Real-life graphsIB MYP Maths Extended: Revision notes

A linear graph is a straight line with equation $y=mx+c$, where $m$ is the gradient and $c$ is the $y$-intercept. To read a graph, check the scales on both axes and the units. To draw one, make a table of values or plot the intercept and use the gradient; two points are enough, but a third checks the line. Label the axes with the quantity and the unit, for example 'Distance (km)' and 'Time (h)'.

A conversion graph changes one unit to another, for example pounds to dirhams. If $0$ of one unit is $0$ of the other, the line goes through the origin. From the point $(10, 46)$: $\text{gradient}=\frac{46}{10}=4.6$ AED per £, so $25$ pounds is $25\times4.6=115$ AED and $92$ AED is $92\div4.6=20$ pounds. To convert using the graph, go across from the value on one axis to the line, then down or across to the other axis. Using the gradient gives the exact answer.

A distance–time graph shows distance from the start on the vertical axis and time on the horizontal axis. The gradient is the speed: $\text{speed}=\frac{\text{distance}}{\text{time}}$. A steeper line means a faster speed. A horizontal line means stationary (speed $0$). A line sloping down means moving back towards the start. For a walk with $5$ km in $1$ h, a $0.5$ h rest and $7$ km in $1$ h, the speeds are $5$ km/h, $0$ km/h and $7$ km/h. Average speed $=\frac{\text{total distance}}{\text{total time}}=\frac{12}{2.5}=4.8$ km/h, and the rest counts in the total time.

In a real-life graph, the gradient is a rate of change: how much the $y$ quantity changes for each $1$ unit of $x$, with units such as km/h, AED per hour or litres per minute. The $y$-intercept is the starting value, when $x=0$. For $C=80+120h$: the intercept $80$ is the call-out fee and the gradient $120$ is the hourly rate in AED per hour. For $V=600-15t$ the gradient $-15$ means the tank loses $15$ litres every minute. A negative gradient means the quantity is decreasing. To find where a graph meets the horizontal axis, put $y=0$: $600-15t=0$ gives $t=40$ minutes.

A good description covers the start, each section (steady, steeper, flat, falling), and the end, with values and units. Compare lines using their gradients and intercepts: Plumber A ($80$ fee, $120$ per hour) is cheaper than Plumber B ($50$ fee, $140$ per hour) for a $4$-hour job, since $560<610$. A comparison needs numbers, and a conclusion must be supported by working, not by words like 'slower' alone: a slower rate does not always mean a longer time if the starting amounts are different.