Gradient of a curve: worked examples and method

A curve keeps changing

A straight line has one gradient. A curve gets steeper or flatter, so its gradient is different at different points.

Average gradient

Join two points on the curve with a straight line, called a chord. Its gradient is the rise over the run, and it is the average gradient between the points.

Gradient at a point

The gradient at a point is the gradient of the tangent there. A tangent is a straight line that touches the curve at that point.

Far apart points

Pick two points on the tangent that are far apart and easy to read. A long rise and run make a small reading error matter less.

Chords close in

Chords that start at the point and get shorter have gradients that settle towards the gradient of the tangent. This is how you estimate from a table of values.

Units tell the story

A gradient has the vertical unit per the horizontal unit. Distance in metres against time in seconds gives metres per second.

Why a curve has no single gradient

On a straight line, moving one step to the right always raises or lowers the line by the same amount. That constant amount is the gradient. On a curve such as y = x^2, one step to the right near the origin raises the curve by very little, while one step to the right far from the origin raises it by a lot. The curve gets steeper as x grows.

That is why questions about curves are always specific. They ask for the gradient between x = 1 and x = 3, or for the gradient at the point where x = 2. Read the wording carefully. Between two points means an average, found from a chord. At a point means an instantaneous gradient, found from a tangent. Using the wrong one is the main way to lose marks, because the numbers can be quite different.

Method: 1. Underline the wording: between two values means a chord, at one value means a tangent. 2. Check the axes and the units before you start, so you can label the answer. 3. Decide whether the curve is getting steeper or flatter, so you know roughly what size of answer to expect.

Tip: Between two points is an average. At a point needs a tangent.

Average gradient between two points

To find the average gradient between two points on a curve, read the coordinates of both points, subtract the y-values for the rise and the x-values for the run, and divide. For y = x^2 between x = 1 and x = 3, the points are (1, 1) and (3, 9). The rise is 9 - 1 = 8, the run is 3 - 1 = 2, and the average gradient is 8 ÷ 2 = 4. Always subtract in the same order on the top and the bottom. Going from the second point to the first on top and from the first to the second on the bottom gives the wrong sign.

In a real context, the unit comes from the axes. If a cyclist is 120 m from the start after 10 s and 200 m after 18 s, the average speed is (200 - 120) ÷ (18 - 10) = 80 ÷ 8 = 10 metres per second. That is the speed the cyclist would need to cover the same distance at a steady pace. It tells you nothing about the speed at any one instant, and the cyclist may have been faster at some times and slower at others.

Method: 1. Read the two points from the graph or the table. 2. Work out the change in y and the change in x, subtracting in the same order. 3. Divide, and write the unit as the y unit per x unit.

Estimating the gradient at a point with a tangent

To estimate the gradient at a point, place a ruler so that it touches the curve at that point and appears to balance there, with the curve on one side of the ruler close to the point and on the other side further along. Draw the line carefully. Then choose two points on the tangent line that are far apart and have easy coordinates, such as points where the line crosses grid intersections. Far apart matters: a small slip in reading a coordinate makes a big difference to a short triangle, and very little difference to a long one.

Now work out the gradient of the tangent in the usual way. For the curve y = x^2 the tangent at (2, 4) passes through (0, -4) and (3, 8). The rise is 8 - (-4) = 12 and the run is 3 - 0 = 3, so the gradient is 12 ÷ 3 = 4. Because the tangent is drawn by eye, an exam answer is an estimate and the mark scheme accepts a range around the true value. Show the two points you used and the division, because the method earns marks even if your line was a little off.

Method: 1. Draw a tangent that touches the curve at the given point only, using a ruler. 2. Pick two points on the tangent that are far apart and easy to read. 3. Work out the rise divided by the run, and state that the answer is an estimate.

Tip: Show the two points on your tangent and the division. A shaky line still earns the method mark.

Chords that close in on the tangent

There is a way to see why a tangent gives the gradient at a point. Fix the point (2, 4) on y = x^2 and join it to a second point a short distance h to the right. When h = 1 the second point is (3, 9) and the chord has gradient 5. When h = 0.5 the second point is (2.5, 6.25) and the gradient is (6.25 - 4) ÷ 0.5 = 4.5. When h = 0.1 the gradient is 4.1, and when h = 0.01 it is 4.01.

The chord gradients 5, 4.5, 4.1, 4.01 are heading towards 4, which is the gradient of the tangent at (2, 4). The shorter the chord, the better it matches the curve near the point. This idea is how you can estimate a gradient from a table of values when no graph is drawn: work out the gradient of chords that start at the point and get shorter, and read off the number they approach. Say “the gradient is approximately 4”, not “exactly 4”, unless you are told how to find an exact value.

Method: 1. Start at the point you want and choose a second point a short distance away. 2. Work out the gradient of the chord, then repeat with a closer second point. 3. Look at the sequence of gradients and state the value they approach.

What the gradient tells you

The gradient of a distance–time graph is speed, so a steeper section means faster travel and a flat section means standing still. The gradient of a speed–time graph is acceleration, in metres per second squared. A negative gradient on a speed–time graph means the object is slowing down. When a question asks you to interpret a gradient, give the number with its unit and say what it means: “the cyclist is travelling at about 10 m/s at that instant”.

Method: 1. Identify what the vertical and horizontal axes measure. 2. Write the unit of the gradient as vertical unit per horizontal unit. 3. Give a sentence that says what the number means in the situation.