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.

In addition to its familiar meaning, the word "slope" has precise mathematical meaning. The slope of a line is the rise over the run, or the change in y divided by the change in x. To find the slope of a line, pick any two points on the line. Then subtract their x-coordinates and subtract their y-coordinates in the same order. Divide the difference of the y-coordinates by the difference of the x- coordinates:

Given two points (x_{1}, y_{1}) and (x_{2}, y_{2}) on a line, the slope of the line is equal to:

m = =

Example 1. Find the slope of the line which passes through the points (2, 5) and (0, 1):

m = = = 2. This means that every time x increases by 1 (anywhere on the line), y increase by 2, and whenever x decreases by 1, y decreases by 2.

Negative Slope

If a line has a positive slope (i.e. m > 0), then y always increases when x increases and y always decreases when x decreases. Thus, the graph of the line starts at the bottom left and goes towards the top right.

Often, however, the slope of a line is negative. A negative slope implies that y always decreases when x increases and y always increases when x decreases. Here is an example of a graph with negative slope:

m = = = - Thus, as x increases by 3, y decreases by 4, and as x decreases by 3, y increases by 4.

Horizontal and Vertical Lines

Sometimes, we will see equations whose graphs are horizontal lines. These are graphs in which y remains constant -- that is, in which y_{1} - y_{2} = 0 for any two points on the line:

m = = = 0. The slope of any horizontal line is 0. In other words, as x increases or decreases, y does not change. x takes every possible value at a specific y value.

We will also see equations whose graphs are vertical lines. These are graphs in which x remains constant -- that is, in which x_{1} - x_{2} = 0 for any two points on the line:

m = = = undefined. We cannot divide a number by zero. The slope of any vertical line is undefined.x does not increase or decrease; rather, y takes every possible value at a specific x value.