Basics of Graphing

We have already used the basic plotting command sequence

plot(expression,x=a..b,optional stuff including the range on the dependent variable);

We have already seen that sometimes the key to getting a nice graph involves analyzing the expression to determine the appropriate range of values for x and y. Here we explore some more of MAPLE's plotting features. First we see how to plot multiple graphs on the same axes. The key thing to remember is that "curly braces", { }, are used to denote a set of objects (without regard to the ordering of the objects). For example, 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

denotes the set containing the functions 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 . Try the MAPLE command

plot({x^2-2,x^2-1,x^2, x^2+1},x=-5..5,y=-5..25);

You just asked MAPLE to plot the set of all 4 functions all at once. Of course we could name the set M

M:={x^2-2,x^2-1,x^2, x^2+1};

and then simply plot M.

plot(M,x=-5..5,y=-5..25);

MAPLE has a sequence generator command that will come in handy. Try the MAPLE command

seq(x^2+i,i=-2..1);

Use this with the above plot command to obtain the same 4 graphs. (Note, you can copy and paste.)

plot({seq(x^2+i,i=-2..1)},x=-5..5,y=-5..25);

1. Why does the sequence need to be included in the curly braces?

Note that the graph of 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 eventually is entirely above the x-axis. Plot the graphs of 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 for the values of i = -3, -2, -1, 0, 1,2,3 .

2. Is it possible to find any value of i so that the graph of 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 is entirely above the x-axis? Explain your answer.

Sketch the graph of 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 .

3. Where are the vertical asymptotes?

4. Is the vertical line really a part of the graph? Explain.

"Click" in the vicinity of the graph to frame the graph and to obtain a set of "buttons" associated with the "style" of the graph. Experiment with these "buttons" to see how they change the style of the graph. In the case of our above rational function, plotting only the data points might be appropriate. Include the graph of the same function plotted in this "points only" style. (Remember, you can copy and paste the MAPLE command.)

In each case below, graph the given set of functions on the same axes.

{k*sin(x), k = -3 .. 3};

5. Explain the effect of changing the parameter k on the graph of 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 .

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

6. Explain the effect of changing the parameter k on the graph of 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 .

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

7. Explain the effect of changing the parameter k on the graph of 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 .

Sketch the graphs of the following functions: 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 , 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 , 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 ).

8. In a, b, c, and d plot the graph of the function and explain what the function is doing.

a. y=abs(x) .

Explanation:

b. y= 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 .

Explanation:

c. y= 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 .

Explanation:

d. y= 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 .

Explanation:

We'll finish by considering some more sophistificated plotting procedures contained in a special plotting package loaded using the command

with(plots);

Notice all the various commands contained in this package. Many of them should "ring a bell" with you.

The command for graphing implicit relations (the conic sections for example) is

implicitplot(relation in x and y, x=a..b,y=c..d, optional stuff)

9. On the same axes plot the graphs of the following conics 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 for i = -3, -2, -1, 0, 1, 2, 3. View the graph in 1-1 perspective.

10. On the same axes plot the graphs of the following circles 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 for i = -3, -2, -1, 0, 1, 2, 3. View in 1-1 perspective.

What is the effect on these circles of changing the parameter k?

For the remainder of the exercises you must use the "on line" help with its accompanying examples to figure out how to sketch the desired graphs.

11. Sketch the 3-d plot, the 2-d contour plot, and the 3-d contour plot of each of the following functions (each on its own axes). You can "click" on the graph, "drag" it around. Experiment to get the "best" picture.

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

12. Plot the graphs of the following 3-d implicit relations. Rotate as necessary to get a good view of the graph. (Choose reasonable values for the ranges of the x and y variables!)

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

13. Plot the following polar graphs in polar coordinates.

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14. Graph the following parametric equations in rectangular coordinates.

(This one is tricky. Use the appropriate plot command (2-d or 3-d) to plot the "vector" [x(t),y(t)]

or [x(t,s),y(t,s),z(t,s)] .)

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This graph is called a "surface". Does it look to you like a "smooth surface?"