Rate of change

The rate of change of a function is how much $\displaystyle y$ changes when $\displaystyle x$ grows by 1.

A constant rate of change: at every step of 1 in $\displaystyle x$ , $\displaystyle y$ changes by the same number.
A varying rate of change: at different steps $\displaystyle y$ changes by different numbers.

A burning candle: a constant rate of change

The

\displaystyle x

-axis: hours. The

\displaystyle y

-axis: the candle’s height in cm.

A candle 20 cm tall is lit, and every hour it gets 3 cm shorter.

Time (hours)	0	1	2	3	4
Height (cm)	20	17	14	11	8

Every hour the height goes down by the same amount. The rate of change is constant: $\displaystyle -3$ cm an hour. The minus says the height goes down. The graph is straight.

\displaystyle y=20-3x

A falling ball: a varying rate of change

A ball falling from the roof of a building goes faster and faster. The table shows how many metres it has fallen after each second:

Time (seconds)	0	1	2	3
Distance (metres)	0	5	20	45

In the first second the ball fell 5 metres, in the next one 15, and in the one after that 25. The rate of change varies – it grows every second – and the graph curves.

\displaystyle y=5x^{2}

A constant rate of change and a straight graph

A function with a constant rate of change has a straight graph: it is a linear function. And the other way round – if the graph is straight, the rate of change is constant.

A positive rate: the line goes up from left to right.
A negative rate: the line goes down.
A rate of 0: the line is horizontal, and the value of $\displaystyle y$ does not change.

A taxi and a car park

A taxi charges 12 ₪ to get in, and 4 ₪ more for every kilometre.

Distance (km)	0	1	2	3	4
Price (₪)	12	16	20	24	28

Every kilometre adds 4 ₪: a constant rate of change, 4 ₪ a kilometre.

A car park charges 10 ₪ for the first hour and 6 ₪ for every further hour, but no more than 34 ₪ a day.

Time (hours)	1	2	3	4	5	6
Price (₪)	10	16	22	28	34	34

Up to the fifth hour every hour adds 6 ₪, and after it – 0 ₪. The rate of change varies.

The rate of change between two columns

Check: between 5 and 10 minutes 250 litres went in as well – the rate is constant.

When $\displaystyle x$ jumps by more than 1 in a table, divide the change in $\displaystyle y$ by the change in $\displaystyle x$ .
A paddling pool is being filled:

Time (minutes)	0	5	10	15
Water (litres)	100	350	600	850

In 5 minutes $\displaystyle {350-100}={250}$ litres went in. In one minute: $\displaystyle {\frac{250}{5}}={50}$ . The rate of change is 50 litres a minute.

Exercise 1

In which table is the rate of change constant, and in which does it vary?
Where the rate is constant, write it.

a. Roni’s savings:

Weeks	0	1	2	3	4
Savings (₪)	50	80	110	140	170

b. The height of a seedling:

Days	1	2	3	4	5
Height (cm)	2	3	5	8	12

$\displaystyle x$	0	2	4	6
$\displaystyle y$	30	26	22	18

$\displaystyle x$	0	1	2	3
$\displaystyle y$	1	2	4	8

Exercise 2

A pool holds 1200 litres of water.
The drain is opened, and every minute 40 litres flow out of the pool.

Complete a table of the amount of water after 0, 5, 10, 15 and 20 minutes.
What is the rate of change?
Write an equation and solve it: after how many minutes will the pool be empty?

The rate of change

/ – 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

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

Exercise 3

Noa and Itai set off together on a bike ride.
The graphs show how many kilometres each of them had ridden ( $\displaystyle y$ -axis) against the time in hours ( $\displaystyle x$ -axis).
$\displaystyle f$ – Noa, $\displaystyle g$ – Itai.

Which of them rode at a constant rate? What was the rate?
How many kilometres did Itai ride in each of the three hours?
After how many hours were they at the same place?

\displaystyle f(x)=15x

\displaystyle g(x)

The constant rate

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

At the same place (hours)

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

Exercise 4

In which of these is the rate of change constant? Explain.

The price of cheese against its weight.
A child’s height against their age.
The distance travelled by a train that goes at 120 km/h all the time, against the time.
The temperature outside during one day, against the hour.

Formulas: algebra, grade 8

Open as a page

The linear function

A linear function – $\displaystyle m$ the slope; the line crosses the $\displaystyle y$ -axis at $\displaystyle (0,b)$
$\displaystyle y=mx+b$
The slope of the line through $\displaystyle (x_1,y_1)$ and $\displaystyle (x_2,y_2)$
$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$
Parallel lines have the same slope
$\displaystyle m_1=m_2$
A line through the origin – $\displaystyle {b}={0}$
$\displaystyle y=mx$
A horizontal line – the slope is 0
$\displaystyle y=b$

Word problems

$\displaystyle p$ percent of $\displaystyle x$
$\displaystyle \frac{p}{100}\cdot x$
$\displaystyle s$ the distance, $\displaystyle v$ the speed, $\displaystyle t$ the time
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$

Inequalities

Adding or subtracting the same number on both sides – the sign stays
$\displaystyle a+c<b+c$ $\displaystyle a<b$
Multiplying or dividing by a positive number – the sign stays
$\displaystyle a\cdot c<b\cdot c$ $\displaystyle a<b,\ c>0$
Multiplying or dividing by a negative number – the sign turns round
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$