Modelling · Topic 3 of 9
Modelling Situations With Graphs
Theory
Graphs are an incredibly useful way to model mathematical situations because they allow us to instantly visualise how changing an input affects the final output.
When analysing or sketching a graphical model, you must pay close attention to three main features:
- The Gradient (Slope): This represents the rate of change of the output. A steeper slope means a faster rate of change. A horizontal line means the rate of change is zero (the value is constant).
- The Intercepts: The y-intercept represents the starting value or initial condition before any changes occur.
- Changes in Behaviour: Look for points where the graph suddenly changes direction, curves, or flattens out, as these represent significant events in the real-life situation.
Independent and Dependent Variables
In any 2D mathematical model, you must identify which variable is which based on the context:
- Independent Variable (x-axis): The variable that is changed or controlled (the cause). It does not depend on the other variable. Time is almost always an independent variable.
- Dependent Variable (y-axis): The variable being tested or measured (the effect). Its value depends entirely on the independent variable.
Worked examples
Example 1
Example 1: Identifying Variables
A local plumber charges a fixed call-out fee of £40, plus an additional £25 for every hour they spend working on the repair. A graph is to be drawn to model the total cost of hiring the plumber.
(a) Identify the independent variable and state which axis it should be plotted on.
(b) Identify the dependent variable and state which axis it should be plotted on.
Solutions:
- (a) The independent variable is the time spent working in hours, and it should be plotted on the x-axis.
- (b) The dependent variable is the total cost, and it should be plotted on the y-axis (because the total cost depends entirely on how many hours the plumber works).
Example 2
Example 2: Interpreting Rates of Change
A conical glass vase (narrow at the bottom, widening out to a large circular opening at the top) is placed under a tap. Water pours into the vase at a completely constant rate. Describe the shape of the graph that would model the depth of the water in the vase over time.
Solution: The graph would start with a steep positive gradient, but the curve would gradually flatten out (become less steep) over time.
Reasoning: Because the bottom of the vase is narrow, it takes very little water to increase the depth, so the depth increases rapidly at first. As the water reaches the wider sections of the vase, it takes much more water to raise the level, so the rate of change of the depth slows down.
Example 3
Example 3: Matching a Graph to a Story
A delivery driver's journey is tracked using GPS. The driver:
- Drives at a steady speed along a motorway.
- Gets stuck in a completely stationary traffic jam.
- Once clear, drives at a very fast, steady speed to make up for lost time.
- Slows down to navigate a residential housing estate at a steady, slow speed.
Describe the four distinct sections of a distance-time graph that models this journey.
Solutions:
- Section 1: A straight line with a moderate positive slope.
- Section 2: A completely flat, horizontal line (time increases, but distance does not).
- Section 3: A straight line with a very steep positive slope (steeper than section 1).
- Section 4: A straight line with a shallow positive slope (less steep than sections 1 and 3).