Graphs (Grade 8) — Mathematics Grades 8–11

Reading and describing graphs, and plotting points on the Cartesian plane.

Gr 8 Term 4 — GraphsTerm 4

Practise Graphs in the app →

What gets asked

Describe a graph as linear or non-linear, increasing or decreasing.
Read off the maximum or minimum value shown on a graph.
Match a story to the global graph that represents it.
Plot a table of ordered pairs on the Cartesian plane.

You must be able to

Recognise that a flat part of a graph means the quantity is not changing.
Tell discrete data apart from continuous data on a graph.
Plot an ordered pair correctly as (x ; y), in that order.
Read a value off a graph using the correct scale on each axis.

Traps that cost marks

Reading how steep a graph is as if it shows how big the quantity is. Steepness shows how fast something is changing, not how much of it there is.
Plotting a point as (y ; x), the wrong way round. The order is always (x ; y). The first number moves you sideways, the second up or down.
Calling any straight-looking part of a graph a full linear graph. A linear graph must be a straight line for its whole length, not just for one section.
Reading the maximum value off the wrong axis of a graph. Check which axis you are reading before you write down a maximum or minimum.

Worked example

Thato fills a water tank at a steady rate, stops for a while, then fills it again. Describe the middle part of the graph.

Height is not changing while the graph is flat, so the tank is not filling.
A flat line still uses up time on the horizontal axis.
The middle part shows a pause, not the tank getting fuller or emptier.

Describe and interpret graphs

Reading the story a graph tells, before a single number is written on the axes.

global graph: A graph that tells a story through its shape, not through exact numbers.
linear: One straight line from the start of the graph to the end.
increasing: Going up as you read from left to right. Going down instead is decreasing.
maximum: The highest value the graph reaches. The lowest one is the minimum.
discrete: Data you count in whole steps, like taxis at a rank.
continuous: Data you measure, which can land on any value, like water in a tank.

A global graph tells a story with its shape. Read it from left to right, like a sentence. The across axis usually shows time. The upright axis shows the thing being measured.

First ask whether it is linear. A graph built from several straight pieces at different slopes is not. Each piece is straight, but the whole graph is not.

Then say whether it is constant, increasing or decreasing. Flat means constant, so nothing changes. Rising means increasing. Falling means decreasing, which is less of the thing, not a negative amount.

The maximum is the highest value the graph reaches, the minimum the lowest. Read it off the upright axis. The question asks how much, not when.

Discrete or continuous comes from the story, not from the drawing. Counting learners gives discrete data, so the points stay separate. Measuring water gives continuous data, so the line is unbroken.

Rules to remember

The upright axis answers how much. The across axis answers when.

Examples

A learner walks to the spaza shop, queues for

4

minutes, then walks home. Describe the middle of her distance-time graph.

Worked answer

The upright axis shows how far she is from home.
While she queues, that distance does not change.
So the middle is flat: she is standing still, not walking backwards.

Answer: a flat section, meaning she is standing still

A flat line means nothing is changing, not that nothing is happening.

A classroom is

1 8^{\circ}

C at 07:00,

2 6^{\circ}

C at 12:00 and

2 1^{\circ}

C at 16:00. Give the maximum temperature.

Worked answer

A temperature is read off the upright axis.
Compare the three values $18$ , $26$ and $21$ .
The highest is $26$ .
The time 12:00 answers a different question.

Answer: $2 6^{\circ}$ C

The maximum is how hot it got, not when it got hot.

A spaza shop counts the loaves it sells each day. A tank loses water steadily all day. Which set of data is discrete?

Worked answer

Loaves are counted, and nobody buys part of a loaf.
So the loaf data is discrete, and its points stay separate.
Water can sit at any level, even $2, 5$ litres.
So water is continuous, and its graph is one unbroken line.

Answer: the loaves are discrete; the water is continuous

The story decided it, not the way the graph was drawn.

