Study Notes: Graphs in Practical Situations (IGCSE Mathematics 0580)
Welcome to the chapter on Graphs in Practical Situations! This chapter takes the algebra and coordinate geometry skills you've learned and applies them to real-world problems involving movement, cost, and rates. Understanding these graphs is crucial because they show us how variables change over time or distance—a key skill not just for maths, but for science, economics, and beyond!
Don't worry if reading graphs feels confusing at first. We will break down what the slope and the area mean, which are the two most important tools in this chapter.
Section 1: The Foundation – The Linear Equation
Most practical graphs in IGCSE Mathematics start with the fundamentals of straight lines. You must be fluent with the equation of a straight line:
\[y = mx + c\]
Key Terms and Components
- \(y\) and \(x\): These are your variables, plotted on the vertical and horizontal axes, respectively. In practical situations, the horizontal axis (\(x\)) is often Time.
- \(c\): The y-intercept. This is the starting value.
- \(m\): The Gradient (or slope). This tells you the Rate of Change.
Understanding the Gradient (\(m\))
The gradient is the single most important concept in practical graphs. It tells you how quickly one variable is changing compared to the other.
The formula for the gradient is:
\[m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{Rise}}{\text{Run}}\]
What the gradient means:
- Positive Gradient (\(m > 0\)): The graph is sloping up. The quantity is increasing (e.g., distance is increasing, cost is increasing).
- Zero Gradient (\(m = 0\)): The graph is a horizontal line. The quantity is staying constant (e.g., speed is constant, distance is not changing).
- Negative Gradient (\(m < 0\)): The graph is sloping down. The quantity is decreasing (e.g., distance is decreasing, returning home).
Quick Review: Gradient
The gradient represents the rate. If the axes are labeled, the units of the gradient will be the units of \(y\) divided by the units of \(x\).
Section 2: Distance-Time Graphs
Distance-Time (D-T) graphs are used to show the relationship between distance travelled and time elapsed.
Axes Setup:
- Vertical Axis (\(y\)): Distance (usually from a fixed point, like home or start).
- Horizontal Axis (\(x\)): Time.
Interpreting D-T Graphs
On a Distance-Time graph:
\[\text{Gradient} = \frac{\text{Change in Distance}}{\text{Change in Time}} = \text{Speed}\]
This means the slope of the line tells you the speed of the object.
1. Constant Speed:
- Represented by a straight line with a positive slope.
- A steeper line means a faster speed.
2. Stationary (Stopped):
- Represented by a horizontal line (gradient = 0).
- The distance is not changing over time.
3. Returning to Start:
- Represented by a straight line with a negative slope.
- The distance from the starting point is decreasing.
4. Accelerating or Decelerating (Non-Constant Speed):
- Represented by a curve.
- If the curve gets steeper, the speed is increasing (acceleration).
- If the curve gets flatter, the speed is decreasing (deceleration).
Step-by-Step: Calculating Average Speed
When asked for the average speed over a period of time, you use the gradient formula on the straight line segment:
- Identify the start point \((x_1, y_1)\) and end point \((x_2, y_2)\) of the segment you are interested in.
- Calculate the gradient: \[ \text{Speed} = \frac{D_2 - D_1}{T_2 - T_1} \]
Did you know? If a question asks for the "average speed for the entire journey," you divide the total distance travelled by the total time taken. Remember to count any distance travelled on the return leg!
Key Takeaway: D-T Graphs
The slope is speed. The steeper the line, the faster the movement.
Section 3: Speed-Time Graphs
Speed-Time (S-T) graphs show how the speed (or velocity) of an object changes over time. These graphs are essential for understanding acceleration and total distance.
Axes Setup:
- Vertical Axis (\(y\)): Speed (e.g., m/s or km/h).
- Horizontal Axis (\(x\)): Time.
