Real-Life and Conversion Graphs

A conversion graph is a straight line that changes an amount in one unit into another unit, such as miles into kilometres, or pounds into euros. A real-life graph shows how one quantity changes as another changes, such as the cost of a taxi ride as the distance goes up.

In the exam you are usually asked to read a conversion graph one way and then the other way, for 1 mark each. A harder part asks you to convert an amount that is too big for the graph, for 2 or 3 marks. Real-life graph questions ask you to read a value, find a fixed charge, work out a rate, or say what the gradient represents, for 1 or 2 marks each.

Travel graphs, which show distance against time, have their own page.

If the amount you know is on the horizontal axis, go straight up to the line, then straight across to the vertical axis and read the value. If it is on the vertical axis, go across to the line, then down. Draw these lines on the graph with a ruler, so you can see your reading.

Before you read, work out what one small square is worth on each axis. If 5 small squares make 10 km, then one small square is 10\div5=2 km. The two axes often have different scales.

Write the unit with every reading, such as 15 miles = 24 km, so you don't mix up the two units.

Most conversion graphs go through (0,0), because 0 miles is 0 km. So the two amounts are in proportion: double the miles and you double the kilometres.

If an amount is too big for the graph, choose an amount on the graph that divides into it exactly. Read that amount from the graph, then multiply. For example, if 20 miles is 32 km, then 100 miles is 5\times32=160 km.

You can also add readings. 35 miles is 25 miles + 10 miles, so it is 40+16=56 km.

Use a reading far along the line, not one close to 0, because any error in a small reading gets multiplied too. Never use a value beyond the end of the drawn line.

To draw a conversion graph you need two points. One is usually (0,0). For the other, use the fact you are given with a large amount, so that the point is near the far end of the axes.

Plot the points with small crosses and join them with one straight line, drawn with a ruler. Then work out a third point and check that it lies on your line.

Many real-life graphs are straight lines that don't go through (0,0). An electrician might charge a fixed charge of £35 for each job, plus £25 for each hour. Even a job of 0 hours costs £35, so the line starts at £35 on the vertical axis.

The gradient (the steepness) of the line is the rate: how much the cost goes up for each hour. Work it out from two points on the line:

\text{rate}=\frac{\text{change in cost}}{\text{change in time}}

A rate is the vertical unit per horizontal unit, such as pounds per hour, pounds per mile, litres per minute or centimetres per hour.

The rate is the same all along a straight line, so each extra hour costs the same. A job that is 2 hours longer costs 2\times\pounds25=\pounds50 more, however long the shorter job is.

The gradient of any real-life graph is a rate of change. If a graph shows the volume of water in a tank against time in minutes, its gradient is the number of litres the volume changes by each minute.

A steeper line means a faster rate. A flat line means nothing is changing. A line that goes down means the amount is getting smaller, such as water draining out of a tank.

If you are asked what the gradient represents, give the rate in words with both units, for example "the number of litres that flow into the tank each minute". Don't just describe the shape of the graph.

When two companies charge in different ways, both lines can be drawn on one graph. For any amount across, the lower line is the cheaper company.

Where the lines cross, both companies charge the same. On one side of the crossing one company is cheaper, and on the other side the other company is.

If one company's charges are given as a rule instead, such as £20 plus £9 for each hour, work out its cost for the value you need. For 4 hours that is \pounds20+4\times\pounds9=\pounds56. Then compare it with the reading from the graph.

• Counting each small square as 1. Work out what one small square is worth on each axis before you read a value.
• Reading from the wrong axis. Check which axis your amount is on, and write the unit with every reading, such as £40 = €46.
• Scaling up a reading taken close to 0. Use a reading far along the line, so a small error is not multiplied many times.
• Using a value beyond the end of the line. Only take readings from the drawn line, then multiply or add them.
• Forgetting the fixed charge. Take it off before you divide to find the rate.
• Describing the shape when asked about the gradient. Say how much the vertical quantity changes for each unit across, such as litres per minute.