Estimate a curve’s gradient
Draw a tangent at the specified point and use two separated points on that tangent.
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A curve does not have one constant gradient
For a straight line, any two distinct points give the same gradient. On a curve, a chord between two points gives an average gradient over that interval. To estimate the local gradient at one point, draw the tangent: the line matching the curve’s direction there.
On a carefully plotted graph of y = x², near the point (2,4), a tangent can be drawn through (1,0) and (3,8). Its gradient is (8-0)/(3-1) = 4. Those two points are on the tangent, not both on the curve. The chord from (0,0) to (2,4) instead has gradient 2 and answers a different question.
A tangent need not lie on just one side of a curve everywhere, and “touches at one point” is not a universal definition. Use its local direction at the stated point.
Estimate with a useful triangle
Choose two well-separated points on the drawn tangent, preferably where coordinates can be read accurately. Read the horizontal and vertical scales separately, then calculate Δ y/Δ x. A larger reading triangle reduces the relative effect of a small plotting error.
If the vertical axis measures distance in metres and the horizontal axis time in seconds, the gradient unit is m/s. A downward slope gives a negative rate of change; do not discard the sign automatically.
Check your understanding 1 Use the drawn line, not a second curve point
At a point on a distance-time curve, a drawn tangent passes through (2 s, 5 m) and (8 s, 23 m). Estimate the rate there.
Show answer
The estimate is (23 − 5)/(8 − 2) = 18/6 = 3 m/s. It describes the tangent’s rate at the chosen curve point. The points used to read the tangent need not be measured positions on the original curve.
Report a graphical estimate as an estimate
If a hand-drawn tangent gives 2.9 instead of 3.0, consider line placement and reading precision. A neat decimal is not proof of exactness. State suitable accuracy for the graph, and keep the mathematical distinction between a tangent estimate and an average gradient.
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