Interpreting Graphs in Context
This topic covers how to interpret different types of graphs to understand real-world scenarios, particularly in physics problems involving motion. You will need to extract data, calculate rates of change (gradients), and understand the relationships represented by linear, reciprocal, and exponential curves.
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
- A straight-line graph indicates a constant rate of change. The gradient 'm' is the rate, and the y-intercept 'c' is the initial value.y = mx + c
- A horizontal line on a distance-time graph means the object is stationary. A horizontal line on a speed-time graph means the object is moving at a constant speed.
- A reciprocal graph shows inverse proportionality; as one variable increases, the other decreases such that their product is constant.y = k/x
- An exponential graph models a quantity being multiplied by a constant factor 'a' in each time period. The value 'k' is the starting amount at x=0.y = k × a^x
- For kinematics graphs, the gradient of a distance-time graph gives speed, and the gradient of a speed-time graph gives acceleration.
- The area under a speed-time graph represents the distance travelled.
Diagram
Formulae
Gradient = (y2 - y1) / (x2 - x1) To find the rate of change from a straight-line section of a graph, such as calculating speed or acceleration.
Speed (m/s) = Gradient of a distance (m) vs. time (s) graph When analysing a distance-time graph to find how fast an object is moving.
Acceleration (m/s2) = Gradient of a speed (m/s) vs. time (s) graph When analysing a speed-time graph to find the rate of change of speed.
Distance (m) = Area under a speed (m/s) vs. time (s) graph To calculate the total distance travelled from a speed-time graph.
Definitions
- Gradient
- The steepness of a line, calculated as the change in the vertical axis divided by the change in the horizontal axis ('rise over run'). It represents a rate of change.
- Y-intercept
- The point where the graph crosses the vertical (y) axis. It represents the starting value or a fixed initial condition when the horizontal variable is zero.
- Piecewise Function
- A function represented by a graph made of several distinct segments, where each segment follows a different rule. Travel graphs are a common example.
Worked example
The distance-time graph shows a drone's flight. It flies from its base to a target, hovers for a period, and then returns to base at a different speed. (a) How long does the drone hover for? (b) What is the drone's speed on its outbound journey in m/s? (c) What is its average speed for the entire flight, excluding the time spent hovering?
- 1
Step 1 (a):
Identify the 'hovering' period.
This is the horizontal section of the distance-time graph, where distance from the base does not change.
This occurs between t=40s and t=70s.
The duration is 70 - 40 = 30 seconds.
- 2
Step 2 (b):
Calculate the speed for the outbound journey.
This is the gradient of the first line segment (t=0 to t=40).
The drone travels 600m in 40s.
Speed = distance / time = 600m / 40s = 15 m/s - 3
Step 3 (c):
Calculate the average speed for the moving parts of the journey.
Total distance travelled = 600m out + 600m back = 1200m.
Total time spent moving = 40s (outbound) + (90s - 70s) = 40s + 20s = 60s.
Average speed = Total distance / Total time moving = 1200m / 60s = 20 m/s.
Answer: (a) 30 seconds, (b) 15 m/s, (c) 20 m/s
Common mistakes
- ×Confusing distance-time and speed-time graphs. A flat line means 'stopped' on a distance-time graph but 'constant speed' on a speed-time graph.
- ×Miscalculating the gradient by swapping the change in x and change in y, or by misreading the scales on the axes.
- ×Finding the average speed by averaging the two speeds (e.g., for the worked example, (15+30)/2), which is incorrect unless the time for each leg is identical.
- ×Forgetting to account for an initial condition (a non-zero y-intercept) when formulating an equation for a straight-line relationship.
No-calculator tips
- ✓To calculate gradient, always choose points that fall exactly on grid intersections to avoid dealing with fractions or decimals.
- ✓When finding the area under a graph, break it down into simple rectangles and triangles. Remember the area of a triangle is (1/2) × base × height.
- ✓Simplify fractions before performing division. For a speed of 600m in 40s, cancel the zeros to get 60/4, which is easier to calculate mentally as 15.
- ✓Before calculating, quickly check the units on the axes. A common trap is to have distance in km and time in minutes, but ask for a speed in m/s.