Stretching and Shrinking Translations

Positive Stretch

If you examine closely any x-value on these two graphs you will notice that the stretched function's y-value is always 3 times further above the x-axis then the original function.  For example (0, 1) is a point on the original graph, while (0,3) is the point on the stretched function.   

Positive Shrink

In this function you will notice that the shrunk function is only 1/4 as high as the original function at each value of x.  For example, at x = 2, the original function has a value of 8.  While the shrunk function has a value of 2.  The points are (2,8) and (2, 2), respectively.

 

Negative Stretch

When the stretch value c  < -1 it has two effects upon the original graph.  First, the graph is reflected through the x-axis.  Second, the original graph is stretched by |c|.  In the above example the stretched graph is going down, and at any x-value that you pick along the x-axis you will see that the stretched function's y-value is twice as far away from the x-axis then the original function. 

For example: (0, 1), and (1, 2) are on the original function, while (0, -2), and (1, -4) are on the negatively stretched function. 

In general, the point (x, 2^x) becomes (x, -2 * 2^x).

 

Negative Shrink

When the stretch value c satisfies -1 < c < 0, it again has two effects upon the original graph.  First, because c is negative, the graph is reflected through the x-axis.  Second, the original graph is shrunk by |c|.  In the above example the shrink factor is 1/5. 

The point (0, 1) on the original becomes (0, -0.2), and the point (1, 2) becomes (1, -2/5). 

In general, the point (x, 2^x) becomes (x, -1/5 * 2^x).

Positive Stretch - Table

x
-3 0.125 0.375
-2 0.25 0.75
-1 0.5 1.5
0 1 3
1 2 6
2 4 12
3 8 24
In the table above notice that all the values in column 3 are three times those in column 2.  The second function multiplies each y-value by 3.

Negative Stretch

If 3 is replaced with -3, we would then have a negative stretch.   All the values in the third column would be negative.

Positive Shrink - Table

x
-3 0.125 0.0625
-2 0.25 0.125
-1 0.5 0.25
0 1 0.5
1 2 1
2 4 2
3 8 4

In the table above notice that all the values in column 3 are half of those in column 2.  The second function multiplies each y-value by 1/2.  If

Negative Shrink

If 1/2 were replaced by -1/2, we would then have a negative shrink.  All the values in the third column would be negative.

Practice Stretch and Shrink Translations

  • Select a base value b.  This must be a positive number.

  • Select a stretch or shrink value c.  This may be any real number.

  • Adjust the window range settings to your liking.

  • Click on Graph to view the graph.  Both the base graph and the shifted graph will be drawn.  (example: 2^x and 3 * 2^x)

  • If you wish to see a grid layout set according to the scale of each axis click on the No-Grid/Grid checkbox.  Then click Graph again.

  • If you wish to erase the window, click on Erase.

Suggested Practice:  Make sure the No-Grid/Grid is set to Grid.  Enter a value for c.  Then predict where the stretch or shrink graph will be drawn.  Then click on Graph and see if you are correct.

 

Practice Stretch and Shrink

Practice Problems

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