Understand the effect of simple transformations on the graph of y = f(x), including sketching associated graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax) and combinations of these transformations

Overview

Graph transformations are a high-return sketching topic because a candidate can earn several marks without expanding an expression. Pearson Edexcel specification point 2.9 requires you to start with a known graph, such as a quadratic, cubic, reciprocal, trigonometric or exponential curve, and create a correct sketch after translations, stretches, reflections and combinations. The mathematical idea is simple: every feature has coordinates, so a transformation tells you exactly how those coordinates move.

In a Pure Mathematics paper, these skills are often embedded inside a larger question. You may be given a graph of (y=f(x)), asked to sketch (y=2f(3x)), identify new asymptotes, or compare (|f(x)|) with (|f(-x)|). Marks are awarded for accurate movement of turning points, roots, intercepts and asymptotes, then for a plausible curve of the right shape. A labelled rough sketch is enough: it does not need a perfect scale, but it must be mathematically unambiguous.

Specification focus: Pearson Edexcel A-Level Mathematics (9MA0), Section 2.9: (y=af(x)), (y=f(x)+a), (y=f(x+a)), (y=f(ax)), their combinations, (|f(x)|) and (|f(-x)|).

Key Concepts

1. Outside the function changes (y)-coordinates

In (y=af(x)), the multiplier is outside (f). For every point ((x,y)) on (y=f(x)), the new point is ((x,ay)). The (x)-coordinate stays fixed. If (|a|>1), the graph is stretched parallel to the (y)-axis. If (0<|a|<1), it is compressed, often called a vertical squash. If (a<0), there is also a reflection in the (x)-axis because every (y)-value changes sign.

For example, if ((3,-2)) lies on (y=f(x)), then it becomes ((3,-6)) on (y=3f(x)), and ((3,1)) on (y=-\tfrac12 f(x)). Any root stays a root under a non-zero vertical stretch because (a\times0=0). A horizontal asymptote (y=c), however, becomes (y=ac). This distinction earns marks in reciprocal and exponential sketches.

Memory hook: outside means outward, up/down. Look outside the bracket and alter (y).

2. Adding outside the function translates vertically

For (y=f(x)+a), every point moves from ((x,y)) to ((x,y+a)). For positive (a), translate the graph up; for negative (a), translate it down. The shape, horizontal positions of points and vertical distances are unchanged.

A common examiner-tested example is (y=3+\sin 2x). The (+3) moves the sine wave’s midline from (y=0) to (y=3). It does not change amplitude or period. A reciprocal graph (y=\frac{1}{x}) becomes (y=\frac{1}{x}+4), so its horizontal asymptote moves from (y=0) to (y=4). When a candidate moves a vertical asymptote for an outside addition, credit is not given: it has no reason to move.

3. Inside the function changes (x)-coordinates, so direction reverses

For (y=f(x+a)), the new point is ((x-a,y)). Positive (a) therefore moves the graph left by (a), not right. The reversal happens because the input must become (a) larger to make the function produce the same old output. If the original feature occurred at input (x=p), solve (x+a=p), giving (x=p-a).

For example, the vertex of (y=x^2) is ((0,0)). The vertex of (y=(x+4)^2=f(x+4)) is ((-4,0)), which is four left. Do not rely on a vague phrase such as “plus means right”. Instead, either use the coordinate rule or ask which new (x) gives the old input. This is the most dependable way to avoid the most common error in this topic.

4. Multiplying the input changes width

For (y=f(ax)), divide every old (x)-coordinate by (a): ((x,y)\mapsto(\frac{x}{a},y)). If (|a|>1), the graph is compressed horizontally by scale factor (\frac1{|a|}), usually described in Edexcel answers as a stretch parallel to the (x)-axis with scale factor (\frac1{|a|}). If (0<|a|<1), the graph stretches horizontally. If (a<0), include a reflection in the (y)-axis.

Take (y=f(4x)). A root at (x=8) moves to (x=2), not (x=32). The graph looks narrower because features are pulled towards the (y)-axis. For (y=f(-2x)), first the factor (-2) sends (x) to (-x/2): there is a horizontal compression by factor (\frac12) and a reflection in the (y)-axis.

5. Combinations: track a few features and work in a declared order

The safest method is to separate the expression into stages. In (y=2f(3(x+1))-1), read the inside first: (f(3x)) compresses horizontally by factor (\frac13), then (f(3(x+1))) translates one left. The outside then stretches vertically by factor 2, followed by a translation one down. A point ((x,y)) on the original curve maps to

[
(x,y)\mapsto\left(\frac{x}{3}-1,\ 2y-1\right).
]

For the useful general form (y=A f(B(x+h))+k), use ((x,y)\mapsto(\frac{x}{B}$-h, Ay+k)), provided (A) and (B) are non-zero. This rule is not a formula-sheet result: memorise it by understanding that inputs are solved for, outputs are substituted directly. In an exam, you do not have to quote the mapping if a clearly correct staged sketch is shown.$

When the original curve has asymptotes, transform them as features. If (f) has vertical asymptote (x=p), then (f(B(x+h))) has vertical asymptote (x=\frac{p}{B}-h). If it has horizontal asymptote (y=q), then (Af(B(x+h))+k) has horizontal asymptote (y=Aq+k). Sketch the asymptotes as dotted lines before the curve.

