2.02 The gradient of an interval

Some lines have increasing slopes, like these:

And some have decreasing slopes, like these:

The slope of a line is a measure of how steep it is. In mathematics we call this the gradient.

A gradient is a single value that describes:

• if a line is increasing (has positive gradient)

• if a line is decreasing (has negative gradient)

• how far up or down the line moves (how the y-value changes) with every step to the right (for every 1 unit increase in the x-value)

You can find the rise and run of a line by drawing a right triangle created by any two points on the line. The line itself forms the hypotenuse.

Sometimes it is difficult to measure how far the line goes up or down (how much the y-value changes) in 1 horizontal unit, especially if the line doesn't line up with the grid points on the xy-plane. In this case we calculate the gradient by using a formula: \text{gradient}= \dfrac{\text{rise}}{\text{run}}

The rise and run can be calculated from using any two points on the line.

In this case, the gradient is positive because, over the 1 unit increase in the x-values, the y-value has increased. If the y-value decreased as the x-value increases, the gradient would be negative.

Examples

If you have a pair of coordinates, such as A (3, 6) and B (7, -2), we can find the gradient of the line between these points using the same formula. It is a good idea to draw a quick sketch of the points, which helps us quickly identify what the line will look like:

The run is the difference in the x-values of the points. We take the x-value of the rightmost point and subtract thex-value of the leftmost point to describe the change in horizontal distance from A to B: \text{run} = 7 - 3 = 4

Notice that we subtracted the x-values and the y-values in the same order - we check our sketch, and it does seem sensible that between A and B there is a rise of -8 and a run of 4. We can now put these values into our formula to find the gradient: \begin{aligned} \text{gradient} &= \dfrac{\text{rise}}{\text{run}} \\ &= \dfrac{-8}{4} \\ &= -2 \end{aligned}

We have a negative gradient, as we suspected. Now we know that when we travel along this line a step of 1 in the x-direction means a step of 2 down in the y-direction.

Examples

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

Since gradient is calculated by \dfrac{\text{rise}}{\text{run}} and there is no rise (i.e. \text{rise} = 0), the gradient of a horizontal is always 0.

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

Vertical lines have no "run" (i.e. \text{run} = 0 ). If we substituted this into the \dfrac{\text{rise}}{\text{run}} equation, we'd have a 0 denominator of the fraction. However, fractions with a denominator of 0 are undefined.

So, the gradient of vertical lines is always undefined.

Examples