Post-Composition and Graphing: Solutions

Here are solutions to the questions on the worksheet accompanying the section Post-Composition and Graphing.  

• Complete the following tables to calculate three points on the graph of f.  For many of the examples we will consider, it is sufficient to compute with the inputs x = -1, 0, 1.

  Corresponding Algebraic formula	x	|x|	2|x|	-2|x|	-2|x| + 1
  Numerical Effect	Take the input	Apply the absolute value function	Multiply by 2	Multiply by -1	Add 1
  Numerical  Results	Inputs	Outputs
  	-1
  	0
  	1

  so that f goes through the points:  

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

  Answer

  Simply following the English instructions gives:

  Corresponding Algebraic formula	x	|x|	2|x|	-2|x|	-2|x| + 1
  Numerical Effect	Take the input	Apply the absolute value function	Multiply by 2	Multiply by -1	Add 1
  Numerical  Results	Inputs	Outputs
  	-1	1	2	-2	-1
  	0	0	0	0	1
  	1	1	2	-2	-1

  Thus, the graph of f goes through the points:  

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

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• Complete the following table to relate the graph of g with the graph of f.   

  Corresponding Algebraic formula	y = |x|	y = 2|x|	y = -2|x|	y = -2|x| + 1
  Geometric Effect	Take the graph of the absolute value function	Stretch vertically by a factor of 2

  Answer

  Using the patterns discovered in the previous section, we can complete the table as follows:

  Corresponding Algebraic formula	y = |x|	y = 2|x|	y = -2|x|	y = -2|x| + 1
  Geometric Effect	Take the graph of the absolute value function	Stretch vertically by a factor of 2	Flip the vertical axis	Shift vertically, up by 1

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• Plot the points you calculated earlier for y = = -2|x| + 1.  Using the the geometric information from the previous Exercise, connect-the-dots to create a fairly accurate sketch of the resulting graph.  Hint: Imagine, in your mind's eye what the arithmetic operations would do geometrically to the graph of the absolute value function; if necessary, make a rough sketch of the graph after each operation.

  Answer

  Normally the two sides of the absolute value function go up at 45° angles.  Stretching this vertically by 2 (away from the x-axis) will make both sides steeper, leaving the vertex at the origin.  Flipping the vertical axis will keep both sides as steep, but the graph will now open downwards.  Finally, shifting up by 1 will keep both sides as steep, but the entire graph will move up 1 unit, as will be clearly visible at the vertex.  This can be seen by selecting the series of Examples from the following applet: 

  Notice that the final graph does go through the points we calculated in the previous Exercise.

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

  Corresponding Algebraic formula	x	|x|
  Geometric Effect		Take the graph of the absolute value function
  Numerical Effect	Take the input	Apply the absolute value function
  Numerical  Results	Inputs	Outputs
  	-1
  	0
  	1

   

  Answer

  Use the Multigraph Utility to verify that at each step goes through the points you have calculated.  Make sure that the graph of the expression at each step has the stated geometric relationship to the graph at the previous step. 

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• Complete the following tables to relate the graph of with the graph of , and calculate three points on the graph of h, using the inputs x = 0, 1, 4.

  Corresponding Algebraic formula	x
  Geometric Effect		Take the graph of the square root function
  Numerical Effect	Take the input	Apply the square root function
  Numerical  Results	Inputs	Outputs
  	0
  	1
  	4

  so that h goes through the points:  

  x
  0 1 4

  Plot the points you calculated for y = h(x) and use the the geometric description to envision how to correctly connect-the-dots to create a fairly accurate sketch of the resulting graph.  Hint: Imagine, in your mind's eye what the arithmetic operations would do geometrically to the graph of the square root function; if necessary, make a rough sketch of the graph after each operation.

  Answer

  Since the same operations appear in the same order as the previous Exercise, the geometric and numerical effects are the same as before; the final values are different only because we start with different values and apply a different function at the first step. 

