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

  1. Identify the input row or column.
  2. Determine the magnitude of the shift.
  3. Add the shift to the value in each input cell.

Shifting a tabular function horizontally

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 the output values of g are the same as the output value of f when the input value is 3 less than the original value. For example, we know that f ( 2 ) = 1. To get the same output from the function g , we will need an input value that is 3 larger . We input a value that is 3 larger for g ( x ) because the function takes 3 away before evaluating the function f .

g ( 5 ) = f ( 5 3 ) = f ( 2 ) = 1

We continue with the other values to create [link] .

x 5 7 9 11
x 3 2 4 6 8
f ( x 3 ) 1 3 7 11
g ( x ) 1 3 7 11

The result is that the function g ( x ) has been shifted to the right by 3. Notice the output values for g ( x ) remain the same as the output values for f ( x ) , but the corresponding input values, x , have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.

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Identifying a horizontal shift of a toolkit function

[link] represents a transformation of the toolkit function f ( x ) = x 2 . Relate this new function g ( x ) to f ( x ) , and then find a formula for g ( x ) .

Graph of a parabola.

Notice that the graph is identical in shape to the f ( x ) = x 2 function, but the x- values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so

g ( x ) = f ( x 2 )

Notice how we must input the value x = 2 to get the output value y = 0 ; the x -values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the f ( x ) function to write a formula for g ( x ) by evaluating f ( x 2 ) .

f ( x ) = x 2 g ( x ) = f ( x 2 ) g ( x ) = f ( x 2 ) = ( x 2 ) 2
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Interpreting horizontal versus vertical shifts

The function G ( m ) gives the number of gallons of gas required to drive m miles. Interpret G ( m ) + 10 and G ( m + 10 ) .

G ( m ) + 10 can be interpreted as adding 10 to the output, gallons. This is the gas required to drive m miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.

G ( m + 10 ) can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than m miles. The graph would indicate a horizontal shift.

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Given the function f ( x ) = x , graph the original function f ( x ) and the transformation g ( x ) = f ( x + 2 ) on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?

The graphs of f ( x ) and g ( x ) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

Graph of a square root function and a horizontally shift square foot function.
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Combining vertical and horizontal shifts

Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output ( y -) values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( x -) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and left or right.

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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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