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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.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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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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