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Special Topic: How Graphics Are OutputRedrawing and Combining Plots

1.9.3 Options

When Mathematica 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, Mathematica 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 Mathematica in making some of its choices.

There is a general mechanism for specifying "options" in Mathematica 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.

Choosing an option for a plot.

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. Section 1.9.4 will also discuss how you can change some of the options, even after you have produced the plot.

Some of the options for Plot. These can also be used in Show.

Here is a plot with all options having their default values.

In[1]:= Plot[Sin[x^2], {x, 0, 3}]


This draws axes on a frame around the plot.

In[2]:= Plot[Sin[x^2], {x, 0, 3}, Frame->True]


This specifies labels for the and axes. The expressions you give as labels are printed just as they would be if they appeared as Mathematica output. You can give any piece of text by putting it inside a pair of double quotes.

In[3]:= Plot[Sin[x^2], {x, 0, 3},

AxesLabel -> {"x value", "Sin[x^2]"} ]


You can give several options at the same time, in any order.

In[4]:= Plot[Sin[x^2], {x, 0, 3}, Frame -> True,

GridLines -> Automatic]


Setting the AspectRatio option changes the whole shape of your plot. AspectRatio gives the ratio of width to height. Its default value is the inverse of the Golden Ratio—supposedly the most pleasing shape for a rectangle.

In[5]:= Plot[Sin[x^2], {x, 0, 3}, AspectRatio -> 1]


Some common settings for various options.

When Mathematica 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.

Settings for the option PlotRange.

The setting for the option PlotRange gives explicit limits for the graph. With the limits specified here, the bottom of the curve is cut off.

In[6]:= Plot[Sin[x^2], {x, 0, 3}, PlotRange -> {0, 1.2}]


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

The function wiggles infinitely often when . Mathematica tries to sample more points in the region where the function wiggles a lot, but it can never sample the infinite number that you would need to reproduce the function exactly. As a result, there are slight glitches in the plot.

In[7]:= Plot[Sin[1/x], {x, -1, 1}]


More options for Plot. These cannot be used in Show.

It is important to realize that since Mathematica can only sample your function at a limited number of points, it can always miss features of the function. By increasing PlotPoints, you can make Mathematica sample your function at a larger number of points. Of course, the larger you set PlotPoints to be, the longer it will take Mathematica to plot any function, even a smooth one.

Since Plot needs to evaluate your function many times, it is important to make each evaluation as quick as possible. As a result, Mathematica usually compiles your function into a low-level pseudocode that can be executed very efficiently. One potential problem with this, however, is that the pseudocode allows only machine-precision numerical operations. If the function you are plotting requires higher-precision operations, you may have to switch off compilation in Plot. You can do this by setting the option Compiled -> False. Note that Mathematica can only compile "inline code"; it cannot for example compile functions that you have defined. As a result, you should, when possible, use Evaluate as described in Section 1.9.1 to evaluate any such definitions and get a form that the Mathematica compiler can handle.

Special Topic: How Graphics Are OutputRedrawing and Combining Plots