A picture is worth a thousand words. When we want to show how a quantity changes, or how two quantities are related, a graph is often the clearest way to communicate the information. Graphs are used everywhere - in newspapers to show prices and temperatures, in hospitals to show a patient's condition, in businesses to show sales and profits, and in science to show experimental results.
In this chapter we will learn what a graph is, how to plot points on a coordinate plane using the x-axis, y-axis and the origin, how to read information from graphs, and how to draw graphs for given situations such as a person walking, a car moving, or temperature changing. We will also learn about bar graphs, line graphs and pie charts, and how to choose the right kind of graph for different types of data.
A Cartesian plane is formed by two perpendicular number lines intersecting at a point called the origin. The horizontal line is called the x-axis and the vertical line is called the y-axis.
The position of a point is given by an ordered pair (x, y), where x is the perpendicular distance of the point from the y-axis (called the x-coordinate or abscissa), and y is the perpendicular distance from the x-axis (called the y-coordinate or ordinate).
For example, the point (3, 5) means x = 3 and y = 5. The origin has coordinates (0, 0).
To plot the point (3, 5), move 3 units to the right of the origin along the x-axis and then 5 units upward. Points are plotted with the x-coordinate first and the y-coordinate second. The axes are divided into equal units, and a proper scale is chosen for each axis.
For example, on a graph of distance against time, the x-axis usually shows time and the y-axis shows distance. The slope of the graph tells us about the speed: a steeper line means faster movement.
A line graph is a graph in which points are joined by line segments. It is used to show how a quantity changes continuously over time.
A linear graph is a line graph whose points lie on a straight line. A linear graph represents a relationship of the form y = mx, which is a direct proportion. For example, if a car travels at 60 km/h, the distance y after x hours is y = 60x, and its graph is a straight line passing through the origin.
Plot the points (0, 0), (1, 60), (2, 120) and (3, 180) and join them. We get a linear graph showing distance = 60 x time.
A bar graph uses rectangular bars of equal width to compare categories.
A pie chart shows how a whole quantity is divided into parts, using sectors of a circle.
| Part | Meaning |
|---|---|
| x-axis | Horizontal number line |
| y-axis | Vertical number line |
| Origin | Intersection point, coordinates (0, 0) |
| x-coordinate (abscissa) | Distance from y-axis, written first |
| y-coordinate (ordinate) | Distance from x-axis, written second |
| Data | Best graph |
|---|---|
| Comparison of categories | Bar graph |
| Change over time | Line graph |
| Parts of a whole | Pie chart |
| Direct relation y = kx | Linear graph |
Suppose we want to draw a graph showing the distance travelled by a cyclist who rides at a steady speed of 15 kilometres per hour. Since the speed is constant, the distance is directly proportional to the time, and the relationship can be written as distance = 15 x time. We begin by choosing the axes: the x-axis represents time in hours and the y-axis represents distance in kilometres. Next we select a convenient scale - for example, one unit on the x-axis for each hour and one unit on the y-axis for every 15 kilometres. We then calculate a few pairs of values: at 0 hours the distance is 0 kilometres, at 1 hour it is 15 kilometres, at 2 hours it is 30 kilometres, and at 3 hours it is 45 kilometres. Plotting the points (0, 0), (1, 15), (2, 30) and (3, 45) and joining them with a straight line gives the required linear graph.
Reading information from a graph is just as important as drawing one. If we want to know the distance travelled after 2.5 hours, we can use the graph by moving from 2.5 on the time axis up to the line and then across to the distance axis, where we read approximately 37.5 kilometres. We can also verify this by using the formula, since 15 x 2.5 = 37.5. This shows how a graph serves as a visual table of values from which we can estimate answers for any point along the line, even values that were not plotted originally. The slope of the line also carries meaning: a steeper line represents a faster cyclist, because more distance is covered in the same amount of time. Understanding this connection between the equation, the graph and the slope gives students a powerful tool for analysing motion, temperature and many other changing quantities.
Graphs turn numerical data into visual understanding. In this chapter we learnt about the Cartesian plane and its parts - the x-axis, y-axis and the origin - and how to plot and read points using coordinates (x, y). We studied line graphs and linear graphs, discovering that a direct relationship like y = kx produces a straight line through the origin. We also revisited bar graphs and pie charts and learnt how to choose the appropriate graph for different kinds of data. Graphs are a universal language of science and business, and the skills learnt in this chapter will be used throughout higher mathematics, physics, economics and statistics.