answers we give exact numbers,
like the ones you get solving equations longhand when you are not
using a calculator. For example, if we have an equation to solve,
say x2 = 2, we give the exact (“symbolic” is
the technical jargon) answer x = ± . We also give the approximate value x = ±1.41 to
two decimal places. When the exact value is not too lengthy, you
can expect to see the exact value. This feature—giving both
exact and approximate answers—helps to make connections between
symbolic values and their magnitude.
graph any function?
designed to analyze and graph functions typically found in math
textbooks, from algebra through precalculus
and first-year calculus. We tested our comprehensive graphing system
with graphing problems from eight calculus textbooks and it handled
all these problems successfully. Of course, some advanced functions
can take longer to analyze and graph and WebGraphing.com is
not designed to spend an inordinate amount of time with any particular
function. For such advanced functions, when analysis takes too
long, we deliver one or both graphs in black with whatever analyses
possible within the time constraints. If you come across a function
for graphing in your math textbook that you believe was not handled
properly, please put it out on our forum (mention the title, author,
edition, and page number for reference) so we can investigate it. We
aim to improve.
print out the graphs and solutions?
Yes, you can use your browser’s print
button to print out any information on the web site. Also, you
can right click on any graph
and save the graph to your computer as a picture that can subsequently
be inserted into your document. Teachers can use this feature for
inserting graphs into their test questions. Students can use this
feature for inserting graphs into their math projects.
How do I use The
In general, algebra and precalculus students will be most interested
in the analyses on the left side of The Analyzer pad
while calculus students will be most interested in the analyses on
the right side of The Analyzer pad. Clicking on, say,
the Intercepts button will give details about x- and y-intercepts
and how to find them. Clicking the Show All Analyzer Solutions button
will enable you to see all the details at once. The latter can also
be printed, together with the graph, for offline reference and study.
What are Tricks of the Trade?
The Tricks of the Trade web page offers practical
advice for graphing functions by specific category: polynomial functions,
rational functions, algebraic functions, and transcendental (nonalgebraics)
functions. It provides useful information for developing both an
overview of properites specific to each type plus pinpointed advice
on what to look for.
How do I post to the forum?
Once you become a member, you are entitled to post questions on the forum and
answer other members' questions. The forum is moderated, so moderators may,
their discretion, edit questions to improve clarity or exclude questions that
are off topic. Along with other members, moderators may also answer questions.
What is Guess
Guess the Graph is a game you can play to see how well
you can spot differences between the four specific categories of
graphs: polynomial, rational, algebraic, and transcendental (nonalgebraic). Strictly
speaking, polynomial functions are also rational functions and
rational functions are also algebraic functions, so being specific
careful consideration. There are over 200 graphs that are selected
randomly while you play.