A line is said to be increasing if it slopes upwards, as we move from left to right. Increasing lines always have a positive gradient.
A decreasing line slopes downwards as we move from left to right. Decreasing lines always have a negative gradient.
The following applet allows us to see how the value of the gradient changes with the steepness of the line. Notice that increasing lines have a positive gradient and decreasing lines have a negative gradient.

The gradient of a line is defined as the vertical change in the $y$ycoordinates ('rise') of two points on the line, divided by the horizontal change in the corresponding $x$xcoordinates ('run').
$\text{Gradient }$Gradient  $=$=  $\frac{\text{change in }y\text{coordinates}}{\text{change in }x\text{coordinates}}$change in ycoordinateschange in xcoordinates 
$=$=  $\frac{\text{rise }}{\text{run }}$rise run 
Find the gradient of the line below:
Solution
We begin by choosing any two points on the line and use them to create a rightangled triangle, where the line itself forms the hypotenuse of the triangle.
In this case we have chosen the points $\left(1,0\right)$(−1,0) and $\left(0,2\right)$(0,2). The 'run' (highlighted red) and the 'rise' (highlighted blue) form the sides of the rightangled triangle.
If we start at the left most point, we see that the run is $1$1 and the rise is $2$2. Both values are positive because we move first to the right $1$1 unit, then up $2$2 units. We calculate the gradient as follows:
$\text{Gradient }$Gradient  $=$=  $\frac{\text{rise }}{\text{run }}$rise run 
$=$=  $\frac{2}{1}$21  
$=$=  $2$2 
When choosing points on the line to calculate the gradient, we try to choose points that line up with the axis scale markings, or gridlines. In this way we don't need to estimate values that may lie between these markings, and it makes our calculation of the gradient more accurate.
The applet below shows that the gradient of a line is not actually affected by the location of the points used to calculate the gradient. We can also see how the value of the gradient changes for lines of different steepness and whether the line is increasing or decreasing.

If we are given the coordinates of two points on the coordinate plane, we can find the gradient of the line that would pass through these points.
Find the gradient of the line between the points $\left(3,6\right)$(3,6) and $\left(7,2\right)$(7,−2).
Solution
It is good practice to first draw a sketch of the two points. A sketch means the location of the points doesn't have to be exact. As long as the points are in the correct quadrant and correctly positioned relative to each other.
We can then add a rightangled triangle that shows the 'rise' and 'run'. This allows us to see immediately whether the line is increasing or decreasing, and if it has a positive or negative gradient.
The run is the horizontal distance between the points. We can see that the leftmost point has an $x$xcoordinate of $3$3 and the rightmost point has an $x$xcoordinate of $7$7, so the horizontal distance between them is $4$4 units.
The rise is the vertical distance between the points. We can see that one point is $6$6 units above the horizontal axis and the other point is $2$2 units below, so the vertical distance between them is $8$8 units.
If we start at the leftmost point, we move $4$4 units to the right and then $8$8 units down to reach the rightmost point. This gives us a run of $4$4 and a rise of $8$−8.
We can now calculate the gradient:
$\text{Gradient }$Gradient  $=$=  $\frac{\text{rise }}{\text{run }}$rise run 
$=$=  $\frac{8}{4}$−84  
$=$=  $2$−2 
Another way to find the gradient, without drawing a sketch, is to consider the changes in the $y$ycoordinates and $x$xcoordinates of the two points.
$\text{Gradient }$Gradient  $=$=  $\frac{\text{change in }y\text{coordinates}}{\text{change in }x\text{coordinates}}$change in ycoordinateschange in xcoordinates 
$=$=  $\frac{26}{73}$−2−67−3  
$=$=  $\frac{8}{4}$−84  
$=$=  $2$−2 
Notice that we always subtract the coordinates of the leftmost point from the coordinates of the rightmost point (the leftmost point will have the lowest value for the $x$xcoordinate).
$x$x  $3$−3  $0$0  $3$3  $6$6 

$y$y  $2$−2  $3$−3  $4$−4  $5$−5 
Solution
Using $\left(3,2\right)$(−3,−2) and $\left(3,4\right)$(3,−4), for example, and subtracting the coordinates of the rightmost point from the coordinates of the leftmost point:
$\text{Gradient }$Gradient  $=$=  $\frac{\text{change in }y\text{coordinates}}{\text{change in }x\text{coordinates}}$change in ycoordinateschange in xcoordinates 
$=$=  $\frac{4\left(2\right)}{3\left(3\right)}$−4−(−2)3−(−3)  
$=$=  $\frac{2}{6}$−26  
$=$=  $\frac{1}{3}$−13 
Note: It doesn't matter which two points we choose. As long as our calculations are correct, the gradient will be the same.
Because the gradient is negative, the line is decreasing.
A horizontal line has a 'run' but no 'rise', therefore:
$\text{Gradient of horizontal line }$Gradient of horizontal line  $=$=  $\frac{\text{rise }}{\text{run }}$rise run 
$=$=  $\frac{0}{\text{run }}$0run  
$=$=  $0$0 
A vertical line has a 'rise' but no 'run', therefore:
$\text{Gradient of vertical line }$Gradient of vertical line  $=$=  $\frac{\text{rise }}{\text{run }}$rise run 
$=$=  $\frac{\text{rise }}{0}$rise 0  
$=$=  $\text{undefined }$undefined 
Remember that division by zero is mathematically 'undefined'.
$\text{Gradient of a horizontal line }$Gradient of a horizontal line  $=$=  $0$0 
$\text{Gradient of a vertical line }$Gradient of a vertical line  $=$=  $\text{undefined }$undefined 
What is the gradient of the line shown in the graph given that Point A $\left(3,3\right)$(3,3) and Point B $\left(6,5\right)$(6,5) both lie on the line.
What is the gradient of the line going through A and B?
Find the gradient of the line that passes through Point A $\left(2,6\right)$(2,−6) and the origin using $m=\frac{\text{rise }}{\text{run }}$m=rise run .
Lines drawn on the $xy$xyplane, extend forever in both directions. If we ignore the special case of horizontal and vertical lines, all other lines will either cross both the $x$xaxis and the $y$yaxis or they will pass through the origin, $\left(0,0\right)$(0,0).
Here are some examples:
We use the word intercept to refer to the point where the line crosses or intercepts with an axis.
The $y$yintercept is the point where the line crosses the $y$yaxis. The coordinates of the $y$yintercept will always have an $x$xcoordinate of zero.
Note: Every line must have at least one intercept but cannot have any more than two intercepts.
Find the $y$yintercept for the straight line below:
The $y$yintercept is $6$−6, and the coordinates of the $y$yintercept are $\left(0,6\right)$(0,−6).
Consider the following graph.
Find the coordinates of the $y$yintercept.
Find the gradient of the line.
Consider the following graph.
Find the coordinates of the $y$yintercept.
Find the gradient of the line.
Consider the following graph.
Find the coordinates of the $y$yintercept.
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