Sign up for your FREE 7-day trial.Get instant access to all the benefits of SparkNotes PLUS! Cancel within the first 7 days and you won't be charged. We'll even send you a reminder.

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.

Step 2 of 4

Choose Your Plan

Step 3 of 4

Add Your Payment Details

Step 4 of 4

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.

Did you know you can highlight text to take a note?x

If a graph does not change when reflected over a line or rotated around a point,
the graph is symmetric with respect to that line or point. The following
graph is symmetric with respect to the x-axis (y = 0). Note that if (x, y) is a point on the graph, then (x, - y) is also a point on the graph.

If a function is symmetric with respect to the x-axis, then f (x) = - f (x).

The following graph is symmetric with respect to the y-axis (x = 0). Note
that if (x, y) is a point on the graph, then (- x, y) is also a point on the
graph.

If a function is symmetric with respect to the y-axis, then f (x) = f (- x).

If a graph can be reflected over a line without altering the graph, then that
line is called the axis of symmetry. In the following graph, x = 2 is the
axis of symmetry. Note that if (2 + x, y) is a point on the graph, then (2 - x, y) is also a point on the graph.

If a function has an axis of symmetry x = a, then f (x) = f (- x + 2a).

The following graph is symmetric with respect to the origin. In other words, it
can be rotated 180^{o} around the origin without altering the graph. Note
that if (x, y) is a point on the graph, then (- x, - y) is also a point on the
graph.

If a function is symmetric with respect to the origin, then f (x) = - f (- x).