Coordinates and Linear Graphs

Syllabus coverage

Use and interpret Cartesian coordinates in two dimensions
Plot a point from its coordinates, and read off the coordinates of a point on a grid
Draw straight-line graphs for linear equations
Draw a straight line from a table of values when the equation is given in that form
Use equations of the form $y = m x + c$ , for example $y = - 2 x + 5$
Interpret and obtain the equation of a straight-line graph in the form $y = m x + c$ , including lines of the form $x = k$
Extended only: draw straight-line graphs for equations such as $3 x + 2 y = 5$ and $y = 7 - 4 x$

The graph of

y = 2 x - 3

. The line crosses the

y

-axis at

(0, - 3)

, and rising 2 units for every 1 unit across shows that the gradient is 2.

The coordinate plane and its four quadrants

The coordinate plane is made by two number lines crossing at right angles. The horizontal line is the x-axis, the vertical one is the y-axis, and they meet at the origin, the point $(0, 0)$ . Any point is fixed by an ordered pair $(x, y)$ : the first number says how far across to go, the second how far up.

The axes cut the plane into four quadrants, numbered anticlockwise from the top right. In the first quadrant both coordinates are positive; in the second $x$ is negative and $y$ positive; in the third both are negative; in the fourth $x$ is positive and $y$ negative. The signs of the two numbers are enough to name the quadrant without plotting anything.

The order of the pair is part of the meaning. $(3, 5)$ is 3 across and 5 up, while $(5, 3)$ is 5 across and 3 up - two different points. Swapping the numbers is the commonest single mistake in this topic.

Quadrant	$x$	$y$	Example
First	positive	positive	$(4, 3)$
Second	negative	positive	$(- 4, 3)$
Third	negative	negative	$(- 4, - 3)$
Fourth	positive	negative	$(4, - 3)$

A point on an axis belongs to no quadrant:

(0, 4)

lies on the

y

-axis and

(- 2, 0)

lies on the

x

-axis.

Plotting a point and reading one back

To plot $(4, - 3)$ , start at the origin, count 4 units to the right along the $x$ -axis, then 3 units down. Mark the point with a small cross rather than a heavy blob, and label it with its letter when the question names it.

Reading a point is the same journey in reverse: go straight down or up to the $x$ -axis for the first coordinate, and straight across to the $y$ -axis for the second. If the grid is numbered every 2 units and the point sits halfway between 2 and 4, its coordinate is 3.

Before plotting anything, work out what one square is worth. Count the squares between two numbered lines: a square may be 2, 5 or 10 units, and the two axes do not have to use the same scale. Reading the scale wrongly shifts every point on the page.

Count across the $x$ -axis first, then up or down for $y$
A small cross is clearer than a dot when two points are close together
Work out the value of one grid square before you plot anything

If one square is 2 units, a point on the third square from the origin is at 6, not 3. Check the scale before you trust a coordinate.

Drawing a straight line from a table of values

When a question gives a table of values, the table decides which points you plot. Take each value of $x$ in turn, substitute it into the equation and write the result in the $y$ row. For $y = 2 x - 3$ with $x = - 1$ : $2 \times (- 1) - 3 = - 5$ .

Plot each pair, then join the points with a straight line drawn with a ruler. Two points fix a line, but plot at least three: if the third point is not on the line you have drawn, one of the substitutions is wrong and you have found it before the examiner does.

Draw the line right across the grid instead of stopping at the last plotted point. The line carries on beyond the table, and questions often ask you to read a value of $x$ that lies outside it.

$x$	Substituting into $y = 2 x - 3$	$y$
-1	$2 \times (- 1) - 3$	-5
0	$2 \times 0 - 3$	-3
1	$2 \times 1 - 3$	-1
2	$2 \times 2 - 3$	1
3	$2 \times 3 - 3$	3

Substitute with the sign attached:

2 \times (- 1)

- 2

, so

x = - 1

gives

y = - 5

, not

- 1

Reading values off the line

To find $y$ for a given $x$ , start on the $x$ -axis at that value, go straight up or down to the line, then across to the $y$ -axis. To find $x$ for a given $y$ , start on the $y$ -axis and work the other way. Read to the accuracy the grid allows: on a grid marked every 1 unit, 3.5 is a sensible answer and 3.47 is not.

Every point on the line satisfies the equation, so a reading can always be checked by substitution. On the line $y = 2 x - 3$ the point $(5, 7)$ is on it because $2 \times 5 - 3 = 7$ , while $(3, 4)$ is not, because $2 \times 3 - 3 = 3$ , not 4.

If an answer has to be exact, solve the equation rather than read the graph. Reading gives $y = 4$ when $x = 3.5$ , and solving $2 x - 3 = 4$ gives the same value exactly, with no grid to misread.

Up from the $x$ -axis to the line, then across to the $y$ -axis
Substitute the coordinates to test whether a point lies on a line
Solve the equation when the question asks for an exact value

A point that misses the line by one square is usually a plotting slip rather than a reading slip, so check the substitution first.

