<< Chapter < Page Chapter >> Page >

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
How can I solve for a domain and a codomains in a given function?
Oliver Reply
Thank you I mean range sir.
proof for set theory
Kwesi Reply
don't you know?
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
the value of 2 sin square 60 Cos 60
Sanjay Reply
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
need help
maybe provide us videos
about complex fraction
What do you mean by a
nothing. I accidentally press it
you guys know any app with matrices?
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
b= p - 4a - 2c
like Deadra, show me the step by step order of operation to alive for b
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.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?