Gradients

Imagine hiking up a mountain. The steepness of the climb can vary significantly from one trail to another. In mathematics, the gradient of a line tells us how "steep" the "climb" is on a graph. Just as knowing the steepness of a trail helps you prepare for a hike, understanding gradients helps you analyse and predict the behaviour of linear equations in $y = mx + c$.

 

What is a Gradient?

The gradient, or slope, of a line is a measure of its steepness, usually represented by the letter $m$. It's calculated as the ratio of the vertical change ($\Delta y$) to the horizontal change ($\Delta x$) between two points on the line:

$$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

This ratio indicates how many units the line moves vertically for every one unit it moves horizontally.

Interpreting Gradients

• Positive Gradient: Indicates the line slopes upwards from left to right. The steeper the line, the greater the gradient.
• Negative Gradient: Indicates the line slopes downwards from left to right. Steeper lines have larger (in absolute value) negative gradients.
• Zero Gradient: Represents a horizontal line, indicating no vertical change as you move along the line.
• Undefined Gradient: Represents a vertical line, where there is no horizontal movement, hence $\Delta x = 0$, making the gradient division by zero, which is undefined.

 

 

Application of Gradients in $y = mx + c$

In the linear equation $y = mx + c$, the gradient $m$ dictates the direction and steepness of the line, while $c$ determines where the line crosses the y-axis. This allows for quick sketching of the line's graph and understanding its relationship with the axes.

 

Example 

Given two points on a line, $(-1, 7)$ and $(4, 11)$, find the gradient.

 

Solution - Use the gradient equation to calculate m by substituting the co-ordinates in:

\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \\ m = \frac{11 - 7}{4 - (-1)}  \\ m = \frac{4}{5}  \\ \text{So the gradient is} \  m = \frac{4}{5} \]