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Shifting of a Graph

To know how a graph would shift, we need to know two things:
  1. The shifting is either on the X-axis or on the Y-axis or on both of them,
  2. The shifting is in the +ve direction or in the –ve direction.
Let us understand this phenomena with the help of an example y = f(x) = |x|

Case 1

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = f(x) – 5 It will shift 5 units downwards,
i.e., 5 units on the –ve Y axis
 

Description: 11-8.tif
y = |x| − 5

Case 2

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = f(x) + 5 It will shift 5 units upwards,
i.e., 5 units on the +ve Y axis
Description: 11-9.tif
y = |x| + 5

Case 3

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = f(x) −5 It will shift 5 units rightwards,
i.e., 5 units on the +ve X axis
Description: 11-10.tif
y = |x − 5|

Case 4

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = f(x) −5 It will shift 5 units leftwards,
i.e., 5 units on the +ve X axis

 

Description: 11-111.tif
y = |x + 5|

Case 5

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = −f(x) It will be reflected along the X-axis as if X-axis is a mirror
 
Description: 11-12.tif
y = − |x|

Case 6

If y = f(x) is changed to Impact on the graph of y = f(x)
Y = |f(x)| Graph below the X-axis will be reflected along the X-axis on its above as if X-axis is a mirror

We can summarize the whole phenomena of the shifting of a graph as:
  1. If there is a change in the value of X-axis, then it will move on the X-axis and similarly for the Y-axis. The graph of y = f(x−5) has a change in the values of x where as graph of y = f(x) – 5, which can be written as y + 5 = f(x) has a change in the values of y.
  2. If there is an addition, then the graph will move on the −ve side, either on the X-axis or on the Y-axis and if there is a subtraction, then graph will move on the +ve side.
Example
The graph of y = |x| is given. Draw the graph of y = ||x|−2|.
Solution
Let us first sketch |x|−2.
Description: 11-13.tif
 
Now ||x|−2| will be having all the negative part of |x| −2 on the positive side of Y axis, or above X-axis.
Description: 11-14.tif
 




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