Real-life graphs

Many real situations give straight-line graphs. Read the units on both axes, then:

• the y-intercept is the starting value, such as a fixed charge
• the gradient is the rate, such as the cost per mile.

Example: the graph shows the cost of a taxi ride. Find the fixed charge, the cost per mile, and the cost of a 6-mile ride.

1. y-intercept: £3 fixed charge
2. Gradient: (18 − 3) ÷ 10 = £1.50 per mile
3. 6 miles: 3 + 6 × 1.50 = £12

A conversion graph changes one unit to another, such as pounds to euros. It is a straight line through the origin.

1. Find the value on one axis.
2. Go across (or up) to the line, then read off the other axis.
3. For a value off the graph, convert a smaller amount and multiply.

Example: use the graph to convert £35 to euros, and €90 to pounds.

1. Up from £35 to the line, then across: €42
2. €90 is off the graph. €90 = €30 × 3, and €30 = £25
3. £25 × 3 = £75

If water flows into a container at a steady rate, a graph of depth against time shows its shape.

• Straight sides (a cylinder): the depth rises steadily, so the graph is a straight line (A).
• Wide at the bottom, narrow at the top: the depth rises slowly, then faster, so the graph gets steeper (B).
• Narrow at the bottom, wide at the top: the graph starts steep and gets less steep.

Example: a cone-shaped glass, narrow at the bottom, is filled at a steady rate. Describe its depth–time graph.

1. Near the bottom, the glass is narrow, so the depth rises quickly.
2. Higher up, it's wider, so the depth rises more slowly.
3. A curve that starts steep and gets less steep

A line graph shows how something changes over time. Plot the points and join them with straight lines. A trend is the general direction: up, down or steady.

Example: describe the trend in visitors, and find the biggest increase between two months.

1. Apart from a dip in month 4, the numbers go up: an increasing trend
2. Changes: +30, +30, −10, +40, +30
3. The biggest increase is from month 4 to month 5: 40 visitors

The lines between points only show the change: there are no readings in between.

On a curved real-life graph, the rate of change varies.

• The average rate between two points is the gradient of the straight line (chord) joining them.
• The rate at an instant is the gradient of the tangent at that point.

Example: a tank's depth is 10 cm after 2 minutes and 30 cm after 6 minutes. Find the average rate of change.

1. Change in depth: 30 − 10 = 20 cm
2. Change in time: 6 − 2 = 4 minutes
3. 20 ÷ 4 = 5 cm per minute

Always give a rate with its units: the y-axis unit "per" the x-axis unit.