Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Algebra
Practise Graphs of functions. unlimited generated questions on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Recognising, sketching, plotting and interpreting graphs of quadratic, cubic and reciprocal functions (both tiers), and on Higher exponential and trigonometric graphs. Includes reading approximate solutions from graphs and interpreting real-life graphs such as distance–time graphs.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
3
Complete a table of values and plotSubstitute each x (with brackets for negatives), plot the points and join them with a smooth curve.
4
Recognise the shapes of standard graphsLinear, quadratic (U or ∩ shape), cubic (S shape) and reciprocal \(y = \frac{1}{x}\) (two separate curves).
4
Interpret distance–time graphsThe gradient is the speed and a horizontal section means the object is stationary.
5
Read roots and turning points from graphsRoots are where the curve crosses the x-axis; the turning point is the minimum or maximum point.
5
Solve equations using a graphDraw a horizontal line such as \(y = 3\) and read the x-coordinates where it meets the curve.
Notes
Plotting graphs
Make a table of values, substitute each x (negatives in brackets) and plot the points accurately.
Join the points of a curve with a smooth curve, not straight line segments. Never join across a gap (reciprocal graphs).
To solve \(x^2 - 2x - 3 = 5\) using the graph of \(y = x^2 - 2x - 3\), draw \(y = 5\) and read the x-coordinates where it crosses the curve.
Shapes to recognise
Quadratic \(y = ax^2 + bx + c\): a U shape if a is positive, a ∩ shape if a is negative. It is symmetrical about a vertical line through its turning point.
Cubic, e.g. \(y = x^3\): an S-shaped curve. \(y = x^3\) passes through the origin and goes from bottom left to top right; \(y = -x^3\) goes from top left to bottom right.
Reciprocal \(y = \frac{1}{x}\): two separate curves in opposite quadrants, getting closer and closer to the axes without touching them. x cannot be 0.
Real-life graphs
Distance–time graph: the gradient is the speed; a horizontal line means stopped; steeper means faster.
Velocity–time graph: the gradient is the acceleration.
Read scales carefully: check what one small square represents on each axis.
Cheatsheet
Quadratic: U shape (a positive) or ∩ shape (a negative), symmetrical
Cubic: S shape
Reciprocal \(y = \frac{k}{x}\): two curves that never touch the axes
Roots: x-coordinates where the curve crosses the x-axis (y = 0)
Substitute each x value, using brackets for negatives.
Plot every point accurately.
Join with one smooth curve.
Example. Complete the table of values for \(y = x^2 - 2x - 3\) for x = −2, −1, 0, 1, 2, 3, 4. Then draw the graph.
Show the model answer
y values: 5, 0, −3, −4, −3, 0, 5 B2 (B1 for at least 4 correct) At least 6 points plotted correctly M1 A smooth U-shaped curve through all 7 points A1
Interpret a distance–time graph
3 marks4
A horizontal section means no movement.
Speed = distance ÷ time, using the right units.
Example. A cyclist rides 12 km in the first 40 minutes, stops for 20 minutes, then rides 8 km more in 30 minutes. (a) Describe the graph between 40 and 60 minutes. (b) Work out the speed, in km/h, for the first 40 minutes.
Show the model answer
(a) A horizontal line: the cyclist is stationary B1 (b) \(12 \div \frac{40}{60}\) M1 = 18 km/h A1
Use a graph to solve an equation
2 marks5
Rearrange so one side is the equation of the graph you have.
Draw the line for the other side.
Read off the x-coordinates of the crossing points.
Example. Use the graph of \(y = x^2 - 2x - 3\) to estimate the solutions of \(x^2 - 2x - 3 = 2\).
Show the model answer
Draw \(y = 2\) and read off where it meets the curve M1 \(x \approx -1.4\) and \(x \approx 3.4\) A1 (a small range is accepted)
Match equations to graphs
1 mark each5
Identify the type of each equation (quadratic, cubic, reciprocal, linear).
Use the sign of the leading term and the y-intercept to decide.
Example. Match each equation to its description. (a) \(y = x^3 - 2\) (b) \(y = \frac{4}{x}\) (c) \(y = 3 - x^2\) P: a ∩-shaped curve crossing the y-axis at 3. Q: two separate curves, in the first and third quadrants. R: an S-shaped curve crossing the y-axis at −2.
Show the model answer
(a) R B1 (b) Q B1 (c) P B1
Shortcuts and memory tricks
Quadratic tables are symmetrical, e.g. 5, 0, −3, −4, −3, 0, 5. A value that breaks the pattern is probably wrong.
If your calculator has a table mode, use it to fill in tables of values, and check one value by hand.
Higher: before finding angles, sketch sin or cos on axes marked 0°, 90°, 180°, 270° and 360°.