Essential Steps: Finding the Gradient of the Curve at a Point

The gradient of a curve at a point can be obtained by drawing a tangent to the curve at that point and finding the gradient of the tangent.

1. Locate the given point \(A\), on the curve.

Place a ruler at the point.

2. Adjust the ruler until its edge follows the local direction of the curve at the point.

The line must pass through the specified point. It may cross the curve or meet it elsewhere.

3. Draw a straight line (tangent) through the given point, \(A\).

4. Locate \(2\) points \(\left( {{x}_{1}},{{y}_{1}} \right)\) and \(\left( {{x}_{2}},{{y}_{2}} \right)\) on the tangent (do not choose points that are too close to each other) and write the coordinates of the points on the graph.

5. Calculate the gradient of the tangent at point, \(A\) using \(m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\).

This method of finding the gradient of tangent at a point is an estimation.

Image check: in Graph A the point label near the vertical axis says \((0,-1)\), but the plotted intercept and calculation use \((0,-2)\); those give gradient \(4\). In Graph B, the quoted line points give \(-4/3\), but that line does not pass exactly through the stated \((-1,3)\). Treat these as approximate teaching illustrations; use a consistently scaled tangent through the actual point when estimating.