Gradients and Area Under Graphs
This topic covers how to find the gradient (rate of change) and the area under a graph, which are key for interpreting physical and financial data without using calculus. For ESAT, this most commonly applies to distance-time and speed-time graphs to find speed, acceleration, and distance.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The gradient of a graph represents the rate of change. For a distance-time graph, this is speed. For a speed-time graph, this is acceleration.
- The area under a graph represents the cumulative total. The area under a speed-time graph is the total distance travelled.
- To find the gradient of a curve at a specific point, draw a tangent to the curve at that point and calculate the gradient of that straight line.
- To estimate the area under a curve, divide the region into simpler shapes like triangles, rectangles, and trapezia, and sum their individual areas.
- The units of the gradient are the units of the y-axis divided by the units of the x-axis.e.g., m / s = m/s
- The units of the area are the units of the y-axis multiplied by the units of the x-axis (e.g., (m/s) × s = m).
Diagram
Formulae
m = (y2 - y1) / (x2 - x1) To calculate the gradient of a straight line (or a tangent to a curve) passing through points (x1, y1) and (x2, y2).
Area = 0.5 × (a + b) × h To find the area of a trapezium, commonly used for one strip when estimating the area under a graph. 'a' and 'b' are the parallel side lengths and 'h' is the perpendicular height (or width) between them.
Definitions
- Gradient
- A measure of the steepness of a line, calculated as the change in the vertical axis ('rise') divided by the change in the horizontal axis ('run').
- Tangent
- A straight line that touches a curve at a single point, matching the curve's instantaneous gradient at that point.
- Area Under a Graph
- The area enclosed between the plotted line or curve and the horizontal axis over a specified interval.
Worked example
A speed-time graph for a car's journey is a straight line from (0s, 5m/s) to (10s, 25m/s), followed by a straight line to (30s, 0m/s). Calculate the total distance travelled by the car.
- 1
Visualise or sketch the graph.
It consists of two straight line segments, forming a triangle on top of a rectangle, or simply a single large trapezium.
- 2
The total area under the graph represents the total distance.
The shape is a trapezium with the time-axis as its base.
- 3
The parallel sides of the trapezium are the total duration of the journey (30s) and the duration of the top flat edge, which is not applicable here.
A better way is to see it as a shape with height from t=0 to t=30.
- 4
Let's split the shape into a triangle and a trapezium for clarity.
Alternative method:
calculate the area of the whole shape, which is a large triangle on top of a rectangle, or just a single large triangle and a smaller triangle.
The easiest method is to treat the whole shape from t=0 to t=30 as two separate shapes:
a trapezium (0-10s) and a triangle (10-30s).
- 5
Area of first shape (trapezium from t=0 to t=10):
Area = 0.5 × (initial speed + final speed) × time = 0.5 × (5 + 25) × 10 = 0.5 × 30 × 10 = 150 m - 6
Area of second shape (triangle from t=10 to t=30):
Area = 0.5 × base × height = 0.5 × (30 - 10) × 25 = 0.5 × 20 × 25 = 10 × 25 = 250 m - 7
Total distance is the sum of the two areas:
150 m + 250 m = 400 m.
Answer: 400 m
Common mistakes
- ×Confusing gradient and area calculations, for example calculating acceleration when asked for distance.
- ×Making simple arithmetic mistakes, especially with decimals or fractions when calculating trapezium areas. Check your work.
- ×Using the wrong formula, such as 'run over rise' for gradient or forgetting to multiply by 0.5 for a triangle's area.
- ×Incorrectly reading coordinates from the graph axes, leading to errors in gradient or area calculations.
- ×When a shape is split into multiple parts, forgetting to sum all the parts to find the total area.
No-calculator tips
- ✓When calculating the area of a trapezium like 0.5 × (a+b) × h, do the addition (a+b) first. If (a+b) is an even number, halve it before multiplying by h to keep numbers smaller.
- ✓Break down complex shapes under a graph into the simplest possible rectangles and triangles. It may seem longer, but it reduces the chance of arithmetic errors with more complex formulae.
- ✓When estimating the gradient of a tangent, extend the line until it hits points with integer coordinates to make the (y2-y1)/(x2-x1) calculation much easier.