  Corresponding Algebraic formula	x
  Geometric Effect		Take the graph of the square root function	Stretch vertically by a factor of 2	Flip the vertical axis	Shift vertically, up by 1
  Numerical Effect	Take the input	Apply the square root function	Multiply by 2	Multiply by -1	Add 1
  Numerical  Results	Inputs	Outputs
  	0	0	0	0	1
  	1	1	2	-2	-1
  	4	2	4	-4	-3

  so that h goes through the points:  

  x
  0 1 4	1 -1 -3

  As before, stretching the graph of the square root function vertically by 2 will make the graph steeper, leaving the vertex at the origin.  Flipping the vertical axis will keep the graph as steep, but the graph will now go downwards from the same vertex.  Finally, shifting up by 1 will simply move the graph up 1 unit; as before, this is most clearly visible at the vertex.  As before, by selecting the series of Examples from the following applet, we can verify this analysis:

  Notice that the final graph does go through the points we calculated.

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• 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	x
  Geometric Effect		Take the graph of the square root function
  Numerical Effect	Take the input	Apply the square root function
  Numerical  Results	Inputs	Outputs
  	0
  	1
  	4

  Answer

  Use the Multigraph Utility to verify that at each step goes through the points you have calculated.  Make sure that the graph of the expression at each step has the stated geometric relationship to the graph at the previous step. 

  Back to Worksheet

• Complete the following tables to relate the graph of s(x) = ëxû with the graph of r(x) = -2ëxû + 1, and calculate three points on the graph of r, using the inputs x = -1, 0, 1.

  Corresponding Algebraic formula	x	ëxû	2ëxû	-2ëxû	-2ëxû + 1
  Geometric Effect		Take the graph of the greatest integer function
  Numerical Effect	Take the input	Apply the greatest  integer function
  Numerical  Results	Inputs	Outputs
  	-1
  	0
  	1

  so that r goes through the points:  

  x	y = r(x) = -2ëxû + 1
  -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

  The same operations appear in the same order as before, so our table is almost identical to the previous Exercise, except for the values.

  Corresponding Algebraic formula	x	ëxû	2ëxû	-2ëxû	-2ëxû + 1
  Geometric Effect		Take the graph of the greatest integer function	Stretch vertically by a factor of 2	Flip the vertical axis	Shift vertically, up by 1
  Numerical Effect	Take the input	Apply the greatest  integer function	Multiply by 2	Multiply by -1	Add 1
  Numerical  Results	Inputs	Outputs
  	-1	-1	-2	2	3
  	0	0	0	0	1
  	1	1	2	-2	-1

  so that r goes through the points:  

  x	y = r(x) = -2ëxû + 1
  -1 0 1	3 1 -1

  This time, stretching the graph of the greatest integer function vertically by 2 will pull the "steps" farther apart (i.e., by 2 units).  When we flip the vertical axis, the "steps" will now go downwards.  Finally, shifting up by 1 will simply move each step up 1 unit.  Unlike the previous examples, this graph does not have a "vertex".  Although we cannot simply "connect-the-dots", if we have a rough picture of the graph, we can sketch the horizontal lines from (-1, 3), (0, 1), and (1, -1), and then extend this pattern to obtain the rest of the graph.  Remember to put the closed circles on the left side of each step, with an open circle on the right.  As usual, we can verify this analysis by examining the Examples in the following applet:

  Notice that the final graph does go through the points we calculated, and does correspond to our verbal description.  Notice also that the computer fails to clearly indicate whether the points at the end of each "step" belong to the graph or not; that is why we must understand how the graph truly looks, without simply relying on the computer.

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

  Answer

  If, for example, you chose r(x) = 3ëxû - 5, your chart should look something like:

  Corresponding Algebraic formula	x	ëxû	3ëxû	3ëxû - 5
  Geometric Effect		Take the graph of the greatest integer function	Stretch vertically by a factor of 3	Shift down 5 units
  Numerical Effect	Take the input	Apply the greatest integer function	Multiply by 3	Subtract 5
  Numerical  Results	Inputs	Outputs
  	-1	-1	-3	-8
  	0	0	0	-5
  	1	1	3	-2

  You should use the Multigraph Utility to verify that at each step goes through the points you have calculated and that the graph has the stated geometric relationship to the graph at the previous step.  In this example, we would plot (see the Examples in the top menu):

  Note: You should repeat this Exercise, even experimenting with different "core" functions, as often as necessary, until you feel confident that you have mastered this graphing technique; if you use another function (such as the cube root or x²), you will need to decide what three points would be important to plot by examining its graph.

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