Gradient of Horizontal and Vertical Lines

We know that the gradient of a line is a measure of its steepness or slope.

Straight lines on the Cartesian Plane can literally be in any direction and pass through any two points.  

This means that straight lines can be:

Horizontal Lines

On horizontal lines, the $y$y value is always the same for every point on the line. In other words, there is no rise- it's completely flat. 

$A=\left(-4,4\right)$A=(−4,4)

$B=\left(2,4\right)$B=(2,4)

$C=\left(4,4\right)$C=(4,4)

All the $y$y-coordinates are the same. Every point on the line has a $y$y value equal to $4$4, regardless of the $x$x-value.

The equation of this line is $y=4$y=4.

Since gradient is calculated by $\frac{\text{rise }}{\text{run }}$rise run ​ and there is no rise (ie. $\text{rise }=0$rise =0), the gradient of a horizontal line is always $0$0.

Vertical Lines

On vertical lines, the $x$x value is always the same for every point on the line.

Let's look at the coordinates for A,B and C on this line.  

$A=\left(5,-4\right)$A=(5,−4)

$B=\left(5,-2\right)$B=(5,−2)

$C=\left(5,4\right)$C=(5,4)

All the $x$x-coordinates are the same, $x=5$x=5, regardless of the $y$y value.

The equation of this line is $x=5$x=5.

Vertical lines have no "run" (ie. $\text{run }=0$run =0). l If we substituted this into the $\frac{\text{rise }}{\text{run }}$rise run ​ equation, we'd have a $0$0 as the denominator of the fraction. However, fractions with a denominator of $0$0 are undefined.

So, the gradient of vertical lines is always undefined.

Examples

Question 1

Question 2