Reading the graph of a process

Science and society, grade 10 (3 units) Lesson 5 of 15Learning

Reading the graph of a process

Rising, falling, maximum and minimum

The graph of a process shows how a quantity changes with time. It shows where the quantity rises, where it falls, and when it peaks.

Where it rises – the times when the graph goes up from left to right.
Where it falls – the times when the graph goes down.
The maximum point – the highest point of the graph; the minimum point – the lowest.

The body temperature of a patient with flu, in degrees, over 24 hours from the start of measuring:

\displaystyle T(x)

\displaystyle (5,39.5)

\displaystyle (24,36.8)

It rises from hour 0 to hour 5 – the temperature went up.
It falls from hour 5 to hour 24 – the temperature went down.
The maximum point: $\displaystyle (5,\ 39.5)$ – after 5 hours the temperature was highest, 39.5 degrees.
The minimum point: $\displaystyle (24,\ 36.8)$ – at the end of the measuring.

The average rate of change

The average rate of change between two points of a graph is the change in the quantity divided by the time that passed. It says by how much the quantity changed, on average, in each unit of time.

\displaystyle \frac{y_2-y_1}{x_2-x_1}

In the temperature graph:

From hour 0 to hour 5: $\displaystyle {\frac{39.5-37}{5-0}}={\frac{2.5}{5}}={0.5}$ – the temperature rose by half a degree an hour on average.
From hour 9 to hour 15: $\displaystyle {\frac{37.5-39}{15-9}}={\frac{-1.5}{6}}={-0.25}$ – the temperature fell by a quarter of a degree an hour on average.

A positive rate – the quantity rises; a negative rate – it falls.

A steep part – a fast change

The steeper a part of the graph, the faster the quantity changes there.

The amount of water in a reservoir, in millions of cubic metres, at the end of each month of one year:

It rises from January to April, in the rainy winter, and again from October to December.
It falls from April to October – in summer water is pumped out and there is no rain.
The maximum: 58 million m³, in April. The minimum: 28 million m³, in October.
From April to October: $\displaystyle {\frac{28-58}{6}}={-5}$ – the reservoir lost 5 million m³ a month on average.
The steepest part is between January and February: a rise of 8 million m³ in one month.
From January to December: $\displaystyle \frac{36-40}{11}\approx-0.36$ – almost no change, although during the year the reservoir rose and fell by tens of millions of m³. An average rate over a long period hides what happened inside it.

Exercise 1

The graph: the temperature in a town, in degrees, over one day – from midnight ( $\displaystyle {x}={0}$ ) to the next midnight ( $\displaystyle {x}={24}$ ).

Mark on the graph the maximum point and the minimum point.

\displaystyle T(x)

In which hours did the temperature rise, and in which did it fall?

Exercise 2

In the graph of Exercise 1:

Work out the average rate of change of the temperature from hour 6 to hour 14, in degrees an hour.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power