Introduction to Graphing: Worksheet

In this Worksheet, we will see how we may use XFunctions to create and manipulate functions This program has a large number of features that we will not use in this course, so we will only give a brief introduction to the basic features that we want to use right away.  We will discuss more advanced features as necessary.  However, you should feel free to read the complete documentation and experiment on your own, if you wish.  In particular, you should notice how the program supports four out of the seven different perspectives on functions, namely as a formula, graph, black-box, or table of values.

We will give step-by-step instructions on using XFunctions.  Please try to carefully follow the directions written in this section.  Each instruction will be numbered separately.  As you work through the Worksheet, try to answer the bulleted questions.  As usual, you should give an honest attempt to answer the question before looking at the solutions.  Warning: If you do want to look at the Answers before completing the Worksheet, make sure to "right-click" on the link to open the link in a new window; otherwise, your XFunctions window will disappear and you will lose any work you may have done.

You should work with a partner.  You should also make sure that both of you try working with the program for some amount of time. 

  1. Start XFunctions by clicking  on this button: .  
  2. Resize your browser window and position both windows so you can read these instructions and work with XFunctions.  Do not resize the XFunctions window, or you will distort the dimensions of the graphing window.

Notice that the XFunctions window initially shows a list of pre-defined functions on the left, a plot in the middle, a pull-down menu at the top, and various control buttons and boxes along the right and bottom.

  1. Hit the "Zoom Out" button once.  Observe carefully what happens to the limits of the plot window.  Now hit it two more times and see if you can detect precisely how it sets the limits.
  2. Now hit the "Restore" button.  Observe carefully what happens to the limits of the plot window.  Now hit the "Zoom In" button once and the "Restore" button again.  

Note: The physical dimensions of the plot window are in an approximate ratio of 6:5; this means if you want the units on the horizontal and vertical axes to look equal, then you need to choose the limits on the axes to be in the same ratio.

  1. Change the values in the "xmin" and "xmax" boxes to -6 and 6, respectively.  Notice how the sides of the absolute value graph now look to be 45°, as they should.  Click on the "Equalize" button to have the horizontal limits to be set exactly to a "true" picture of the graph.

The Main Screen

The Main Screen allows you to obtain a quick graph of a single function and examine values of that function.

  1. Click anywhere on the graph.

Notice how, when you click on the graph, a pair of crosshairs (which looks like a red plus sign) appears on the graph at the same horizontal position but on a plotted value of the function.  Also, the x-value (i.e., the horizontal coordinate) appears in the box at the bottom, with the corresponding functional evaluation displayed directly above.  Note: In this way, XFunctions allows you to treat a function as a machine taking inputs to outputs, without needing to pay attention to the formulas that may be involved in evaluation.

  1. Click-and-drag over the graph.

Notice how the crosshairs move along the graph, directly above or below where you are pointing (i.e., with the same x-value), and the values at the bottom of the window change automatically, as well.

  1. Enter the number -3 and hit the "Set" button (or hit the Enter key).
  1. Select the floor function from the list on the left.  Now enter several different decimal numbers, such as -2.5, 3, 4.12, etc. in the "x =" box and record the results.

Making New Functions

Now we will move on to define our own functions.  Specifically, the buttons in the lower-left of the window allow us to define new functions either by a formula, as a graph, or as a table of values (i.e., three of the other different perspectives on functions).  While XFunctions will allow us to define a function either algebraically, graphically, or numerically, we will focus primarily on the algebraic approach.  

We will show how to enter three different functions: f(x) = 3x2, , and:

  1. Click on the "New Expr." button.

This will change the window to allow you to enter the name of the function, a formula (which may be defined by up to four cases), and the limits on the plot window.

  1. If we begin with f, then you should enter "f" in the "Name of function" box.  Likewise, enter "3*x^2" in the "y = " box.  Notice how we have to explicitly put in a "*" sign for multiplication, and how we use "^" for powers.   When you are finished, click the "Done" button.

Now we enter g.  Notice that a "sqrt" function is already built into XFunctions in the original list of functions.

