To estimate the gradient of a curve at a specific point, the first step is to draw a tangent line at that point and determine its slope.

The gradient of a curve at a given point indicates the steepness of the curve at that location. To estimate this gradient, you need to draw a tangent line. A tangent is a straight line that touches the curve at the point of interest without crossing it.

After you have drawn the tangent line, the next step is to calculate its slope. The slope of a line is defined as the ratio of the change in the vertical direction (often referred to as “rise”) to the change in the horizontal direction (commonly known as “run”). This relationship can be expressed mathematically as:

To compute this, select two points on the tangent line, ensuring they are as far apart as possible to enhance accuracy. Label these points as $(x_1, y_1)$ and $(x_2, y_2)$. The rise is determined by the difference in the $y$-coordinates, given by $y_2 - y_1$, while the run is the difference in the $x$-coordinates, expressed as $x_2 - x_1$.

Thus, the gradient $m$ of the tangent line, which also represents the gradient of the curve at that point, can be calculated using the formula:

This method yields an estimate of the gradient. For more precise calculations, particularly in advanced mathematics, you would utilize calculus to find the derivative of the function that describes the curve. Nonetheless, for GCSE Mathematics, drawing a tangent and calculating its slope is a practical and effective method for estimating gradients.