Function transformations in A Level Math Paper 1: common mistakes and examiner advice

Transformation questions look simple, and many students understand the idea. The marks are lost in two places: the order of the transformations, and the language used to describe them. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Vague wording. Descriptions such as "move", "squeeze", "compress", "up", "down", "along", "across", "forwards" or "backwards" are not enough on their own. Reports say that descriptions using these words without reference to the x- or y-axis are usually insufficient.
• Translations without a direction. Give a translation as a column vector, or say clearly "left 2 units horizontally" or "down 5 units vertically". Students often swapped the x- and y-directions.
• Stretches without a scale factor and direction. A stretch needs the scale factor and the axis it is parallel to. A common mistake is to write the reciprocal of the scale factor, or to call a horizontal stretch vertical.
• Wrong order. Many students apply the three operations in the wrong order. A translation that comes after a stretch has a vector that depends on the stretch. For example, a translation carried out after a horizontal stretch of scale factor 2 would be a different vector from the same translation before it.
• Reflection in the wrong axis. A reflection in the x-axis is confused with a reflection in the y-axis, and the reflection is often the hardest of the three transformations to get right.
• Errors when writing the new equation. Typical examples are placing the multiplying factor inside the function instead of outside, placing the minus sign outside the function instead of inside, multiplying only some of the terms by the factor, or squaring 2x as 2x² rather than 4x².
• Not including the function name or "y =". When the equation of a graph is wanted, the answer needs to start with y = (or the name of the function), not only the expression.
• Graphs that do not match earlier answers. Students who did not see the link between two parts of a question drew a graph that contradicted their earlier work.

What examiners advise

• Work in two stages. Reports note that candidates who dealt with the stretch first, wrote down the result, and only then applied the translation were usually more successful than those who tried to do everything at once.
• Use a vector for translations, or state the direction clearly in terms of the axes.
• Describe each transformation separately and fully. Give the type of transformation, its size, and its direction or axis.
• Show the result as an equation when the question asks for one, starting with y = or the function's name.

Habits that help

• Take one point on the original graph, move it through each step, and see if it lands where your equation says it should.
• Know the standard graphs that appear (such as sine, cosine, tangent, exponentials and reciprocals) so that you can sketch the result.

Before you move on

1. Does each description include the type, the size and the direction or axis?
2. Did I apply the transformations in the order the question states?
3. If the equation is wanted, have I written y = or the name of the function?
4. Did I check one point?

Questions students ask

How do I describe a transformation so that I get full marks?

Give the type, the size and the direction or axis. Words like "move", "squeeze", "compress", "up" or "along" on their own are usually not enough, according to examiner reports. A translation should be a column vector or a clear statement such as "left 2 units horizontally".

What do I need to write for a stretch?

A stretch needs the scale factor and the axis it is parallel to. Common mistakes are writing the reciprocal of the scale factor and calling a horizontal stretch vertical.

Does the order of transformations matter?

Yes. Many students apply them in the wrong order. A translation carried out after a stretch has a vector that depends on the stretch, so the same translation before the stretch would be a different vector.

Why is reflection the transformation students get wrong most often?

Reflection in the x-axis is confused with reflection in the y-axis, and the minus sign is placed outside the function when it belongs inside, or the other way round. Check by taking one point on the original graph and seeing where your equation sends it.

When a question asks for the equation of the new graph, how should I start my answer?

With y = (or the name of the function). Examiners report that answers giving only the expression lose the mark. Take care with factors too: 2x squared is 4x², not 2x².