Representations of a function

Four representations

The same function can be shown in four ways – four representations. Each one shows something else easily:

A story – what the variables are, and what they mean.
A table – some exact values, ready to read.
A formula – the $\displaystyle y$ of any $\displaystyle x$ , by substituting.
A graph – the whole function at a glance: where it goes up, where it goes down, where it is highest.

Example: a water tank

Only

\displaystyle x

from 0 to 8 has a meaning: after 8 minutes the tank is empty.

A tank holds 40 litres of water.
Every minute 5 litres flow out of the tank.

The variables: $\displaystyle x$ – the time in minutes, $\displaystyle y$ – the water in the tank in litres.
The formula: after $\displaystyle x$ minutes $\displaystyle 5x$ litres have flowed out, so $\displaystyle {y}={40-5x}$ .
The table: substitute in the formula.

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 2$	$\displaystyle 4$	$\displaystyle 6$	$\displaystyle 8$
$\displaystyle y$	$\displaystyle 40$	$\displaystyle 30$	$\displaystyle 20$	$\displaystyle 10$	$\displaystyle 0$

\displaystyle y=40-5x

The graph goes down: as time passes, less water is left in the tank. It meets the $\displaystyle x$ -axis at $\displaystyle (8,0)$ – the moment the tank is empty.

Clues for matching representations

To recognise the same function in different representations, look for the same two things in each:

The starting value – $\displaystyle y$ when $\displaystyle {x}={0}$ . In the tank: 40 litres. In a table – the column of $\displaystyle {x}={0}$ ; in a graph – the point on the $\displaystyle y$ -axis; in a formula – substitute $\displaystyle {x}={0}$ .
The change – by how much $\displaystyle y$ grows or shrinks when $\displaystyle x$ grows by 1. In the tank: it shrinks by 5. In a graph you see whether the line goes up or down.

Exercise 1

Match each story with its formula, its table and its graph. In every story $\displaystyle x$ is the independent variable and $\displaystyle y$ the dependent one.

The stories:

Savings: Dan’s money box holds 10 shekels, and every week he adds 2 shekels. $\displaystyle x$ – the number of weeks.
The candle: a candle is 20 cm tall, and every hour it gets 2 cm shorter. $\displaystyle x$ – the number of hours.
Cookies: a cookie costs 3 shekels. $\displaystyle x$ – the number of cookies.

The formulas: $\displaystyle {y}={20-2x}$ , $\displaystyle {y}={3x}$ , $\displaystyle {y}={10+2x}$ .

The tables:

	$\displaystyle {x}={0}$	$\displaystyle {x}={1}$	$\displaystyle {x}={2}$	$\displaystyle {x}={3}$
Table 1	$\displaystyle 20$	$\displaystyle 18$	$\displaystyle 16$	$\displaystyle 14$
Table 2	$\displaystyle 0$	$\displaystyle 3$	$\displaystyle 6$	$\displaystyle 9$
Table 3	$\displaystyle 10$	$\displaystyle 12$	$\displaystyle 14$	$\displaystyle 16$

The graphs: $\displaystyle f$ , $\displaystyle g$ , $\displaystyle h$ .

\displaystyle f(x)

\displaystyle g(x)

\displaystyle h(x)

Exercise 2

The points $\displaystyle A$ , $\displaystyle B$ and $\displaystyle C$ are on the graph.

Write the points in a table.
Find the rule, and write the function’s formula.

\displaystyle y=\,?

\displaystyle A=(0,1)

\displaystyle B=(1,4)

\displaystyle C=(2,7)

Exercise 3

Yael’s money box holds 50 shekels.
Every week she adds 20 shekels to it.

Write a formula: $\displaystyle y$ – the money in the box after $\displaystyle x$ weeks.
After how many weeks will the box hold 250 shekels? Write an equation and solve it.

Exercise 4

$\displaystyle x$	$\displaystyle 0$	$\displaystyle 1$	$\displaystyle 2$	$\displaystyle 3$
$\displaystyle y$	$\displaystyle 5$	$\displaystyle 8$	$\displaystyle 11$	$\displaystyle 14$

Write a story the table fits. What are $\displaystyle x$ and $\displaystyle y$ in your story?
Write the function’s formula.

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
$\displaystyle a\cdot c>b\cdot c$ $\displaystyle a<b,\ c<0$