When the Wolfram Language plots a graph for you, it has to make many choices. It has to work out what the scales should be, where the function should be sampled, how the axes should be drawn, and so on. Most of the time, the Wolfram Language will probably make pretty good choices. However, if you want to get the very best possible pictures for your particular purposes, you may have to help the Wolfram Language in making some of its choices.

There is a general mechanism for specifying "options" in Wolfram Language functions. Each option has a definite name. As the last arguments to a function like Plot, you can include a sequence of rules of the form name->value, to specify the values for various options. Any option for which you do not give an explicit rule is taken to have its "default" value.

A function like Plot has many options that you can set. Usually you will need to use at most a few of them at a time. If you want to optimize a particular plot, you will probably do best to experiment, trying a sequence of different settings for various options.

Each time you produce a plot, you can specify options for it. "Redrawing and Combining Plots" will also discuss how you can change some of the options, even after you have produced the plot.

When the Wolfram Language makes a plot, it tries to set the and scales to include only the "interesting" parts of the plot. If your function increases very rapidly, or has singularities, the parts where it gets too large will be cut off. By specifying the option PlotRange, you can control exactly what ranges of and coordinates are included in your plot.

The Wolfram Language always tries to plot functions as smooth curves. As a result, in places where your function wiggles a lot, the Wolfram Language will use more points. In general, the Wolfram Language tries to adapt its sampling of your function to the form of the function. There is a limit, however, which you can set, to how finely the Wolfram Language will ever sample a function.

It is important to realize that since the Wolfram Language can only sample your function at a limited number of points, it can always miss features of the function. The Wolfram Language adaptively samples the functions, increasing the number of samples near interesting features, but it is still possible to miss something. By increasing PlotPoints, you can make the Wolfram Language sample your function at a larger number of points. Of course, the larger you set PlotPoints to be, the longer it will take the Wolfram Language to plot any function, even a smooth one.

The Wolfram Language saves information about every plot you produce, so that you can later redraw it. When you redraw plots, you can change some of the options you use.

By using Show with a sequence of different options, you can look at the same plot in many different ways. You may want to do this, for example, if you are trying to find the best possible setting of options.

You can also use Show to combine plots. All of the options for the resulting graphic will be based upon the options of the first graphic in the Show expression.

All Wolfram Language graphics are expressions and can be manipulated in the same way as any other expression. Doing these kinds of manipulations does not require the use of Show.

Using Show[plot₁,plot₂,…] you can combine several plots into one. GraphicsGrid allows you to draw several plots in an array.

GraphicsGrid by default puts a narrow border around each of the plots in the array it gives. You can change the size of this border by setting the option Spacings->{h,v}. The parameters h and v give the horizontal and vertical spacings to be used. The Spacings option uses the width and height of characters in the default font to scale the h and v parameters by default, but it is generally more useful in graphics to use Scaled coordinates. Scaled scales widths and heights so that a value of 1 represents the width and height of one element of the grid.

When you make a plot, the Wolfram Language saves the list of points it used, together with some other information. Using what is saved, you can redraw plots in many different ways with Show. However, you should realize that no matter what options you specify, Show still has the same basic set of points to work with. So, for example, if you set the options so that the Wolfram Language displays a small portion of your original plot magnified, you will probably be able to see the individual sample points that Plot used. Options like PlotPoints can only be set in the original Plot command itself. (the Wolfram Language always plots the actual points it has; it avoids using smoothed or splined curves, which can give misleading results in mathematical graphics.)

Three-dimensional graphics can be rotated in place by dragging the mouse inside the graphic. Dragging inside the graphic causes the graphic to tumble in a direction that follows the mouse, and dragging around the borders of the graphic causes the graphic to spin in the plane of the screen. Dragging the graphic while holding down the Shift key causes the graphic to pan. Use the Ctrl key to zoom.

There are many options for three‐dimensional plots in the Wolfram System. Some are discussed here; others are described in "The Structure of Graphics and Sound".

The first set of options for three‐dimensional plots is largely analogous to those provided in the two‐dimensional case.

Probably the single most important issue in plotting a three‐dimensional surface is specifying where you want to look at the surface from. The ViewPoint option for Plot3D and Show allows you to specify the point in space from which you view a surface. The details of how the coordinates for this point are defined are discussed in "Coordinate Systems for Three-Dimensional Graphics". When rotating a graphic using the mouse, you are adjusting the ViewPoint value.

The human visual system is not particularly good at understanding complicated mathematical surfaces. As a result, you need to generate pictures that contain as many clues as possible about the form of the surface.

View points slightly above the surface usually work best. It is generally a good idea to keep the view point close enough to the surface that there is some perspective effect. Having a box explicitly drawn around the surface is helpful in recognizing the orientation of the surface.

