Pearson Edexcel · A-Level · Mathematics
Course version: 9MA0 · Version from Mar 2026
Understand and use graphs of functions; sketch curves defined by simple…
Full specification wording
Algebra and functions: 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs.
Sketching shows shape, intercepts, turning points, asymptotes and behaviour as x → ±∞.
Checked against the Pearson Edexcel 9MA0 specification (opens in a new tab) on .
- Point 7 of 11
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Sketching shows shape, intercepts, turning points, asymptotes and behaviour as x → ±∞. For a polynomial like y = x³ − 3x, find intercepts, differentiate for stationary points, and note opposite end behaviour. For y = |mx + c|, sketch the V-shape by reflecting the negative part of the line in the x-axis. For y = a/x, the axes are asymptotes (x = 0, y = 0), with branches in opposite quadrants. For y = a/x², branches lie on opposite sides of the vertical asymptote x = 0, but on the same side of the horizontal asymptote y = 0 (both above the x-axis if a > 0). Solving f(x) = g(x) graphically means reading x-coordinates of intersection points. Proportional relationships give straight-line graphs through the origin: y = kx.
What earns the marks
- Sketch shows correct general shape and labels all axis intercepts found by setting x = 0 or y = 0.
- Stationary points of a polynomial are located by solving f′(x) = 0 and their coordinates labelled.
- Modulus graph y = |mx + c| is drawn as a V with vertex where mx + c = 0, and the negative branch reflected above the x-axis.
- For y = a/x, both axes are drawn as asymptotes and the two branches occupy opposite quadrants.
- For y = a/x², the vertical asymptote x = 0 and horizontal asymptote y = 0 are shown, with branches on opposite sides of the y-axis but the same side of the x-axis.
- Intersection points of two curves are used to state the x-values solving f(x) = g(x), with the number of intersections matching the number of solutions.
- A proportional relationship is identified as y = kx and its graph drawn as a straight line through the origin with gradient k.
Common mistakes and how to put them right
Drawing y = a/x² with branches in opposite quadrants.
Put it right by testing x = 1 and x = −1, which give the same positive value when a > 0, so both branches lie on the same side of the x-axis.
Treating an asymptote as a line the curve crosses.
Put it right by checking that the function is undefined at the vertical asymptote and tends to the horizontal value without reaching it.
Reading only the y-coordinate at an intersection to solve f(x) = g(x).
Put it right by stating the x-coordinate, since the solution of the equation is the input value.
Assuming every straight-line graph represents proportion.
Put it right by checking the line passes through the origin before writing y = kx.
Exam tips
- Label asymptotes with their equations, for example x = 0 and y = 0, because unlabelled dashed lines lose clarity.
- Substitute one positive and one negative x-value to confirm the quadrant placement of reciprocal and reciprocal-square branches.
- When a question says 'hence' or 'graphically', link the number of intersection points explicitly to the number of solutions.
- Check whether the domain is restricted; a sketch over a given interval should stop at the endpoints rather than extend indefinitely.
Try the lesson
The start of the Algebra and functions: 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs. lesson, made from checked teaching notes for this specification point.
Have a go first
The curve y = 6/x has a horizontal asymptote. State the equation of this asymptote.
Listen
Sketching shows shape, intercepts, turning points, asymptotes and behaviour as x → ±∞. For a polynomial like y = x³ − 3x, find intercepts, differentiate for stationary points, and note opposite end behaviour. For y = |mx + c|, sketch the V-shape by reflecting the negative part of the line in the x-axis. For y = a/x, the axes are asymptotes (x = 0, y = 0), with branches in opposite quadrants. For y = a/x², branches lie on opposite sides of the vertical asymptote x = 0, but on the same side of the horizontal asymptote y = 0 (both above the x-axis if a > 0). Solving f(x) = g(x) graphically means reading x-coordinates of intersection points. Proportional relationships give straight-line graphs through the origin: y = kx. A strong answer earns marks for these points. Sketch shows correct general shape and labels all axis intercepts found by setting x = 0 or y = 0. Stationary points of a polynomial are located by solving f′(x) = 0 and their coordinates labelled. Modulus graph y = |mx + c| is drawn as a V with vertex where mx + c = 0, and the negative branch reflected above the x-axis. For y = a/x, both axes are drawn as asymptotes and the two branches occupy opposite quadrants. A common slip: Drawing y = a/x² with branches in opposite quadrants. Instead, testing x = 1 and x = −1, which give the same positive value when a > 0, so both branches lie on the same side of the x-axis. Exam tip: Label asymptotes with their equations, for example x = 0 and y = 0, because unlabelled dashed lines lose clarity.
See it
Key words
Say what each word means before you turn the card.
Asymptote
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Modulus graph
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Intersection points
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Proportional relationship
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See every definition
- Asymptote
- A line that the curve approaches but does not cross, such as the axes for y = a/x.
- Modulus graph
- A V-shape drawn by reflecting the negative part of the line in the x-axis, with vertex where mx + c = 0.
- Intersection points
- Points where two curves meet, whose x-coordinates give the solutions of f(x) = g(x).
- Proportional relationship
- A relationship giving a straight-line graph through the origin, written as y = kx.
Label the diagram
Some boxes have lost their names. Pick the right name for each number.
1Box 1
2Box 2
3Box 3
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Test your understanding of Algebra and functions: 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, the modulus of a linear function, y = a/x and y = a/x² (including their vertical and horizontal asymptotes); interpret algebraic solution of equations graphically; use intersection points of graphs to solve equations. Understand and use proportional relationships and their graphs.
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