Rates of Change of Graphs

Calculating Linear Gradients: Identify two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line. Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ to find the constant rate of change.

The Tangent Method for Curves: To find the rate at a specific point on a curve, draw a straight line that just touches the curve at that single point. Calculate the gradient of this straight line to estimate the instantaneous rate.

Interpreting Intercepts: The $y$-intercept often represents a 'fixed' or 'initial' value (such as a call-out fee or starting distance), while the $x$-intercept represents the point where the dependent variable has been exhausted or returned to zero.

Check the Axes First: Before performing any calculations, identify exactly what is being measured on each axis. Calculating the gradient of a distance-time graph gives speed, but doing the same on a speed-time graph gives acceleration.

Unit Consistency: Always state the units in your final answer. If the $y$-axis is in 'liters' and the $x$-axis is in 'minutes', your rate must be in 'liters per minute'.

Reasonableness Check: If a graph represents a car braking, the gradient should be negative. If your calculation results in a positive number, re-check your coordinate subtraction order.

Tangent Precision: When drawing tangents, ensure the line is balanced on the curve. Use a long ruler to extend the tangent line as far as possible to make the 'rise over run' calculation more accurate.

Confusing Speed and Distance: Students often see a downward slope on a distance-time graph and assume the object is slowing down. In reality, it is moving at a constant speed back toward the starting point.

Ignoring the Origin: Not all graphs start at $(0,0)$. Forgetting to account for a non-zero $y$-intercept can lead to incorrect proportional reasoning.

Scale Errors: Misreading the scale of the axes (e.g., assuming one grid square equals one unit when it actually equals five) is a frequent source of calculation errors.