Gradients as rates of change

#### Tier: #Higher

Description

The gradient of a graph at any point represents the rate of change of the $y$-quantity with respect to the $x$-quantity at that instant.

For a straight-line graph: the gradient is constant throughout. $$\text{Gradient} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

For a curve: the gradient changes at every point. To find it at a specific point, draw a tangent at that point and calculate its gradient.

Practical interpretations:

• On a distance-time graph: gradient = speed (m/s or km/h)
• On a speed-time graph: gradient = acceleration (m/s²)
• On a temperature-time graph: gradient = rate of temperature change

Drawing a tangent: use a ruler to draw a straight line that just touches the curve at the point, extending either side. Then pick two clearly readable points on the tangent to calculate the gradient.

Common error: using two points on the curve rather than on the tangent line, or reading the gradient with the axes in the wrong order.

Links

Gradients and intercepts Speed time graphs