Key Points

Graphs are visual representations of numerical data that help in understanding trends, making comparisons, and presenting information clearly and quickly. They are often easier to interpret than tables of data.

A graph has two perpendicular lines: the horizontal axis (usually called the $x$-axis) and the vertical axis (usually called the $y$-axis). Their point of intersection is called the origin, with coordinates $(0,0)$.

The independent variable is a quantity that is controlled or changes on its own, like time. It is plotted on the $x$-axis. The dependent variable, which changes in response, like distance or temperature, is plotted on the $y$-axis.

Any point on a graph is located by an ordered pair of numbers called coordinates, written as $(x, y)$. The $x$-coordinate represents the horizontal distance from the origin, and the $y$-coordinate represents the vertical distance.

A line graph connects a series of data points with line segments. It is very useful for showing how a quantity changes continuously over a period of time, such as a patient's temperature recorded every hour.

To find a value, locate the known quantity on one axis, move perpendicularly to the graph line, and then move perpendicularly to the other axis to read the corresponding value. This method works for finding either $x$ or $y$.

An upward sloping line from left to right indicates an increase in the value on the $y$-axis. A downward sloping line indicates a decrease. A horizontal line signifies that the value is constant over that interval.

In a distance-time graph, the steepness of the line indicates speed. A steeper line means a greater speed. A horizontal line segment means the distance is not changing, so the speed is zero (the object is stationary).

A line graph that is a single, unbroken straight line is called a linear graph. This type of graph represents a constant rate of change between the two variables.

When two quantities are in direct proportion (e.g., cost and number of items), their graph is a straight line that passes through the origin $(0,0)$. This shows that if one quantity is zero, the other must also be zero.

To plot a graph from a table of values: 1. Draw and label the axes. 2. Choose a suitable and uniform scale for each axis. 3. Mark the points $(x, y)$ from the table. 4. Join the points with line segments.

A graph must represent a realistic situation. For example, a time-temperature graph cannot be a vertical line, as this would imply multiple different temperatures at the exact same instant of time, which is not possible.