Home → Curriculum Hub → Maths Lessons → Schemes of Work → A-Level Mathematics → Graphs and Transformations
This scheme of work for A-Level graphs and transformations builds on earlier work at GCSE with quadratic sketches and simple translations by extending these skills to cubics, quartics, reciprocal graphs and stretches. It develops the skill of sketching a curve from its factors and moving it with function notation, and these ideas are used throughout the AS Pure course.
Students learn to sketch a cubic, then a quartic, then a reciprocal. They find points of intersection algebraically, then translate and stretch a non-linear graph.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
Each linear factor gives an intercept. A repeated factor touches the axis and turns back. The sign of the leading term decides whether the curve rises or falls for large x. Crossing a repeated root, or ignoring the leading term, writes the wrong shape.
y equals a over x, and y equals a over x squared, have branches that approach the axes. Joining those branches across x equals 0 invents points that are not on the graph. A calculator table can check. The sketch still needs the gap left clear.
Set the two expressions equal, then solve. A line can meet a cubic more than twice. Check each solution back on a sketch. Reading an intersection from a turning point, rather than from a crossing, is the wrong point.
f(x) plus a shifts vertically. f(x plus a) shifts horizontally, opposite to the sign. af(x) stretches vertically. f(ax) stretches horizontally, and the factor looks backwards. f(2x) is a horizontal stretch by one half.
| Misconception | Teaching focus |
|---|---|
| Treating a repeated root as two crossings, so the curve cuts the axis twice at the same x. | A squared factor touches and turns. Mark that intercept as a turning point on the axis. |
| Sketching a cubic or quartic with the wrong end behaviour. | Look at the leading term. For large positive x, that term decides whether the curve is up or down. |
| Joining the two branches of a reciprocal across x equals 0. | Leave a gap at the y-axis. The function is not defined at x equals 0. |
| Moving f(x plus a) to the right, or treating f(ax) as a vertical stretch. | Adding inside the brackets moves horizontally, opposite to the sign. Multiplying inside scales x. |
| Leaving intercepts unlabelled, so the sketch cannot be checked. | Mark every axis crossing with its coordinates before the sketch is accepted. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all six lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn how to sketch cubic graphs by finding the roots of the function.
Later, learning progresses, and they find the equation when given the sketch graph.
Students learn how to sketch a quartic graph by considering its polynomials.
Later, as learning progresses, they derive a quartic equation from the roots of a sketched graph.
Students learn how to sketch and compare the properties of reciprocal functions.
Later, as learning progresses, they sketch graphs in the form y = k/x2.
Students learn how to find the points of intersection between various types of functions, including linear and quadratic equations. The lesson begins with basic exercises that involve sketching pairs of equations on the same graph and calculating their points of intersection.
As the lesson progresses, the problems increase in complexity. From finding the intersection of linear and quadratic functions to advanced problems involving parameters, where they determine the range of possible values for constants to ensure specific intersection conditions are met.
Points of Intersection between a Line and Curve Part 1
Points of Intersection between Lines and Curves Part 2
Students learn to translate non-linear functions in the form y = f(x).
Later, as learning progresses, they use function notation to find the equation of the transformed graph
Students learn to use function notation to stretch non-linear functions in the form y = f(x).
Later, as learning progresses, they use function notation to find the equation of the transformed graph
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