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C HA PTE R 4
Two-Dimensional Graphics
4.1 Plotting Functions of a Single Variable
Anyone who has ever tried to plot a graph using one of the standard programming languages will appreciate the ease with which graphs can be produced in Mathematica. In many instances, only one simple instruction is all that is needed to produce a pictorial representation of a function or a more general relationship between two variables. Although Mathematica s defaults work well in most instances, there are many options available to control subtleties. We shall describe the more common ones in this section and present a variety of examples that illustrate the ease with which graphs may be constructed. The basic command for drawing the graph of a function is Plot. Although x is used as the independent variable in the following description, any symbol may be used in its place.
Plot[f[x], {x, xmin, xmax} plots a two-dimensional graph of the function f(x) on the interval xmin x xmax.
EXAMPLE 1 Plot the parabola f(x) = x2 from 3 to 3.
Plot[x2, {x, 3, 3}]
Two functions can be plotted on the same set of axes. Mathematica draws each in a different color.
Plot[{f[x], g[x]}, {x, xmin, xmax}] plots the graphs of f(x) and g(x) from xmin to xmax on the same set of axes. This command can be generalized in a natural way to plot three or more functions.
EXAMPLE 2 Plot f(x) = x2 and g(x) = 9 x2 from 3 to 3.
Two-Dimensional Graphics
Plot[{x2, 9 x2}, {x, 3, 3}]
When plotting points over a specified interval, Mathematica makes a decision on the range of points to plot in order to produce a pleasing graph. PlotRange is an option that allows the user to override Mathematica s default.
PlotRange Automatic is Mathematica s default. Any points whose vertical coordinates appear to be too large (e.g., outliers) are omitted from the graph. PlotRange All forces Mathematica to plot all points. PlotRange {ymin, ymax} plots only those points whose vertical coordinates fall between ymin and ymax. PlotRange {{ xmin, xmax } , { ymin, ymax }} plots those points whose horizontal coordinates fall between xmin and xmax and whose vertical coordinates fall between ymin and ymax.
EXAMPLE 3
1 /; x < 2.9 || x > 3.1 (x 3)2 f[x_] 100 /; 2.9 x 3.1 f[x_] Plot[f[x], {x, 0, 6}]
Plot[f[x],{x,0,6},PlotRange All]
40 20
1 2 3 4 5 6
The Show command is useful for plotting several graphs simultaneously, particularly when their domains are different intervals.
Show[g1, g2, . . .] plots several graphs on a common set of axes.
Two-Dimensional Graphics
EXAMPLE 4 Suppose we wish to plot the graph of y = x2 9 on the interval [ 4, 4] and the graph of y = sin x on the interval [0, 2 ], but wish to plot them on one set of axes. We define two graphics objects, g1 and g2.
g1 = Plot[x2 9, {x, 4, 4}]
g2 = Plot[Sin[x], {x, 0, 2 o}]
1 0.5
1.0
Now we apply the Show command. Note how the axes are adjusted to exhibit both graphs: Show[g1, g2,PlotRange All]
Two-Dimensional Graphics
You will notice that in defining g1 and g2, each curve was drawn individually on its own axis. To suppress this output, a semicolon (;) can be placed at the right side of each plot command.
EXAMPLE 5
g1 = Plot[x2 9]; g2 = Plot[Sin[x] {x, 0, 2 o}]; , Show[g1, g2, PlotRange All]
Only the combined graph is drawn, not g1 and g2.
A useful command for drawing multiple graphs is GraphicsArray.
GraphicsArray[{g1, g2, ...}] plots a row of graphics objects. GraphicsArray[{g11, g12, ...},{g21, g22, ...}}] plots a two-dimensional array of graphics objects.
EXAMPLE 6
g1 = Plot[x, {x, 2, 2}]; g2 = Plot[ x, {x, 2, 2}]; g3 = Plot[x2, {x, 2, 2}]; g4 = Plot[ x2, {x, 2, 2}]; GraphicsArray[{g1, g2, g3, g4}]
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