Traps

Calling any straight-looking part of a graph a full linear graph. A linear graph must be a straight line for its whole length, not just for one section.
Reading the maximum value off the wrong axis of a graph. Check which axis you are reading before you write down a maximum or minimum.
Calling data continuous only because the graph was drawn with a joined line. Ask what is being measured. Counted things stay discrete, however the graph is drawn.
Reading how steep a graph is as if it shows how big the quantity is. Steepness shows how fast something is changing, not how much of it there is.

Learn and practise “Describe and interpret graphs” in the app →

Draw and match graphs

Turning a rule into a table, a table into points, and a story into the right shape.

Cartesian plane: The grid made by two number lines crossing at right angles.
origin: The point where the two axes cross, written $(0; 0)$ .
ordered pair: Two numbers written in a set order, like $(3; 5)$ , giving one point.
table of values: A table pairing each $x$ you choose with the $y$ the rule gives back.
scale: How much one square on an axis is worth.

The Cartesian plane is two number lines crossing at right angles. The across one is the $x$ -axis, the upright one the $y$ -axis, and they meet at the origin. A point is an ordered pair, $(x; y)$ : the first number moves you sideways, the second up or down.

Build a table of values first. Choose the $x$ values, put each into the rule, and write down the $y$ that comes back. Each column gives one ordered pair to plot.

Check the scale before you count squares. If one square is worth $2$ , then $y = 10$ sits five squares up, not ten. Counting squares as ones puts the point in the wrong place.

Join the points only when the story allows it. Counted things are discrete, so the points stay separate. When the rule squares $x$ , the points lie on a curve. Draw it smoothly, and do not call it linear.

To match a story to a global graph, take it part by part. A graph is not a picture of the thing. A hill walk drawn as distance against time does not look like the hill. Show every flat and falling part the story mentions.

Rules to remember

Plot across first, then up or down.

Examples

Complete a table for

y = 2 x + 1

using

x = 0; 1; 2; 3

, then write the ordered pairs.

Worked answer

Put each $x$ into the rule in turn.
$2 (0) + 1 = 1$ and $2 (1) + 1 = 3$ .
$2 (2) + 1 = 5$ and $2 (3) + 1 = 7$ .
Pair each $x$ with its own $y$ , across first.

Answer: $(0; 1)$ , $(1; 3)$ , $(2; 5)$ and $(3; 7)$

The rule built the table, and the table built the points.

On a grid, each square on the

y

-axis is worth

5

. How far up does the point

(2; 20)

sit?

Worked answer

The second number is the $y$ value, so $y = 20$ .
Each square is worth $5$ , not $1$ .
$20 \div 5 = 4$ .
So the point sits four squares above the $x$ -axis.

Answer: $4$ squares up

Reading the scale first stops you counting twenty squares.

Make a table for

y = x^{2} + 3

x = 0; 1; 2; 3

. Is the graph linear?

Worked answer

$0^{2} + 3 = 3$ and $1^{2} + 3 = 4$ .
$2^{2} + 3 = 7$ and $3^{2} + 3 = 12$ .
The $y$ values climb by $1$ , then $3$ , then $5$ .
The steps are not equal, so the points bend away from a straight line.

Answer: not linear; the points lie on a curve

Unequal steps mean a curve, so the points are joined smoothly.

Traps

Plotting a point as (y ; x), the wrong way round. The order is always (x ; y). The first number moves you sideways, the second up or down.
Counting squares as ones when each square on the axis is worth more. Check the scale first. If one square is worth $5$ , then $20$ is four squares, not twenty.
Drawing a graph that copies the picture of the situation, like the shape of a hill. A graph plots one quantity against another. Distance against time is not a drawing of the hill.
Joining the points with a solid line when the data is counted. Discrete data stays as separate points. A joined line would claim values in between.

Learn and practise “Draw and match graphs” in the app →

Grade 8 Graphs

What gets asked

You must be able to

Traps that cost marks

Worked example

Describe and interpret graphs

Rules to remember

Examples

Traps

Draw and match graphs

Rules to remember

Examples

Traps

About this material