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Given a tabular function, create a new row to represent a vertical shift.

  1. Identify the output row or column.
  2. Determine the magnitude of the shift.
  3. Add the shift to the value in each output cell. Add a positive value for up or a negative value for down.

Shifting a tabular function vertically

A function f ( x ) is given in [link] . Create a table for the function g ( x ) = f ( x ) 3.

x 2 4 6 8
f ( x ) 1 3 7 11

The formula g ( x ) = f ( x ) 3 tells us that we can find the output values of g by subtracting 3 from the output values of f . For example:

f ( 2 ) = 1 Given g ( x ) = f ( x ) 3 Given transformation g ( 2 ) = f ( 2 ) 3 = 1 3 = 2

Subtracting 3 from each f ( x ) value, we can complete a table of values for g ( x ) as shown in [link] .

x 2 4 6 8
f ( x ) 1 3 7 11
g ( x ) −2 0 4 8
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The function h ( t ) = 4.9 t 2 + 30 t gives the height h of a ball (in meters) thrown upward from the ground after t seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function b ( t ) to h ( t ) , and then find a formula for b ( t ) .

b ( t ) = h ( t ) + 10 = 4.9 t 2 + 30 t + 10
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Identifying horizontal shifts

We just saw that the vertical shift is a change to the output, or outside, of the function. We will now look at how changes to input, on the inside of the function, change its graph and meaning. A shift to the input results in a movement of the graph of the function left or right in what is known as a horizontal shift , shown in [link] .

Figure_01_05_004
Horizontal shift of the function f ( x ) = x 3 . Note that h = + 1 shifts the graph to the left, that is, towards negative values of x .

For example, if f ( x ) = x 2 , then g ( x ) = ( x 2 ) 2 is a new function. Each input is reduced by 2 prior to squaring the function. The result is that the graph is shifted 2 units to the right, because we would need to increase the prior input by 2 units to yield the same output value as given in f .

Horizontal shift

Given a function f , a new function g ( x ) = f ( x h ) , where h is a constant, is a horizontal shift    of the function f . If h is positive, the graph will shift right. If h is negative, the graph will shift left.

Adding a constant to an input

Returning to our building airflow example from [link] , suppose that in autumn the facilities manager decides that the original venting plan starts too late, and wants to begin the entire venting program 2 hours earlier. Sketch a graph of the new function.

We can set V ( t ) to be the original program and F ( t ) to be the revised program.

V ( t ) = the original venting plan F ( t ) = starting 2 hrs sooner

In the new graph, at each time, the airflow is the same as the original function V was 2 hours later. For example, in the original function V , the airflow starts to change at 8 a.m., whereas for the function F , the airflow starts to change at 6 a.m. The comparable function values are V ( 8 ) = F ( 6 ) . See [link] . Notice also that the vents first opened to 220  ft 2 at 10 a.m. under the original plan, while under the new plan the vents reach 220  ft 2 at
8 a.m., so V ( 10 ) = F ( 8 ) .

In both cases, we see that, because F ( t ) starts 2 hours sooner, h = 2. That means that the same output values are reached when F ( t ) = V ( t ( 2 ) ) = V ( t + 2 ) .

Figure_01_05_005a
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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