Graphs Revision Notes

Interpreting Axes and Units

Always start by looking at the labels on the x-axis and y-axis. These tell you what quantities are being plotted. For example, a graph might show 'Time (minutes)' on the x-axis and 'Distance (km)' on the y-axis. Understanding the units is essential for correctly interpreting the scale and making calculations. Ensure you can identify the independent variable (x-axis) and the dependent variable (y-axis).

Analysing Graph Shapes and Slopes

The shape of the line or curve on a real-world graph provides key information about the relationship between the quantities. A straight line indicates a constant rate of change. A steeper line means a faster rate of change. A horizontal line means no change. A curved line indicates a changing rate of change. For example, on a distance-time graph, a horizontal line means stationary, a straight upward slope means constant speed, and a curve means changing speed (acceleration/deceleration).

Distance-Time Graphs Explained

In a distance-time graph, the y-axis represents the distance from a starting point, and the x-axis represents time. The gradient (slope) of the line represents speed. A horizontal line means the object is stopped (speed = 0). A line sloping upwards indicates movement away from the start at a constant speed. A line sloping downwards indicates movement towards the start. The steeper the line, the faster the speed. If the graph is curved, the speed is changing.

Extracting Information and Calculating Values

You need to be able to read specific values from graphs by drawing lines to the axes. For example, 'What was the distance after 30 minutes?' or 'How long did it take to travel 15 km?'. You also need to calculate rates of change from the gradient, often using 'change in y / change in x'. For instance, speed from a distance-time graph, or acceleration from a speed-time graph. The area under a speed-time graph gives the total distance travelled.

Reading and Interpreting Graphs

Graphs in real-world contexts have labelled axes showing quantities and their units (e.g., 'Time (seconds)', 'Distance (km)', 'Cost (£)'). To read a value from a graph, find the given value on one axis, move horizontally or vertically to the line, and then move perpendicular to the other axis to read the corresponding value. For example, on a distance-time graph, if you want to find the distance travelled after 30 minutes, find '30 minutes' on the time axis, go up to the line, and then across to the distance axis. Always pay attention to the scale on each axis to read values accurately.

Types of Real-World Graphs

Common real-world graphs include distance-time graphs, speed-time graphs, conversion graphs, and cost graphs. Each type visualises a specific relationship. A distance-time graph shows cumulative distance over time; a horizontal line means no movement, a steeper line means higher speed. Conversion graphs, like currency exchange rates, show how one quantity relates to another. The line on a conversion graph is usually straight and passes through the origin if zero of one unit corresponds to zero of the other. Cost graphs show how total cost changes with quantity, often including a fixed initial cost (y-intercept) and a per-unit cost (gradient).

Calculating Gradient and Its Meaning

The gradient of a straight line on a graph represents the rate of change of the variable on the vertical axis with respect to the variable on the horizontal axis. It is calculated as 'change in y' (vertical change) divided by 'change in x' (horizontal change). On a distance-time graph, the gradient represents speed (distance/time). A positive gradient means an increase, a negative gradient means a decrease, and a zero gradient (horizontal line) means no change. For example, a gradient of 10 km/h on a distance-time graph means the object is moving at 10 kilometres per hour.

What is a Gradient?

The gradient of a straight line measures its steepness. It describes how much the vertical distance changes for every unit change in the horizontal distance. Think of it as 'rise over run'. A positive gradient indicates an upward slope from left to right, while a negative gradient indicates a downward slope. The larger the absolute value of the gradient, the steeper the line.

Calculating the Gradient

To calculate the gradient (m) between two points (x1, y1) and (x2, y2), use the formula: m = (change in y) / (change in x) = (y2 - y1) / (x2 - x1). It doesn't matter which point you choose as (x1, y1) or (x2, y2), as long as you are consistent. For example, if you subtract the y-coordinate of the first point from the y-coordinate of the second point, you must also subtract the x-coordinate of the first point from the x-coordinate of the second point.

Special Cases for Gradients

A horizontal line has a gradient of 0 because there is no change in y (the rise is 0). For example, the line y = 3 has a gradient of 0. A vertical line has an undefined gradient because there is no change in x (the run is 0), which would involve division by zero. For example, the line x = 2 has an undefined gradient. Understanding these special cases is crucial for interpreting graphs correctly.

What is a Gradient?

The gradient of a straight line measures its steepness. It tells you how much the 'rise' (vertical change) there is for a given 'run' (horizontal change). A positive gradient means the line slopes upwards from left to right, like climbing a hill. A negative gradient means it slopes downwards, like going down a hill. The larger the absolute value of the gradient, the steeper the line. For example, a gradient of 3 is steeper than a gradient of 1, and a gradient of -4 is steeper than a gradient of -2.

Calculating Gradient from Two Points

To find the gradient of a line, you need two points that lie on it. Let these points be (x₁, y₁) and (x₂, y₂). The formula for the gradient, often represented by the letter 'm', is: m = (y₂ - y₁) / (x₂ - x₁). This can also be thought of as 'rise over run'. Ensure you subtract the coordinates in the same order for both the numerator (change in y) and the denominator (change in x). For instance, if you use y₂ - y₁, you must also use x₂ - x₁.

Special Cases for Gradients

Horizontal lines have a gradient of 0 because there is no change in the y-coordinate (y₂ - y₁ = 0). For example, the line y = 3 has a gradient of 0. Vertical lines have an undefined gradient. This is because there is no change in the x-coordinate (x₂ - x₁ = 0), which would lead to division by zero in the gradient formula. For example, the line x = 2 has an undefined gradient.

Gradients in Practice

Gradients are used in various real-world applications to describe rates of change. For instance, in physics, the gradient of a distance-time graph represents speed. In economics, it can represent the rate of change of one variable with respect to another. When working with graphs, pick two points with clear integer coordinates if possible to avoid errors in calculation. Always check the sign of your gradient to make sure it matches the direction of the line on the graph.