Transformations change a graph in predictable ways. A change outside the function acts vertically: f(x) + a moves the graph up by a, and af(x) stretches it vertically. A change inside acts horizontally and in the opposite sense: f(x + a) moves the graph left by a, and f(ax) stretches it horizontally by a factor of 1/a. A minus sign outside reflects in the x-axis; inside, in the y-axis.
Transformations of y = f(x)
- y = f(x) + a: translation a units up
- y = f(x + a): translation a units left
- y = af(x): vertical stretch, scale factor a
- y = f(ax): horizontal stretch, scale factor 1/a
- y = −f(x): reflection in the x-axis
- y = f(−x): reflection in the y-axis
Translations
Adding a number outside the function moves the graph vertically: y = f(x) + a moves it up by a. Adding inside the bracket moves it horizontally, in the opposite direction to the sign: y = f(x + 2) moves the graph 2 units left, and y = f(x − 2) moves it 2 units right.
For a reciprocal graph, a translation moves the asymptotes too: y = 1/(x − 2) has a vertical asymptote at x = 2 instead of x = 0.
Stretches
Multiplying outside the function stretches the graph vertically: y = af(x) multiplies every y-coordinate by a. Multiplying inside stretches it horizontally by the reciprocal factor: y = f(ax) multiplies every x-coordinate by 1/a.
A vertical stretch leaves the roots where they are, because 0 × a = 0. A horizontal stretch moves the roots, but leaves the y-intercept unchanged.
Reflections
A minus sign outside the function, y = −f(x), reflects the graph in the x-axis, so every y-value changes sign and a maximum becomes a minimum. A minus sign inside, y = f(−x), reflects it in the y-axis, so every x-value changes sign.
Combined transformations and tracking points
To find where a point ends up, apply each transformation to its coordinates. Under y = f(x − a) + b, a point (p, q) moves to (p + a, q + b). If an inside expression has a factor, take it out first: f(2x − 4) = f(2(x − 2)) is a horizontal stretch and a translation, which are easier to handle separately.
Finding a cubic, then transforming it
A cubic crossing the x-axis at p, q and r has equation y = a(x − p)(x − q)(x − r). One more point, often the y-intercept, fixes a. Once the equation is known, transform it as required. If the cubic touches the axis at a repeated root, that factor appears squared.
Worked examples
Example 1. Describe the transformation from y = x³ to y = (x − 2)³.
- x is replaced by x − 2, a change inside the function.
- f(x − 2) moves the graph 2 to the right.
Answer: A translation 2 units to the right.
Marks: B1 for horizontal, B1 for right 2.
Example 2. Describe the transformation from y = x³ to y = 2x³.
- The 2 multiplies the whole function, a change outside.
- So the graph is stretched vertically by a factor of 2, and the root at the origin stays fixed.
Answer: A vertical stretch, scale factor 2.
Marks: B1 for vertical, B1 for factor 2.
Example 3. Describe the transformation from y = x³ − x to y = −(x³ − x).
- The minus sign is outside the function.
- Every y-value changes sign; the roots stay the same.
Answer: A reflection in the x-axis.
Marks: B1 for the reflection.
Example 4. The point (2, 3) lies on y = f(x). Find its image on y = f(x − 1) + 4.
- f(x − 1) moves the graph right 1, so x goes from 2 to 3.
- +4 moves it up 4, so y goes from 3 to 7.
Answer: (3, 7).
Marks: M1 for right 1 and up 4, A1 for (3, 7).
Example 5. A cubic crosses the x-axis at −1, 2 and 3, and passes through (0, 6). Find its equation.
- y = a(x + 1)(x − 2)(x − 3).
- At (0, 6): a(1)(−2)(−3) = 6, so 6a = 6 and a = 1.
Answer: y = (x + 1)(x − 2)(x − 3).
Marks: M1 for the form, M1 for 6a = 6, A1 for a = 1.
Common mistakes
- Moving the wrong way for f(x + a). f(x + 2) moves the graph left, not right.
- Using a instead of 1/a for f(ax). f(2x) halves every x-coordinate; it does not double it.
- Forgetting to move the asymptotes. A translation of a reciprocal graph moves its asymptotes by the same amount.
- Mixing up the two reflections. −f(x) reflects in the x-axis; f(−x) reflects in the y-axis.
- Not taking a factor out first. Write f(2x − 4) as f(2(x − 2)) before describing the transformations.
How graph transformations are examined
Graph transformations are Pure 1 content (WMA11) in Edexcel International A-Level, in the graphs and transformations chapter, and first-year Pure in Edexcel A-Level (9MA0). Questions commonly:
- describe a single transformation in words
- sketch a transformed cubic or reciprocal graph, labelling roots and asymptotes
- find the image of a given point, such as a turning point
- find a cubic from its roots, then transform it
Questions with sketches and mark schemes are among the Pure 1 graphs and transformations exam questions. Use the graphs and transformations worksheet and the algebra revision engine for more.
The same rules apply to trig graphs and to the modulus function. For the GCSE version, see transformations.
Frequently asked questions
Why does f(x + a) move the graph left?
Because the x-value must be a less than before to give the same output. Every point arrives a units earlier, so the graph shifts left.
What is the difference between af(x) and f(ax)?
af(x) stretches the graph vertically by a. f(ax) stretches it horizontally by 1/a.
How do transformations affect asymptotes?
Translations move asymptotes by the same amount. Stretches and reflections move them in the same way as every other point.