To add an extra element of realism to three‐dimensional graphics, the Wolfram System by default colors three‐dimensional surfaces using a simulated lighting model. In the default case, the Wolfram System assumes that there are four point light sources plus ambient lighting shining on the object. "Lighting and Surface Properties" describes how you can set up other light sources, and how you can specify the reflection properties of an object.

Lighting can also be specified using a string that represents a collection of lighting properties. For example, the option setting Lighting->"Neutral" uses a set of white lights, and so can be faithfully reproduced on a black and white output device such as a printer.

The Wolfram Language can be used to make plots of functions. You give the Wolfram Language a function, and it builds up a curve or surface by evaluating the function at many different points.

Here we describe how you can make plots from lists of data, instead of functions. ("Importing and Exporting Data" discusses how to read data from external files and programs.) The Wolfram Language commands for plotting lists of data are direct analogs of the ones for plotting functions.

"Basic Plotting" describes how to plot curves in the Wolfram Language in which you give the coordinate of each point as a function of the coordinate. You can also use the Wolfram Language to make parametric plots. In a parametric plot, you give both the and coordinates of each point as a function of a third parameter, say .

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max}] is the direct analog in three dimensions of ParametricPlot[{f_x,f_y},{t,t_min,t_max}] in two dimensions. In both cases, the Wolfram Language effectively generates a sequence of points by varying the parameter t, then forms a curve by joining these points. With ParametricPlot, the curve is in two dimensions; with ParametricPlot3D, it is in three dimensions.

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max},{u,u_min,u_max}] creates a surface, rather than a curve. The surface is formed from a collection of quadrilaterals. The corners of the quadrilaterals have coordinates corresponding to the values of the f_i when t and u take on values in a regular grid.

In general, it is possible to construct many complicated surfaces using ParametricPlot3D. In each case, you can think of the surfaces as being formed by "distorting" or "rolling up" the - coordinate grid in a certain way.

You should realize that when you draw surfaces with ParametricPlot3D, the exact choice of parametrization is often crucial. You should be careful, for example, to avoid parametrizations in which all or part of your surface is covered more than once. Such multiple coverings often lead to discontinuities in the mesh drawn on the surface, and may make ParametricPlot3D take much longer to render the surface.

As discussed in "The Structure of Graphics and Sound", the Wolfram Language includes a full graphics programming language. In this language, you can set up many different kinds of plots.

Some of the plotting functions available are defined in the Standard Extra Packages included with the Wolfram System.

On most computer systems, the Wolfram System can produce not only graphics but also sound. The Wolfram System treats graphics and sound in a closely analogous way.

For example, just as you can use Plot[f,{x,x_min,x_max}] to plot a function, so also you can use Play[f,{t,0,t_max}] to “play” a function. Play takes the function to define the waveform for a sound: the values of the function give the amplitude of the sound as a function of time.

Sounds produced by Play can have any waveform. They do not, for example, have to consist of a collection of harmonic pieces. In general, the amplitude function you give to Play specifies the instantaneous signal associated with the sound. This signal is typically converted to a voltage, and ultimately to a displacement. Note that amplitude is sometimes defined to be the peak signal associated with a sound; in the Wolfram System, it is always the instantaneous signal as a function of time.

Play is set up so that the time variable that appears in it is always measured in absolute seconds. When a sound is actually played, its amplitude is sampled a certain number of times every second. You can specify the sample rate by setting the option SampleRate.

In general, the higher the sample rate, the better high‐frequency components in the sound will be rendered. A sample rate of r typically allows frequencies up to hertz. The human auditory system can typically perceive sounds in the frequency range 20 to 22000 hertz (depending somewhat on age and sex). The fundamental frequencies for the 88 notes on a piano range from 27.5 to 4186 hertz.

The standard sample rate used for compact disc players is 44100. The effective sample rate in a typical telephone system is around 8000. On most computer systems, the default sample rate used by the Wolfram System is around 8000.

You can use Play[{f₁,f₂,…] to produce stereo sound. In general, the Wolfram System supports any number of sound channels.

The function ListPlay allows you simply to give a list of values which are taken to be sound amplitudes sampled at a certain rate.

When sounds are actually rendered by the Wolfram System, only a certain range of amplitudes is allowed. The option PlayRange in Play and ListPlay specifies how the amplitudes you give should be scaled to fit in the allowed range. The settings for this option are analogous to those for the PlotRange graphics option discussed in "Options for Graphics".

While it is often convenient to use the setting PlayRange->Automatic, you should realize that Play may run significantly faster if you give an explicit PlayRange specification, so it does not have to derive one.

A Sound object in output is typically formatted as a button which contains a visualization of the sound and which plays the sound when pressed. Sounds can be played without the need for user intervention or producing output by using EmitSound. In fact, the internal implementation of Sound buttons uses EmitSound when the button is pressed.

The internal structure of Sound objects is discussed in "The Representation of Sound".