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C3.1 Coordinates

A Cartesian coordinate $(x,y)$ locates a point using two directed movements from the origin $(0,0)$ : move horizontally by $x$ , then vertically by $y$ .

Coordinate sign	Movement
$x>0$	right
$x<0$	left
$y>0$	up
$y<0$	down

To read a plotted point, project it to the horizontal $x$ -axis first and record that value; then project it to the vertical $y$ -axis and record the second value. For a point three units right and two units down, write $(3,-2)$ .

To plot $(-2,4)$ , start at the origin, move 2 units left to $x=-2$ , then 4 units up to $y=4$ . Mark the point only where these two coordinate lines meet.

Point location	Coordinate fact
on the $x$ -axis	$y=0$
on the $y$ -axis	$x=0$
at the origin	$x=0$ and $y=0$

Read the scale on each axis independently before counting grid spaces: one square may represent more than one unit, and the two axes can use different scales.

Coordinate order never changes: $(x,y)$ means horizontal first, vertical second. For example, $(-2,4)$ and $(4,-2)$ are different points.

C3.2 Drawing linear graphs

Every pair $(x,y)$ satisfying a linear equation lies on one straight line. Generate accurate pairs, plot them, then use a ruler to draw the line.

Read both axis scales; choose at least two well-separated $x$ -values in the required interval; substitute each into the equation; plot every resulting $(x,y)$ pair; check the points align; draw one ruled line through them across only the stated interval.

$y=8-2x,\qquad 0\le x\le4$

$x$	$0$	$2$	$4$
$y=8-2x$	$8$	$4$	$0$

Plot $(0,8)$ , $(2,4)$ and $(4,0)$ . A ruled segment through all three points is the graph for $0\le x\le4$ . The middle point checks that the endpoint calculations are consistent.

Equation	What to draw
$y=k$	a horizontal line through $y=k$
$y=x$	points with equal coordinates, such as $(-2,-2)$ and $(3,3)$

For $y=mx+c$ , setting $x=0$ gives the point $(0,c)$ . Increasing $x$ by 1 changes $y$ by $m$ , so the plotted points should follow that repeated change.

Do not join points freehand or stop at isolated dots when a graph is requested. Use the axis scales—not the number of grid squares—to position each coordinate.

C3.3 Gradient of linear graphs

The gradient measures how much a straight line changes vertically for each horizontal change. Read both changes from two clear points on the grid.

$\text{gradient}=\frac{\text{vertical change}}{\text{horizontal change}}=\frac{y_2-y_1}{x_2-x_1}$

Choose two well-separated points that lie exactly on the line, preferably grid intersections; read each axis scale; move from the left point to the right point; record the signed vertical change and the positive horizontal change; divide and simplify.

$A(-2,4),\ B(4,1):\qquad \text{gradient}=\frac{1-4}{4-(-2)}=\frac{-3}{6}=-\frac12$

Line direction from left to right	Gradient
rises	positive
falls	negative
horizontal	$0$
vertical	undefined because horizontal change is $0$

A larger triangle usually makes grid readings more reliable. Using any other pair of exact points on the same straight line must give the same ratio.

For Core C3.3, obtain the gradient from the grid. Do not count squares without checking the axis scales, and do not discard the negative sign when the line falls from left to right.

C3.5 Equations of linear graphs

In $y=mx+c$ , $m$ is the line’s gradient and $c$ is its $y$ -intercept value. These two numbers completely determine a non-vertical straight line.

Feature	How to read it
gradient	coefficient $m$ of $x$
$y$ -intercept	$c$ , giving point $(0,c)$
horizontal line	$y=k$
vertical line	$x=k$

For $y=6x+3$ , the gradient is $6$ and the line crosses the $y$ -axis at $(0,3)$ . Rewrite an equation into fully simplified $y=mx+c$ form before reading these values.

To obtain an equation from a graph: read the axis scales; choose two exact points and calculate $m=\dfrac{ ext{vertical change}}{ ext{horizontal change}}$ ; read the value $c$ where the line crosses the $y$ -axis; substitute both into $y=mx+c$ .

$m=\frac{6-1}{0-1}=-5,\qquad c=6\quad\Longrightarrow\quad y=-5x+6$

Check the equation against another point on the graph. For example, $(1,1)$ satisfies $y=-5x+6$ because $-5(1)+6=1$ . In a context, use the variables named in the question, such as $c=2d+5$ .

Do not write a vertical line as $y=mx+c$ : its equation is $x=k$ . Always include $y=$ (or the named dependent variable) and give the final equation in fully simplified form.

C3.6 Parallel lines

Distinct parallel straight lines have the same gradient but different intercepts. Keep the given line’s gradient, then use the new point to find its intercept.

Given line	Parallel-line form
$y=mx+c$	$y=mx+k$ , with a different intercept
vertical line $x=a$	$x=b$ , with $b
e a$

Write the new line as $y=mx+k$ using the known gradient $m$ ; substitute the coordinates of the point it passes through; solve for $k$ ; write the complete equation in fully simplified form; substitute the point once more to check it.

$y=4x-1,\quad (1,-3):\qquad -3=4(1)+k\quad\Longrightarrow\quad k=-7$

$\therefore\ y=4x-7$

If the given point lies on the $y$ -axis, its $x$ -coordinate is $0$ , so its $y$ -coordinate is immediately the new intercept. For example, a line parallel to $y=5x+6$ through $(0,-7)$ is $y=5x-7$ .

If the original line is shown on a grid or specified by two points, find its gradient first; only that gradient transfers to the parallel line. The original intercept does not.

Keeping both the same gradient and the same intercept reproduces the original line, not a distinct parallel line. Do not change the sign or take the reciprocal of the gradient.

3. Coordinate geometry

Exam analysis

Most tested topics

C3.1 Coordinates

Read and plot Cartesian coordinates

C3.2 Drawing linear graphs

Draw a straight-line graph from its equation

C3.3 Gradient of linear graphs

Find a straight line’s gradient from a grid

C3.5 Equations of linear graphs

Interpret and find a straight-line equation

C3.6 Parallel lines

Find the equation of a parallel line