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Graphing Equations

Slope

Problems

Slope, page 2

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Slope

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) :

Slope
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:

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:

Graph of a Horizontal 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:

Graph of a Vertical 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.

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