Options for Graphics
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.
Plot[f,{x,x_{min},x_{max}},option>value] 
 make a plot, specifying a particular value for an option 
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.
"Redrawing and Combining Plots" 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.
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This draws axes on a frame around the plot.
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This specifies labels for the x and y axes. The expressions you give as labels are printed just as they would be if they appeared as TraditionalForm Mathematica output. You can give any piece of text by putting it inside a pair of double quotes.
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You can give several options at the same time, in any order.
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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.
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Some common settings for various options.
When
Mathematica makes a plot, it tries to set the
x and
y 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
x and
y coordinates are included in your plot.
Automatic  show at least a large fraction of the points, including the "interesting" region (the default setting) 
All  show all points 
{y_{min},y_{max}}  show a specific range of y values 
{xrange,yrange}  show the specified ranges of x and y values 
Settings for the option PlotRange.
The setting for the option PlotRange gives explicit y limits for the graph. With the y limits specified here, the bottom of the curve is cut off.
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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 x0. 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.
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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.
Mathematica 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
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.
More options for Plot. These cannot be used in Show.
When plotting multiple functions, PlotStyle settings in a list are used sequentially for each function.
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By default nothing is indicated when the PlotRange is set so that it cuts off curves.
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This specifies filling between the curve and the x axis.
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The filling can be specified to extend to an arbitrary height, such as the bottom of the graphic. Filling colors are automatically blended where they overlap.
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This specifies a specific filling to be used only for the first curve.
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This shows a filling from the first curve to the second using a nondefault filling style.
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