Algebra - Graphs: calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts (this does not include calculus) (Higher tier only)

The idea

This is Higher tier only, and no calculus is involved. To find the gradient of a curve at a point, draw a tangent that touches the curve at that point, extend it well across the grid, then divide its rise by its run. On a distance-time graph that gradient is the speed at that moment, and on a velocity-time graph it is the acceleration. For the area under a curve, split the region into strips of equal width and treat each strip as a trapezium. Two strips of width 3, with heights 0, 6 and 10, give ½ × 3 × (0 + 6) + ½ × 3 × (6 + 10) = 9 + 24 = 33. Under a velocity-time graph that area is the distance travelled, and in a financial context it can be a total cost. Trapezium strips overestimate when the curve bends upwards, since each chord sits above the curve, and underestimate when it bends downwards.

What earns the marks

• A tangent drawn touching the curve at the point the question names.
• The rise and the run taken from that tangent, even if the division goes wrong.
• The gradient interpreted in context, such as an acceleration in metres per second squared.
• Splitting the region into strips of equal width with heights read off the curve.
• The total area, and one for saying what it represents, such as the distance travelled.

Common mistakes and how to put them right

• Drawing a chord between two points on the curve instead of a tangent at one point.

• Counting squares under a curve without allowing for what one square is worth on each axis.

• Giving the area as a bare number with no units, which loses the interpretation mark.

• Treating the gradient of a distance-time graph as an acceleration.

Exam tips

• Extend the tangent right across the grid before measuring, because a short tangent gives a small rise and run and a rough answer.
• Say whether your estimate is too big or too small and why, since that reason is often worth a mark.
• Build the units of a gradient from the axes: the vertical unit per horizontal unit.