Sec 1 · G3 (Express)

Coordinates and linear graphs

Practise coordinates and linear graphs with step-by-step help. Free account.

1. Coordinates

Left: points A(3, 2), B(−4, 1), C(−2, −3) and D(4, −2) plotted in the four quadrants. Right: the line y = 2x + 1 through (1, 3) and (3, 7), with a run of 2 and a rise of 4, and the y-intercept 1.

Write	Means
$(x, y)$	across first, then up or down
$(3, 5)$ and $(5, 3)$	different points

📖 Worked example

$ABCD$ is a rectangle. The points $A$ , $B$ and $C$ are shown. Find the coordinates of $D$ .

Step 1 of 2

D is the same move from A as C is from B. Read B and C off the grid: the move from B to C

Across:

-3 - (-3) = 0

Up:

1 - (-5) = 6

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2. Functions and tables of values

$x$	$-1$	$0$	$1$	$2$	$3$
$y = 2x + 1$	$-1$	$1$	$3$	$5$	$7$

Substitute a negative in brackets, x = −1

\displaystyle y = 2x + 1

e.g.

\displaystyle 2(-1) + 1

\displaystyle = -1

📖 Worked example

Complete the table of values for $y = 3x + 3$ .

Step 1 of 1

Put each missing x into y = mx + c (negatives in brackets)

When

x = -2

y = 3(-2) + 3

y = -3

When

x = 0

y = 3(0) + 3

y = 3

When

x = 1

y = 3(1) + 3

y = 6

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3. Gradient and the equation y = mx + c

Name	Formula	Example: $y = 2x + 1$
gradient	$m = \dfrac{\text{rise}}{\text{run}}$	$\dfrac{4}{2}$ , so $2$
$y$ -intercept	$c$ : where the line crosses the $y$ -axis	$1$

Line	Gradient
slopes up to the right	positive
slopes down to the right	negative
horizontal, $y = 3$	$0$
vertical, $x = 2$	not defined

📖 Worked example

The line passes through the points $A$ and $B$ shown. Find its gradient.

Step 1 of 2

Read A and B off the grid. Gradient = rise ÷ run, from A to B (the rise is negative if the line goes down)

A(-1, 4)

and

B(3, -2)

Rise:

-2 - 4 = -6

Run:

3 - (-1) = 4

To find	Method	Example
the line through two points	$m$ from rise $\div$ run, then put one point into $y = mx + c$ for $c$	$(1, 5)$ , $(3, 9)$ : $m = 2$ , $5 = 2 + c$ , $y = 2x + 3$
where a line meets the $x$ -axis	put $y = 0$	$2x + 3y = 12$ : $x = 6$
where it meets the $y$ -axis	put $x = 0$	$2x + 3y = 12$ : $y = 4$

📖 Worked example

A straight line passes through the points $A(5, 2)$ and $B(2, 8)$ . Find the equation of the line, in the form $y = mx + c$ .

Step 1 of 3

Gradient = rise ÷ run, from A to B

m = \dfrac{8 - 2}{2 - 5}

\phantom{m} = \dfrac{6}{-3}

\phantom{m} = -2

📖 Worked example

The line $3x + 2y = 18$ meets the $x$ -axis at $P$ and the $y$ -axis at $Q$ . $O$ is the origin. Find the area of triangle $OPQ$ .

Step 1 of 3

P is on the x-axis: put y = 0 and solve

3x = 18

x = 6

📖 Worked example

The graph shows the taxi fare, $ $y$ , for a trip of $x$ km. Use the graph to find the fare for a trip of $2$ km.

Step 1 of 1

Go up from 2 on the horizontal axis to the line, then across to the vertical axis

Fare: $

9

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Sec 1 · G3 (Express)

Coordinates and linear graphs

Practise coordinates and linear graphs with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Coordinates

Write	Means
$(x, y)$	across first, then up or down
$(3, 5)$ and $(5, 3)$	different points

📖 Worked example

$ABCD$ is a rectangle. The points $A$ , $B$ and $C$ are shown. Find the coordinates of $D$ .

Step 1 of 2

D is the same move from A as C is from B. Read B and C off the grid: the move from B to C

Across:

-3 - (-3) = 0

Up:

1 - (-5) = 6

Practise this →

2. Functions and tables of values

$x$	$-1$	$0$	$1$	$2$	$3$
$y = 2x + 1$	$-1$	$1$	$3$	$5$	$7$

Substitute a negative in brackets, x = −1

\displaystyle y = 2x + 1

e.g.

\displaystyle 2(-1) + 1

\displaystyle = -1

📖 Worked example

Complete the table of values for $y = 3x + 3$ .

Step 1 of 1

Put each missing x into y = mx + c (negatives in brackets)

When

x = -2

y = 3(-2) + 3

y = -3

When

x = 0

y = 3(0) + 3

y = 3

When

x = 1

y = 3(1) + 3

y = 6

Practise this →

3. Gradient and the equation y = mx + c

Name	Formula	Example: $y = 2x + 1$
gradient	$m = \dfrac{\text{rise}}{\text{run}}$	$\dfrac{4}{2}$ , so $2$
$y$ -intercept	$c$ : where the line crosses the $y$ -axis	$1$

Line	Gradient
slopes up to the right	positive
slopes down to the right	negative
horizontal, $y = 3$	$0$
vertical, $x = 2$	not defined

📖 Worked example

The line passes through the points $A$ and $B$ shown. Find its gradient.

Step 1 of 2

Read A and B off the grid. Gradient = rise ÷ run, from A to B (the rise is negative if the line goes down)

A(-1, 4)

and

B(3, -2)

Rise:

-2 - 4 = -6

Run:

3 - (-1) = 4

To find	Method	Example
the line through two points	$m$ from rise $\div$ run, then put one point into $y = mx + c$ for $c$	$(1, 5)$ , $(3, 9)$ : $m = 2$ , $5 = 2 + c$ , $y = 2x + 3$
where a line meets the $x$ -axis	put $y = 0$	$2x + 3y = 12$ : $x = 6$
where it meets the $y$ -axis	put $x = 0$	$2x + 3y = 12$ : $y = 4$

📖 Worked example

A straight line passes through the points $A(5, 2)$ and $B(2, 8)$ . Find the equation of the line, in the form $y = mx + c$ .

Step 1 of 3

Gradient = rise ÷ run, from A to B

m = \dfrac{8 - 2}{2 - 5}

\phantom{m} = \dfrac{6}{-3}

\phantom{m} = -2

📖 Worked example

The line $3x + 2y = 18$ meets the $x$ -axis at $P$ and the $y$ -axis at $Q$ . $O$ is the origin. Find the area of triangle $OPQ$ .

Step 1 of 3

P is on the x-axis: put y = 0 and solve

3x = 18

x = 6

📖 Worked example

The graph shows the taxi fare, $ $y$ , for a trip of $x$ km. Use the graph to find the fare for a trip of $2$ km.

Step 1 of 1

Go up from 2 on the horizontal axis to the line, then across to the vertical axis

Fare: $

9

Practise this →

Try a question: Coordinates and linear graphs

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Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

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