Pre-Composition and Graphing: Worksheet

In this Worksheet, we will practice a technique of analyzing a function to create a sketch by hand, and checking our results using XFunctions.  As before, we will give step-by-step instructions, with each instruction numbered separately.  You should work with a partner and make sure that both of you work with the program for some amount of time.  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.

For our first example, we will analyze and graph the function , based on our knowledge of the graph of .   Remember from the previous worksheet how it is important to choose the right points to plot so that your sketch displays the most important features of the graph.  In this example, the most important features of the graph of g:

are:

1. It has a kind of "vertex" at (0, 0), coming almost straight down to that point.
2. The graph curves downwards as it goes to the right.
3. It goes "off to infinity" up and to the right, but rising rather slowly.

We mentioned before that these features can be clearly displayed by connecting-the-dots through the points:

These are easily calculated points near the vertex and will display the curvature of the graph, if we connect-the-dots based on our knowledge of the shape of the graph.  Likewise, in order to accurately plot , we must make sure to plot the points corresponding to these three, otherwise, we will not know how to appropriately connect-the-dots.  The procedure that we describe in this worksheet is guaranteed to produce the desired points.

As before, we begin by decomposing f as:

1. Multiply the input by 2, then
2. multiply by -1 (i.e., negate), 
3. add 1, and
4. apply the square root function, .

We then translate each of these algebraic steps into a list of steps to calculate the corresponding table of points on the graph of f numerically:

1. Start with a table of points of ,
2. subtract 1 from the inputs, then
3. divide by -1 (i.e., negate), and
4. divide by 2.

Notice how we had to reverse each operation, as well as their order, to compute the correct, corresponding input values, in accordance with the graphing principles in the text.  We can summarize this analysis in a chart, similar to that used in the previous worksheet.

• Complete the following tables to calculate the three points on the graph of f, which correspond to these same three points; that is, calculate numerical values of the three inputs that yield the same outputs.  Also, record the geometric effect of each step.  Remember: Since we view each change as affecting the inputs of the function, all geometric effects will be in the horizontal direction.

  Corresponding Algebraic formula
  Geometric Effect	Original	Shift left 1
  Numerical Effect		Subtract 1	Divide by -1	Divide by 2
  Numerical  Results	Inputs				Outputs
  	0				0
  	1				1
  	4				2

  so that f goes through the points:  

  x
  	0 1 2

  Note: In the process, you are actually determining the graphs of each intermediate expression.  Answer.

• Plot the points for from the previous exercise, and use the the geometric description of the graph to connect-the-dots appropriately to create a fairly accurate sketch of the resulting graph.  Hint: Imagine (and sketch, if necessary) how the graph at each step would look.  Answer.
• Make up your own function, by filling in the blanks, .  Fill in the following chart and sketch a graph for your function.

  Corresponding Algebraic formula
  Geometric Effect	Original
  Numerical Effect
  Numerical  Results	Inputs				Outputs
  	0				0
  	1				1
  	4				2

  Answer.

We can perform a similar analysis to sketch a plot of h(x) = |-2x + 1|, based on our knowledge of the graph of k(x) = |x|:

Remember that the important features of this graph are:

1. It has a vertex at x = 0.
2. Its "sides" are straight lines.
3. The graph is "symmetric" with respect to flips of the x-axis.

and that three "important" points in this graph are:

since they display the vertex and the symmetry of the graph (and they are easy with which to compute).

• Complete the following tables to calculate the three points on the graph of h, which correspond to the three points in the previous table, and to determine how the graphs of k and h are related geometrically.

  Corresponding Algebraic formula	|x|	|x + 1|	|-x + 1|	|-2x + 1|
  Geometric Effect	Original
  Numerical Effect
  Numerical  Results	Inputs				Outputs
  	-1				1
  	0				0
  	1				1

  so that h goes through the points:  

  x	y = h(x) = |-2x + 1|
  	1 0 1

  Plot the points you calculated for y = h(x) and use your geometric description to envision how to correctly connect-the-dots to create a fairly accurate sketch of the resulting graph.  Answer.

• Make up your own function, by filling in the blanks, h(x) = |__x + __|.  Construct a chart similar to the previous exercise and sketch a graph.  Answer.

As a final example, we will apply this technique to sketch a graph of r(x) = ë-2x + 1û.  This will emphasize the importance of knowing how the graph is transformed geometrically, in addition to simply plotting points.  Remember that the graph of s(x) = ëxû:

is characterized by the fact that:

1. It consists of a series or horizontal lines that are closed on one side and open on the other.
2. The graph is "symmetric" with respect to a shift up and to the right.

Remember also how it is generally sufficient to carefully plot the points:

as long as we remember how the remainder of the graph may be sketched in relation to these three points.

• Complete the following tables to relate the graph of s(x) = ëxû with the graph of r(x) = ë-2x + 1û, and to calculate the three points on the graph of r, corresponding to the above table of points.

  Corresponding Algebraic formula	ëxû	ëx + 1û	ë-x + 1û	ë-2x + 1û
  Geometric Effect	Original
  Numerical Effect
  Numerical  Results	Inputs				Outputs
  	-1				-1
  	0				0
  	1				1

  so that r goes through the points:  

  x	y = r(x) = ë-2x + 1û
  	-1 0 1

  Plot the points you calculated for y = r(x) and use the the geometric description to envision how to draw in the rest of the resulting graph.  In particular, you should be careful to correctly place the open and closed circles on the graph.  Answer.

• Make up your own function, by filling in the blanks, r(x) = ë __x + __ û .  Fill in the following chart and sketch a graph for your function.

  Corresponding Algebraic formula	ëxû
  Geometric Effect	Original
  Numerical Effect
  Numerical  Results	Inputs				Outputs
  	-1				-1
  	0				0
  	1				1

  Answer.

Notice that the strategy given earlier accurately describes the process we used in this worksheet as well.  We simply need to remember that:

• the geometric and numerical effects are directly related to the reverse of each algebraic operation, 
• the effects are applied horizontally/to the inputs, and 
• we should write the operations in the reverse order of composition.

In the next section, we will show how to combine both types of analysis into a single table.

Go to Transformational Graphing

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Send questions or comments to jwicks@northpark.edu

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