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Graphs

Revise Graphs for Mathematics A 4MA1 (O Level) — revision notes, 468 practice questions and instant AI marking.

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  Pearson Edexcel International GCSE Mathematics A (4MA1)

Graphs

The big idea: A graph turns an equation or a situation into a picture. The gradient tells you how fast one quantity changes compared with the other, and where two graphs meet tells you where their equations are both true.

Summary — What This Chapter Covers
  • Interpret real-life graphs, including distance–time and speed–time graphs.
  • Plot and read coordinates in all four quadrants; find midpoints.
  • Draw and use straight-line conversion graphs.
  • Gradient = increase in y ÷ increase in x; y = mx + c has gradient m and y-intercept (0, c).
  • Plot linear and quadratic graphs from a table of values.
  • Higher: cubic, reciprocal and trig graphs; transformations; tangents; intersections; parallel and perpendicular lines.
1. Real-Life Graphs
Foundation & Higher

Distance–time and speed–time

On a distance–time graph the gradient is the speed; a horizontal section means stationary. On a speed–time graph the gradient is the acceleration (negative gradient means deceleration) and the area under the graph is the distance travelled.

Worked Example

A car starts from rest and accelerates steadily to 12 m/s in 8 seconds. It travels at 12 m/s for 20 seconds, then decelerates steadily to rest in 6 seconds. (a) Find the acceleration in the first 8 seconds. (b) Find the total distance travelled.

Practice Question 1

A distance–time graph shows Sara walking 3 km in 45 minutes, then a horizontal line for 15 minutes. Find her walking speed in km/h and describe the horizontal section.

2. Coordinates and Midpoints
Foundation & Higher

Four quadrants

Coordinates are written (x, y): across first, then up or down. The midpoint of a line segment is the mean of the x-coordinates and the mean of the y-coordinates. Geometric facts (opposite sides of a parallelogram are equal and parallel; a square has equal sides at right angles) let you find a missing vertex.

Midpoint of (x1, y1) and (x2, y2) ( (x1 + x2)/2 , (y1 + y2)/2 )
Worked Example

Find the midpoint of the line segment joining (−3, 5) and (7, −1).

Practice Question 2

ABCD is a parallelogram with A(1, 2), B(5, 2) and C(6, 5). Find the coordinates of D.

Practice Question 3

M(4, 1) is the midpoint of AB. A is (−2, 3). Find B.

3. Conversion Graphs
Foundation & Higher

Straight lines through the origin

A conversion graph (currency, units) is a straight line, usually through the origin. Read across from one axis to the line, then down or across to the other. For values off the graph, read a value that is on it and multiply.

Worked Example

A conversion graph between US dollars (horizontal axis, 0 to 40) and Hong Kong dollars (vertical axis) is a straight line through (0, 0) and (40, 312). (a) Convert US$25. (b) Convert HK$1000 to US dollars, to the nearest cent.

4. Gradient and y = mx + c
Foundation & Higher

Steepness and intercept

5. Plotting Linear and Quadratic Graphs
Foundation & Higher

Tables of values

6. Equations of Lines, Parallel and Perpendicular
Higher tier

From two points; m1 × m2 = −1

Parallel Perpendicular

Gradient through (x1, y1) and (x2, y2) 2 1 2 1
Memorise
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Also in the full note
  • Recognise, plot and draw
  • f(x) + a, f(x + a), af(x), f(ax)
  • Gradient of a curve; solving equations graphically
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More Mathematics A topics

Integers
Graphical representation of data
Vectors
Angles, lines and triangles
Sequences
Use of symbols
Fractions
Function notation
Polygons
Algebraic manipulation