SparkNotes: Free Study Guides No Fear Shakespeare: The Bard made easy SparkCharts: Just the facts TestPrep: SAT, ACT, and more 101s: College texts condensed Subject Finder: Browse by subject SparkCollege: Get in! SparkLife: 100% study-free home_bottom home_top BN_link
 
◄ PREVIOUS
Problems
NEXT ►
Problems
 
 

Graphs

 
 

Horizontal and Vertical Shifts

 
Among the variations on the graphs of the trigonometric functions are shifts--both horizontal and vertical. Such shifts are easily accounted for in the formula of a given function.
 
Take function f, where f (x) = sin(x). The graph of y = sin(x) is seen below.
Figure 3.1: The Graph of sine(x)
 

Vertical Shifts

 
To shift such a graph vertically, one needs only to change the function to f (x) = sin(x) + c, where c is some constant. Thus the $y$-coordinate of the graph, which was previously sin(x), is now sin(x) + 2. All values of y shift by two. The graph of y = f (x) = sin(x) + 2 looks like this:
 
Figure 3.2: By adding a constant to a function, like sine, the graph is shifted vertically

Horizontal Shifts

 
To shift a graph horizontally, a constant must be added to the function within parentheses--that is, the constant must be added to the angle, not the whole function. Compare the to the graph of y = f (x) = sin(x + ).
 
Figure 3.3: Horizontal shift
The graph of sine is shifted to the left by units. Does it look familiar? The graph is the same as the cosine graph. Sine and cosine are both periodic functions, and have the same domain and range. They are separated only by a phase shift: . This is the constant that must be added to create the necessary horizontal shift to make the graphs directly overlap each other. Because the cosine graph is only a phase shift of the sine graph, one or the other can be used to model motion, but both are not necessary. Most often sine is used.
 
 
 
Help | Feedback | Make a request | Report an error | Send to a friend

◄ PREVIOUS
Problems
NEXT ►
Problems
 
 
 
Advanced Math message board
Ask a question or post an answer on the community boards.
 
Writing Help
A blog about grammar, writing, and your papers.
 
Study On Your Way to Class
 
iPod (read)
Download this sparknote to your iPod on iPREPpress.com.
 
 
 
Can't face the work right now? Waste a few minutes with us.
Life
It's already July
Better get cracking on that summer fling
 
Life
"You look tired"
Translation: "You look absolutely horrible."
 
Books
James pulls the old mom's-voice-playing-on-a-VHS trick
And Dan can't believe Bella falls for it
 
 
Test Prep Books
Take the next step in test prep.
  • SAT Subject Test: Math Level 1
  • SAT Subject Test: Math Level 2
  •  
    Test Prep Centers
    Take a practice exam. Do better.
  • SAT Subject Test: Math Level 1 Test Center
  • SAT Subject Test: Math Level 2 Test Center
  •  
    Contact Us | Privacy Policy | Terms and Conditions | About | Sitemap
    ©2009 SparkNotes LLC, All Rights Reserved.