6. Modulus transformations: decide which axis controls the operation

For (y=|f(x)|), retain every part of (f(x)) that is already on or above the (x)-axis. Reflect only the parts below the (x)-axis in the (x)-axis. Roots remain where they are, but a negative minimum becomes a positive maximum with the same magnitude. The finished graph cannot go below (y=0).

For (y=|f(-x)|), first reflect the original graph in the (y)-axis to form (f(-x)). Then apply the modulus, lifting any negative part in the (x)-axis. You can also take (|f(x)|) first and then reflect it in the (y)-axis: the final graph is the same. State a clear route and make the final sketch satisfy both operations.

7. A mark-winning sketch routine

1. Name the parent graph and list its features. Record roots, turning points, intercepts, asymptotes, period or endpoints.
2. Write the coordinate movement. Use the point rule or show the transformations in stages. This secures method marks even if a final curve has a drawing slip.
3. Transform at least three decisive features. For a quadratic, a vertex and two roots are ideal. For a reciprocal, use both asymptotes and one point on each branch.
4. Draw the shape last. Preserve the original curve’s continuity, branch position and orientation, unless a reflection or modulus changes it.
5. Label transformed features. In a 3- or 4-mark sketch question, labels are the evidence that the construction is intentional rather than guessed.

Mathematical Relationships

All of the following must be memorised. They are not supplied as a dedicated transformation formula on the Edexcel formula booklet.

• Vertical stretch/reflection:(y=af(x)), ((x,y)\mapsto(x,ay)). A negative (a) reflects in the (x)-axis.
• Vertical translation:(y=f(x)+a), ((x,y)\mapsto(x,y+a)).
• Horizontal translation:(y=f(x+a)), ((x,y)\mapsto(x-a,y)). The sign appears reversed in the movement.
• Horizontal stretch/reflection: (y=f(ax)), ((x,y)\mapsto(\frac{x}{a},y)). A negative (a) reflects in the (y)-axis.
• Combined transformation: (y=A f(B(x+h))+k), ((x,y)\mapsto(\frac{x}{B}-h, Ay+k)).
• Modulus: (y=|f(x)|) replaces each negative output (y) by (-y); (y=|f(-x)|) also reverses the (x)-coordinate first.

For trigonometric graphs, remember that (\sin(Bx)) and (\cos(Bx)) have period (\frac{2\pi}{|B|}), while (\tan(Bx)) has period (\frac{\pi}{|B|}). An outside coefficient affects amplitude for sine and cosine, while an inside coefficient affects period. This is a frequent synoptic link between transformations and trigonometric modelling.

Practical Applications and Useful Context

Transformations are the grammar of mathematical models. A tide model may start as a sine curve; changing its mean height uses an outside addition, changing its range uses an outside multiplier, and changing the time scale uses an inside multiplier. In a growth model (y=Ae^{kx}+c), (A) changes vertical scale or reflection, while (c) changes the long-run horizontal asymptote. The same visual reasoning helps you check whether a model makes sense before using algebra or a calculator.

Graph and Data Skills: What a Successful Sketch Shows

Use the scale intelligently rather than mechanically. Start by placing the axes, then choose a few exact features that establish the result. For a quadratic, these are normally the vertex and roots. For a cubic, use roots, turning points and end behaviour. For a sine or cosine curve, mark the midline, maximum, minimum, period and at least one complete cycle. For a reciprocal function, draw the vertical and horizontal asymptotes first, identify which quadrants or regions contain the branches, then add a check point. If the question says sketch, numerical equality of every line spacing is not required, but an intercept on the wrong side of an asymptote is a mathematical error, not a harmless drawing issue.

Coordinate labels should carry the marks. Write ((-1,3)), not merely a dot near the expected place. If a feature is an asymptote, label the whole line as (x=-1) or (y=3), not just one point on it. If an operation leaves a feature unchanged, use that fact efficiently: roots remain at the same (x)-values after (y=af(x)) for non-zero (a), and a vertical translation leaves every (x)-coordinate unchanged. This prevents unnecessary calculation and helps you check your answer.

When a graph is not supplied, sketch the parent function from its known properties before transforming. For (y=e^x), use the point ((0,1)) and horizontal asymptote (y=0). For (y=\frac1x), use its two branches and asymptotes (x=0), (y=0). For (y=\tan x), include vertical asymptotes at (x=\pm\frac\pi2) and the central crossing at the origin. The question is still testing transformations, so recognisable parent-graph features are your starting evidence.

For examination purposes, do not spend time making a display-quality plot. Pearson Edexcel accepts a sketch with clear axes, asymptotes where needed and correctly positioned labelled features. Use a ruler for asymptotes only when it improves clarity. The calculator may help you check a point, but candidates are rewarded for transformation reasoning, not for plotting many decimal coordinates.

Audio Revision Podcast

Use this ten-minute recap after attempting the practice questions. Pause after the quick-fire questions, answer aloud, then replay the relevant section if a direction, axis or asymptote rule was not secure.