Algebra - Graphs: plot and interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration (Higher tier)

The idea

Higher tier graphs cover several families. On a distance-time graph the gradient is speed; on a velocity-time graph the gradient is acceleration and the area between graph and time axis is displacement (signed area), while distance travelled is the total area ignoring sign, so negative velocity means motion in the opposite direction. Estimate curved areas by splitting into trapeziums. Exponential graphs y = abˣ model growth (b > 1) or decay (0 < b < 1), e.g. £2000 at 5% is y = 2000(1.05)ˣ. Reciprocal graphs y = k/x have two separate branches with asymptotes at x = 0 and y = 0, so never join across the discontinuity. For non-standard functions, plot points, join with a suitable smooth curve, then read approximate solutions using construction lines, e.g. draw y = 3 and read where it meets the curve.

What earns the marks

• Calculating speed as the gradient of a distance-time graph, and acceleration as the gradient of a velocity-time graph, using change in y divided by change in x.
• Estimating displacement as the signed area under a velocity-time graph, and distance travelled as the total area ignoring sign (e.g. using triangles and trapeziums).
• Determining the multiplier for exponential growth or decay, such as 1.05 for 5% growth or 0.95 for 5% decay.
• Plotting points accurately for a given function and joining them with a suitable line or smooth curve, keeping reciprocal branches separate.
• Finding an approximate solution by drawing a suitable line on the graph and reading the x-coordinate of the intersection, e.g. drawing y = 3 to solve f(x) = 3.
• Interpreting a reading from a graph in context, stating correct units and explaining what the value represents.

Common mistakes and how to put them right

• Treating the area under a velocity-time graph as distance even when velocity is negative.

  Put it right: Area gives displacement; for distance, add the absolute areas of each section separately.

• Using straight line segments for an exponential curve, giving inaccurate estimates.

  Put it right: Exponential functions produce smooth curves, so join plotted points with a curve rather than a ruler.

• Joining the two branches of a reciprocal graph across x = 0.

  Put it right: y = k/x has separate branches with asymptotes at x = 0 and y = 0; draw each branch separately.

• Forgetting to split the area under a curve into regular shapes.

  Put it right: Divide into strips of equal width and sum the trapezium or triangle areas.

Exam tips

• To estimate the area under a curved graph, split it into several trapeziums; using more strips generally improves the estimate.
• To find the gradient of a curve at a point, draw a tangent there and calculate the gradient of that straight line.
• For approximate solutions, draw the required horizontal or slanted line accurately with a ruler and read the intersection carefully against the grid.