  1. Click on the "New Expr." button.  Enter "g" in the "Name of function" box, and "4*sqrt(-x+3)-5" in the "y = " box.  Notice how we use the pre-defined "sqrt" function to take square roots.   When you are finished, click the "Done" button.
  2. Click on the "Zoom Out" button to get a better view of this function.

Once we learn some basic graphing principles, we will be able to sketch this graph quickly by hand, even without XFunctions.  XFunctions will then be useful in helping us to learn and practice these principles.

Finally, we will enter h.  This is defined in three cases.

  1. Click on the "New Expr." button.  Enter "h" in the "Name of function" box.  Now enter "f(x)" in the first "y = " box, with "x<-1" in the corresponding "provided:" box (Note: In this context, "provided" means the same as "in case").  Notice again how we can use our previously defined function, f, instead of typing out the formula "3*x^2".   Then, enter "2*x-1" in the second "y = " box, with "-1<=x and x<=1" in the corresponding "provided:" box.  Notice how, instead of simply typing -1 ≤ x ≤ 1, we have to enter "<=" for "≤", and how we must specify both ends of the interval separately (cf. the discussion of interval notation).  Finally, enter "sqrt(x)" in the third "y = " box, with "3<x" in its "provided:" box. When you are finished, click the "Done" button.

Now we will briefly look at creating functions given by a table or graph, such as:

x k(x)
-2
-1
0
1
2		 1
-2
-1
3
1
or
  1. Click on the "New Table" button.  Enter "k" in the "Name of Function" box.  Now enter "-2" and "1 " in the "Input x" and "Input y" boxes, respectively, and hit the "Add (x, y) to Table", or hit the "Enter" key.  Notice how this point appears as the first line in the table to the right.  Now repeat the process to add the other four points.  Finally, notice that XFunctions insists on trying to extend your function definition over an entire interval, from the smallest x-value to the largest; select the "Piecewise Linear Function" option.  Finally, set the horizontal limits on your plot to [-2, 2], with the vertical limits as [-4, 4], and click the "Set Axis Limits" button. When you are finished, click the "Done" button.
  1. Click on the "New Graph" button.  Enter "q" in the "Name of function" box.  Set the horizontal limits on your plot to [-2, 2], with the vertical limits as [-4, 4], and click the "Set Axis Limits" button. Notice how the graph already includes two points (with a "handle" on the "inner" side) at x = -2 and 2.  Double-click on the x-axis to add new points at x = -1, 0, 1.  Notice how each new point has "handles" on both sides.  For each point you insert, click and drag the point so the graph goes through the following points:
    -2
    -1
    0
    1
    2    		 1
    -2
    -1
    3
    1

    Note: You will need to estimate the correct vertical position, since the axis is not completely marked.  Then, click and drag on the "handles" to try and reshape the graph to make it look like:

    When you are finished, click the "Done" button.

The Multigraph Utility

The last feature of XFunctions that we will discuss, and which we will use extensively in the upcoming sections, is the Multigraph Utility.  This allows us to plot up to eight functions on the same axes for comparison purposes.

  1. Select "Multigraph Utility" from the menu at the top of the XFunctions Main Screen.  Enter "floor(x)" in the "y = " box, and hit the "Graph!" button or the "Enter" key.
  2. Hit the "Equalize" button to obtain a truer picture of the "floor" function.  You will notice that the dimensions of the plot in the Multigraph Utility are in an approximate ratio of 8:5.
  3. Now select "No. 2" in the menu in the lower left, and enter "2*floor(x)" in the "y = " box, and graph this.

Notice how the two different graphs appear on the same axes, but in different colors.  The corresponding color of each function is displayed to the right of the selection menu.  Note: Where the graphs overlap, only the color of the last function will be visible.

  1. Hit the "Clear All" button.  Notice how it erases both graphs.
  2. Select each of the two functions, in turn, and re-graph both of them.
  3. Select the "No. 2" function and hit the "Remove" button.  Notice how it deletes the previous function definition, as well as the graph.  Now enter "-floor(x)" in the "y = " box, and graph this.

In answering these last two questions, you may have just discovered for yourself two of the graphing principles which we will discuss in more detail in the next section.

Go to Arithmetic and Graphing

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