Graphs of Linear FunctionsEdexcel GCSE Maths: Revision notes

The equation y = mx + c is the standard form for a straight-line graph, where:

• m is the gradient (or slope) — it tells you how steep the line is and the direction it travels
• c is the y-intercept — the point where the line crosses the y-axis, always written as the coordinate (0, c)

The gradient can be calculated as:

m = rise/run or m = (change in y)/(change in x)

A positive gradient means the line slopes upwards from left to right; a negative gradient means it slopes downwards. A gradient of zero produces a horizontal line, and an undefined gradient (vertical line) cannot be expressed in the form y = mx + c.

For example, in the equation y = 2x + 3, the gradient is 2 and the y-intercept is 3, meaning the line crosses the y-axis at (0, 3) and rises 2 units for every 1 unit moved horizontally to the right.

To plot a straight-line graph from an equation:

1. Identify the y-intercept (c) and plot it on the y-axis at (0, c)
2. Use the gradient (m) to find another point: move right by 1 unit (or a convenient number) and move up or down by m units (or the appropriate number)
3. Plot this second point and draw a straight line through both points, extending it across the grid as needed

To interpret a straight-line graph:

• Read the y-intercept directly from where the line crosses the y-axis
• Calculate the gradient by choosing two clear points on the line and using m = (y₂ − y₁)/(x₂ − x₁)
• Check that the line is truly straight and extends consistently in both directions

When points are given in a table or as coordinates, verify they lie on the line by substituting into y = mx + c. If the line doesn't pass through a point exactly, the point is not on the line.

Finding the equation through two given points:

1. Calculate the gradient using m = (y₂ − y₁)/(x₂ − x₁)
2. Substitute m and one of the points into y = mx + c to find c
3. Write the final equation in the form y = mx + c

Finding the equation through one point with a known gradient:

1. Use the point and the given gradient m in the formula y = mx + c
2. Substitute the x and y values of the point to find c
3. Write the final equation

Alternative method (point-slope form):

You can also use y − y₁ = m(x − x₁) directly and then rearrange to y = mx + c form.

Always check your answer by substituting both original points (or the given point) back into your final equation — they should satisfy it exactly.

Parallel lines have the same gradient. If two lines are parallel, their m values in y = mx + c are identical, but their c values (y-intercepts) differ.

For example, y = 2x + 3 and y = 2x − 5 are parallel because both have gradient 2.

Perpendicular lines have gradients that are negative reciprocals of each other. If one line has gradient m, a perpendicular line has gradient −1/m.

The relationship is: m₁ × m₂ = −1

For example:

• A line with gradient 2 is perpendicular to a line with gradient −1/2 (since 2 × −1/2 = −1)
• A line with gradient 3/4 is perpendicular to a line with gradient −4/3

Finding equations of parallel and perpendicular lines through a given point:

1. For a parallel line: use the same gradient as the given line, plus the new point, to find c
2. For a perpendicular line: find the negative reciprocal of the given gradient, use it with the new point to find c
3. Write the equation in the form y = mx + c

These conditions apply only to straight lines; vertical and horizontal lines need special consideration (a vertical line is perpendicular to a horizontal line, both with undefined/zero gradient respectively).

Distance–time graphs:

• The horizontal axis represents time, and the vertical axis represents distance from a starting point
• A straight line with positive gradient shows constant speed (moving away at a steady rate)
• A horizontal line (zero gradient) shows the object is stationary (not moving)
• A steeper gradient indicates faster speed; a gentler gradient indicates slower speed
• The gradient of the line equals the speed — calculated as distance ÷ time

Speed–time graphs:

• The horizontal axis represents time, and the vertical axis represents speed
• A straight line with positive gradient shows acceleration (speed increasing at a constant rate)
• A straight line with negative gradient shows deceleration (speed decreasing at a constant rate)
• A horizontal line shows constant speed (no acceleration)
• The area under the line represents the distance travelled

Interpreting these graphs:

• For distance–time: the gradient is speed, and you can find how far an object has travelled at any time
• For speed–time: the gradient is acceleration, and the area under the graph gives total distance
• Real-life examples include cars accelerating, journeys with stops, and objects moving at constant velocity

Always check units (e.g., km/h, m/s, distance in metres) and interpret what the gradient or area means in context.