Ideas
Gradient
Some lines have increasing slopes, like these:
And some have decreasing slopes, like these:
Exploration
This applet will let you explore lines with positive and negative gradients:
Move the sliders, then observe the sign of m and the form of the line.
If the gradient is positive, then the line is increasing. If the gradient is negative, then the line is decreasing.
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)
If the gradient is positive, the line is increasing. If the gradient is negative, the line is decreasing.
Gradient from a graph
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.
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.
Exploration
This applet allows you to see the rise and run between two points on a line of your choosing.
Move the sliders and notice how the rise affects the gradient and the form of the line.
If the \text{rise} is positive, the line is increasing. If the \text{rise} is negative, the line is decreasing.
Examples
Example 1
What is the gradient of the line shown in the graph given that A(3,3) and B(6,5) both lie on the line.
If the \text{rise} is positive, so the gradient and it makes the line increasing.
If the \text{rise} is negative, so the gradient and it makes the line decreasing.
We calculate the gradient by using a formula:
Gradient from a pair of coordinates
The rise is the difference in the y-values of the points. We take the y-value of the rightmost point and subtract the y-value of the leftmost point to describe the change in vertical distance from A to B:
\text{Rise}=-2-6=-8
The run is the difference in the x-values of the points. We take the x-value of the rightmost point and subtract the x-value of the leftmost point to describe the change in horizontal distance from A to B:
\text{Rise}=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:
| \displaystyle \text{Gradient} | \displaystyle = | \displaystyle \dfrac{\text{Rise}} {\text{Run}} |
| \displaystyle = | \displaystyle \dfrac{-8}{4} | |
| \displaystyle = | \displaystyle -2 |
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.
Let's just remind ourselves how we calculated the rise and run again.
| \displaystyle \text{Rise} | \displaystyle = | \displaystyle y_2-y_1 |
| \displaystyle \text{Run} | \displaystyle = | \displaystyle x_2-x_1 |
This means we can generate a new rule for finding the gradient if we are given two points.
Examples
Example 2
What is the gradient of the line going through A and B?
For any two points (x_1,y_1) and (x_2,y_2), we can find the gradient using the formula:
Gradient of horizontal and vertical lines
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.
On vertical lines, the x-value is always the same for every point on the line.
Examples
Example 3
What is the gradient of any line parallel to the x-axis?
The gradient of a horizontal line is 0, while the gradient of a vertical line is undefined.