SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Payment Details

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Another common way that the graphs of trigonometric
functions are altered is by
stretching the graphs. Stretching a graph involves introducing a
coefficient into the function, whether that coefficient fronts the equation as
in y = 3 sin(x) or is acted upon by the trigonometric function, as in
y = sin(3x). Though both of the given examples result in stretches of the graph
of y = sin(x), they are stretches of a certain sort. The first example
creates a vertical stretch, the second a horizontal stretch.

Vertical Stretches

To stretch a graph vertically, place a coefficient in front of the function.
This coefficient is the amplitude of the function. For example, the
amplitude of y = f (x) = sin(x) is one. The amplitude of y = f (x) = 3 sin(x)
is three. Compare the two graphs below.

The amplitude of the graph of any periodic function is one-half the
absolute value of the sum of the maximum and minimum values of the function.

Horizontal Stretches

To horizontally stretch the sine function by a factor of c, the function must be
altered this way: y = f (x) = sin(cx) . Such an alteration changes the
period of the function. For
example, continuing to use sine as our representative trigonometric function,
the period of a sine function is , where c is the coefficient of
the angle. Usually c = 1, so the period of the
sine function is 2Π. Below are pictured the sine curve, along with the
following functions, each a horizontal stretch of the sine curve:
y = f (x) = sin(2x) and y = f (x) = sin().