The shape of the graph of $y = m x + c$

Every straight line that is not vertical can be written as $y = m x + c$ . The number $m$ is the gradient: it fixes how steep the line is and which way it slopes. The number $c$ is the y-intercept: the value of $y$ where the line crosses the $y$ -axis, at the point $(0, c)$ .

Changing $m$ tilts the line. A bigger $m$ is steeper, and a negative $m$ slopes down from left to right. Changing $c$ slides the whole line up or down without tilting it, so $y = 3 x + 1$ and $y = 3 x - 4$ are parallel, passing through $(0, 1)$ and $(0, - 4)$ respectively.

On the graph of $y = 2 x - 3$ the line crosses the $y$ -axis at $(0, - 3)$ , and from that point one step to the right needs two steps up to land back on the line, so the gradient is 2.

Equation	Gradient $m$	Crosses the $y$ -axis at
$y = 2 x - 3$	2	$(0, - 3)$
$y = - 2 x + 5$	-2	$(0, 5)$
$y = 3 x$	3	$(0, 0)$
$y = 4$	0	$(0, 4)$

y = 5 - 2 x

the

x

-term is

- 2 x

, so the gradient is

- 2

, not 2. Read the sign together with the number.

The graph of

y = 2 x - 3

: the line crosses the

y

-axis at

(0, - 3)

and rises 2 units for every 1 unit across.

Vertical and horizontal lines, and the marks for drawing

Two kinds of equation do not fit $y = m x + c$ . $y = c$ is horizontal: every point on it has $y$ -coordinate $c$ , so the line runs across at height $c$ . $x = k$ is vertical: every point has $x$ -coordinate $k$ , so the line runs up and down through $(k, 0)$ .

A graph question is marked in stages: the axes labelled and drawn to a sensible scale, the points plotted correctly, and a straight line drawn through them with a ruler. A line that is drawn freehand or stops short of the edge of the grid loses the last of those marks.

Choosing the scale is part of the work. Find the largest and smallest values you have to plot, then choose a scale that uses most of the grid and lands on whole squares - 2, 5 or 10 units to a square rather than 3 or 7.

$y = c$ is horizontal, $x = k$ is vertical
Label both axes, and label the line with its equation
Plot at least three points before drawing the line
Extend the line across the grid, not just between the plotted points

Key terms you must be able to define

Coordinates: The pair of numbers that fixes the position of a point: the $x$ -coordinate first, then the $y$ -coordinate.

Ordered pair: Two numbers written in a fixed order, $(x, y)$ . Swapping them describes a different point.

x-axis: The horizontal number line of a coordinate grid.

y-axis: The vertical number line of a coordinate grid.

Origin: The point where the two axes cross, with coordinates $(0, 0)$ .

Quadrant: One of the four regions the axes divide the plane into, numbered anticlockwise from the top right.

x-coordinate: The first number of a coordinate pair: the distance across from the $y$ -axis, positive to the right.

y-coordinate: The second number of a coordinate pair: the distance up from the $x$ -axis, positive upwards.

Cartesian coordinates: Coordinates measured from a pair of perpendicular axes, one horizontal and one vertical.

Linear equation: An equation whose graph is a straight line, such as $y = 2 x - 3$ .

Table of values: A table listing values of $x$ with the matching values of $y$ , used to plot a graph.

Gradient: How steep a straight line is: the vertical change divided by the horizontal change.

y-intercept: The value of $y$ where a line crosses the $y$ -axis, written $c$ in $y = m x + c$ .

Horizontal line: A line of the form $y = c$ , running left to right at height $c$ . Its gradient is 0.

Vertical line: A line of the form $x = k$ , running straight up and down through $(k, 0)$ . Its gradient is undefined.

In context

Cambridge sets 0580 in real situations - a recipe, a bank statement, a map, a garden, a graph of a car's journey. These show how the mathematics appears in a question, then you write your own working for the numbers you are given.

Plotting from coordinates

Plot $A (3, 4)$ , $B (- 2, 1)$ , $C (- 3, - 2)$ and $D (4, - 1)$ on a grid. $A$ is in the first quadrant, because both coordinates are positive. $B$ is in the second, with negative $x$ and positive $y$ . $C$ is in the third, with both negative, and $D$ is in the fourth, with positive $x$ and negative $y$ .

Table of values

Complete the table for $y = 2 x - 3$ using $x = - 1, 0, 1, 2, 3$ . The five values are $- 5, - 3, - 1, 1, 3$ . Plot them, join them with a ruler, and label the line $y = 2 x - 3$ .

Reading off a line

Using the line $y = 2 x - 3$ , when $x = 3.5$ the graph gives $y = 4$ , and when $y = 3$ it gives $x = 3$ . Check the first reading by substitution: $2 \times 3.5 - 3 = 4$ .

The shape of $y = mx + c$

The lines $y = 3 x + 1$ and $y = 3 x - 4$ have the same gradient, 3, so they are parallel and the second is the first shifted 5 units down. The line $y = - x + 6$ has gradient $- 1$ , so it falls from left to right and crosses the $y$ -axis at $(0, 6)$ .