Interpreting S-T Graphs: The Two Main Rules
Unlike D-T graphs, S-T graphs have two critical interpretations:
Rule 1: Gradient = Acceleration
\[\text{Gradient} = \frac{\text{Change in Speed}}{\text{Change in Time}} = \text{Acceleration}\]
- Positive Gradient: Speed is increasing (acceleration).
- Zero Gradient (Horizontal Line): Speed is constant (zero acceleration). This is the "cruising" phase.
- Negative Gradient: Speed is decreasing (deceleration or retardation).
Rule 2: Area Under the Graph = Distance Travelled
The total distance travelled is found by calculating the area between the line and the time axis.
- Often, the area under the graph will form shapes like triangles, rectangles, or trapeziums.
- If the shape is complex, break it down into these standard shapes (e.g., a trapezium can be split into a rectangle and a triangle).
Step-by-Step: Calculating Distance from S-T Graphs
If the graph is a trapezium (often the case for constant acceleration followed by constant speed):
\[\text{Area of Trapezium} = \frac{1}{2} (a + b) h\]
Where \(a\) and \(b\) are the parallel lengths (time durations) and \(h\) is the height (speed).
If the graph segments are simple:
1. Acceleration Phase (Triangle): Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)
2. Constant Speed Phase (Rectangle): Area = \(\text{width} \times \text{height}\)
Common Mistake Alert!
Students often confuse D-T and S-T graphs. Remember:
A horizontal line on a D-T graph means the object is stopped.
A horizontal line on a S-T graph means the object is moving at a constant speed.
Memory Aid: Rates and Areas
Think of the three quantities: Distance (D), Speed (S), and Acceleration (A).
To go D \(\to\) S \(\to\) A, you take the GRADIENT.
To go A \(\leftarrow\) S \(\leftarrow\) D, you take the AREA (or integral, but we use geometric area here).
Key Takeaway: S-T Graphs
The slope is acceleration, and the area is distance.
Section 4: Interpreting General Practical Graphs (Non-Motion)
Graphs aren't just for movement! They are used across practical situations like finance, economics, and health. The general rules of interpretation still apply.
1. Gradients in Other Contexts (Rate of Change)
In any graph, the gradient still represents the rate.
- Example: A graph showing Cost (\(y\)) vs. Hours worked (\(x\)). The gradient (\(m\)) represents the hourly rate of pay.
- Example: A graph showing Water Volume (\(y\)) vs. Time (\(x\)). The gradient (\(m\)) represents the flow rate (litres per minute). (Referencing Syllabus C1.11 / E1.11: Rates).
2. Finding Intersections and Solutions
When you draw two graphs on the same axis (for example, comparing two different phone tariffs or rental car prices), the point where they cross is the intersection point.
- The coordinates of the intersection \((x, y)\) mean that for that specific value of \(x\), both quantities have the same value of \(y\).
- Example: If Graph A shows the cost of Company A and Graph B shows the cost of Company B, the intersection point tells you when the cost is equal for both companies.
3. Maxima, Minima, and Zeros
For curved graphs (like quadratic graphs: \(f(x) = ax^2 + bx + c\)), practical applications involve identifying key points (often done using a Graphic Display Calculator, as noted in Syllabus C3.2 / E3.2):
- Maximum/Minimum: The highest or lowest point (the turning point or vertex). In a practical context, this might be the maximum height a ball reaches or the minimum cost required.
- Zeros (x-intercepts): The points where the graph crosses the \(x\)-axis (where \(y=0\)). In a height-time graph, the zero might represent the moment the object hits the ground.
When reading values from a graph, always pay attention to the scale on the axes. You are expected to read values to an accuracy of within half of the smallest square on the grid.
4. Interpolation vs. Extrapolation
- Interpolation: Estimating a value within the range of your plotted data points. This is generally reliable.
- Extrapolation: Estimating a value outside the range of your plotted data points (e.g., predicting the value for the next year based on the current graph). This is often less reliable, as real-world trends may change.
Key Takeaway: General Graphs
Interpret the graph by identifying what the gradient, the intercepts, and the intersections mean in the context of the problem given.