Pre-Composition and Graphing: Solutions

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

• 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

  Simply following the English instructions gives:

  Corresponding Algebraic formula
  Geometric Effect	Original	Shift left 1	Flip horizontally	Shrink by a factor of 2 horizontally
  Numerical Effect		Subtract 1	Divide by -1	Divide by 2
  Numerical  Results	Inputs				Outputs
  	0	-1	1	1/2	0
  	1	0	0	0	1
  	4	3	-3	-3/2	2

  Thus, the graph of f goes through the points:  

  x
  1/2 0 -3/2	0 1 2

  Back to Worksheet

• 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

  If we start with the square root graph and shift left by 1, the vertex will move from (0, 0) to (-1, 0), but it will still rise up and to the right.  After we flip this horizontally, the vertex will flip to (1, 0) and the graph will then rise up to the left.  Finally, when we shrink horizontally (towards the origin), the graph will become "compressed"; this will "shrink" the vertex to (1/2, 0) and the graph will get "narrower".  Note: Because of the basic shape of the graph, this could also be described as steeper; we will explain this in more detail in the next section.  We can see each of these steps by selecting the Examples from the following applet: 

  Notice how all the graphs do go through the points we have calculated.

  Back to Worksheet

• 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

  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 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

  Since the same operations appear in the same order as the previous Exercise, the geometric and numerical effects are the same as before.  However, since the graph has a different shape, we chose to start with a different table of values. 

  Corresponding Algebraic formula	|x|	|x + 1|	|-x + 1|	|-2x + 1|
  Geometric Effect	Original	Shift left 1	Flip horizontally	Shrink horizontally by a factor of 2
  Numerical Effect		Subtract 1	Divide by -1	Divide by 2
  Numerical  Results	Inputs				Outputs
  	-1	-2	2	1	1
  	0	-1	1	1/2	0
  	1	0	0	0	1

  so that h goes through the points:  

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

  As before, shifting the graph of the absolute value function to the left by 1 will shift the vertex to (-1, 0), but it will still rise at a 45° angle to both sides.  After we flip this horizontally, the vertex will flip to (1, 0).  Finally, when we shrink horizontally (towards the origin), the graph will become "compressed"; this will "shrink" the vertex to (1/2, 0) and the graph will get "narrower".  Note: It is no accident that this could equally well be described as "steeper"; we will explore this further in the next section.  We can verify this analysis by selecting the Examples from the following applet:

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

  Back to Worksheet

• 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

  If, for example, you chose h(x) = |(1/3)x - 2|, your chart should look something like:

  Corresponding Algebraic formula	|x|	|x - 2|	|(1/3)x - 2|
  Geometric Effect	Original	Shift right 2 units	Stretch horizontally by a factor of 3
  Numerical Effect		Add 2	Multiply by 3
  Numerical  Results	Inputs			Outputs
  	-1	1	3	1
  	0	2	6	0
  	1	3	9	1

  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):

  Back to Worksheet

• 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 1/2 0	-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	ëxû	ëx + 1û	ë-x + 1û	ë-2x + 1û
  Geometric Effect	Original	Shift left 1	Flip horizontally	Shrink horizontally by a factor of 2
  Numerical Effect		Subtract 1	Divide by -1	Divide by 2
  Numerical  Results	Inputs				Outputs
  	-1	-2	2	1	-1
  	0	-1	1	1/2	0
  	1	0	0	0	1

  so that r goes through the points:  

  x	y = r(x) = ë-2x + 1û
  1 1/2 0	-1 0 1

  Shifting left by 1 is easy to visualize; each "step" moves left 1 unit.  When we flip horizontally, the steps will now go down as we move to the right.  Moreover, the closed circles will now appear on the right side of each step.  Finally, shrinking the graph horizontally by 2 will shrink each "step" to 1/2 a unit long, and bring them closer to the vertical axis; this will make the graph look somewhat "steeper".  Again, we cannot simply "connect-the-dots", but, since we have a rough picture of the graph, we can sketch the horizontal lines from (1, -1), (1/2, 0), and (0, 1), and then extend this pattern to obtain the rest of the graph.  Remember to put the closed circles on the right side of each step, with an open circle on the left.  As usual, we can verify this analysis by selecting the Example from 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.

  Back to Worksheet

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

  Answer

  If, for example, you chose r(x) = ë(-1/2)x - 1û, your chart should look something like:

  Corresponding Algebraic formula	|x|	ëx - 1û	ë-x - 1û	ë(-1/2)x - 1û
  Geometric Effect	Original	Shift right 1 unit	Flip horizontally	Stretch horizontally by a factor of 2
  Numerical Effect		Add 1	Negate	Multiply by 2
  Numerical  Results	Inputs				Outputs
  	-1	0	0	0	-1
  	0	1	-1	-2	0
  	1	2	-2	-4	1

  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 an other 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.

  Back to Worksheet

Go to Transformational Graphing

Table of Contents

Send questions or comments to jwicks@northpark.edu

Glossary