Flashcards

Click a card to flip it. Mark honestly — the tracker uses this.

Q1What does the point

(4, - 3)

tell you to do?

Go 4 units right along the

x

-axis, then 3 units down. The first number is always

x

Q2Which quadrant has both coordinates negative?

The third, at the bottom left. Quadrants are numbered anticlockwise from the top right.

Q3What are the coordinates of the origin?

(0, 0)

, the point where the two axes cross.

Q4In

y = m x + c

, what does

m

tell you?

The gradient: how steep the line is and which way it slopes.

Q5In

y = m x + c

, what does

c

tell you?

The

y

-intercept: the line crosses the

y

-axis at

(0, c)

Q6What is the gradient of

y = 5 - 2 x

- 2

. The

x

-term is

- 2 x

, so the sign travels with the number.

Q7What shape is the graph of

y = 4

A horizontal line at height 4. Every point on it has

y

-coordinate 4, and the gradient is 0.

Q8What shape is the graph of

x = 3

A vertical line through

(3, 0)

. Every point on it has

x

-coordinate 3.

Q9How do you check that

(5, 7)

lies on the line

y = 2 x - 3

Substitute:

2 \times 5 - 3 = 7

, which matches the

y

-coordinate, so the point is on the line.

Q10What are the values of

y = 2 x - 3

for

x = - 1, 0, 1, 2, 3

- 5, - 3, - 1, 1, 3

Q11Why plot at least three points when drawing a straight line?

A third point that misses the line shows you a substitution error before the examiner sees it.

Q12Where does

y = 3 x + 1

cross the

y

-axis?

(0, 1)

. Put

x = 0

into the equation and

y = 1

Q13How far across the grid should a drawn straight line go?

Right across it, past the plotted points at both ends, drawn with a ruler and labelled with its equation.

0 of 13 marked as known

Exam technique for this topic

Work out the scale of each axis before you plot anything, and write it beside the axis. Counting squares instead of units is the commonest reason a correctly plotted shape lands in the wrong place.
Plot with a small, sharp cross exactly on the grid intersection, and label each point with its letter. A point drawn by eye, a centimetre from the right position, scores nothing however neat it looks.
When a table of values is given, use the values of $x$ from the table. Substituting your own numbers wastes time and may miss the range of values the question wants plotted.
Draw the line with a ruler right across the grid, and write its equation beside it. If the question says to draw the graph, a line that stops at the last plotted point can lose the drawing mark.

Why students lose marks here

A student wroteThe point

(2, 5)

is the same as the point

(5, 2)

Why it loses marksThe pair is ordered: the first number is the

x

-coordinate and the second is the

y

-coordinate.

(2, 5)

is 2 across and 5 up, while

(5, 2)

is 5 across and 2 up - a different point in the same quadrant.

The answer that scores

(2, 5)

and

(5, 2)

are reflections of each other in the line

y = x

, not the same point. Read the first number along the

x

-axis and the second up the

y

-axis.

A student wroteFor

x = - 2

y = 2 x - 3

, the value of

y

is 1.

Why it loses marksThe student has worked out

2 \times 2 - 3

, ignoring the sign of

x

. Multiplying a negative number by a positive one gives a negative answer, so

2 \times (- 2) = - 4

The answer that scores

y = 2 \times (- 2) - 3 = - 4 - 3 = - 7

. Substitute with the sign attached, then add the negative numbers as normal.

A student wroteThe line

x = 3

is horizontal.

Why it loses marksThe equation fixes the

x

-coordinate at 3 and says nothing about

y

, so the line runs vertically through

(3, 0)

. Horizontal lines have the form

y = c

The answer that scores

x = 3

is vertical and passes through

(3, 0)

. The horizontal line at the same height as the point

(3, 4)

y = 4

A student wroteThe line

y = 2 x - 3

crosses the

y

-axis at

(2, - 3)

Why it loses marksThe 2 is the gradient, not the intercept. In

y = m x + c

the intercept is

c

, the number standing on its own, which here is

- 3

The answer that scoresThe line crosses the

y

-axis where

x = 0

, so at

y = - 3

. The intercept is

(0, - 3)

, and the 2 tells you the line rises 2 for every 1 across.

Syllabus coverage

The coordinate plane and its four quadrants

Plotting a point and reading one back

Drawing a straight line from a table of values

Reading values off the line

The shape of the graph of $y = m x + c$

Vertical and horizontal lines, and the marks for drawing

Key terms you must be able to define

In context

Flashcards

Exam-style questions

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Exam technique for this topic

Why students lose marks here

Coordinates and Linear Graphs

Syllabus coverage

The coordinate plane and its four quadrants

Plotting a point and reading one back

Drawing a straight line from a table of values

Reading values off the line

The shape of the graph of y=mx+c

Vertical and horizontal lines, and the marks for drawing

Key terms you must be able to define

In context

Flashcards

Exam-style questions

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Mark scheme — tick each point you actually made

Exam technique for this topic

Why students lose marks here

The shape of the graph of $y = m x + c$