All functions can be transformed, which means they are altered in a certain way. These alterations can be described through geometrical transformations.
What are the types of graphical transformations?
There are three main transformations of graphs that you need to be aware of at A-Level: Translations, Stretches and Reflections.
Translations
Translations are a type of graphical transformation where the function is moved. To explain a translation, you use a vector in the form \(\left( \begin{array} {c} x \\ y \end{array} \right) \), where the top part of the vector shows how the function has been translated horizontally and the bottom part of the vector shows the function has moved vertically.
The direction of the translation depends on whether each variable is positive or negative.
| variable | Positive | negative | 0 |
| x | moves right | moves left | does not move |
| y | moves up | moves down | does not move |
If you find this difficult to remember, just think about how coordinates work, it's a similar principle. For instance, a negative x coordinate would be on the left side of the graph. Equally, a positive y coordinate would be the top part of the graph.
A function can be expressed as \(f(x) = x^2\). It is, then, translated by \(\left( \begin{array} {c} 5 \\ -3 \end{array} \right) \). Sketch the new translated function.
The original function can be seen graphically:
Original graph - StudySmarter Originals
The vector tells you the function will be translated to the right by 5 and down by 3.
If you sketch this, the new function should look like this:
Transformed graph - StudySmarter Originals
As it is a sketch, it is important to label key points on the graph like turning points
Expressing translated functions
You might be asked to write the new translated function using the vector. Translating the function \(f(x) = x^2\) by \(\left( \begin{array} {c} a \\ b \end{array} \right) \), the translated function can be written as\(f(x) = (x-a)^2 + b\). Notice how a becomes negative in the function but b stays the same.
The function \(g(x) = x\) is translated by \(\left( \begin{array}{c} 4 \\ 3 \end{array} \right)\). What is the new translated function?
According to the vector, the function is translated 4 to the right and 3 down.
\(g(x) = (x-4) + 3\)
\(g(x) = x-1\)
Stretch
A function can be stretched both horizontally and vertically.
- Vertical stretch: \(y = af(x)\) where a is the scale factor the function is stretched by. For instance, if function \(y =f(x)\) is stretched vertically by 5, the function becomes \(y = 5f(x)\).
- Horizontal stretch: \(y = f(\frac{1}{a} x)\) where a is the scale factor the function is stretched by. As you can see the reciprocal of a is used in the function equation. For instance, if the function \(y = f(x)\) is stretched horizontally by 2, the function becomes \(y = f(\frac{1}{2} x)\).
Function \(y = h(x)\) has a turning point at (2, 9) and (10, -6). The function is stretched so that the new function can be expressed as \(y = h(\frac{1}{4}x)\). What would be the new turning point?
As the scale factor is within the brackets, the function is being stretched horizontally. Furthermore, as the function is being stretched horizontally, the actual scale factor of the stretch of \(\frac{1}{4}\) is 4.
Therefore, we multiply each of the x-coordinate by 4. The (2, 9) becomes (8, 9) and (10, -6) becomes (40, -6). As you can see, the y-coordinate is unaffected as it is not being stretched.
Reflections
Reflections are where the whole functions are inverted in a line of reflection.
All horizontal in the x-axis and vertical reflections can be expressed as a function.
A function that has been reflected in the x-axis can be written as \(y = -i(x)\). For turning points of the function, the x-coordinate stays the same but y-coordinates becomes inverted. For instance, if a function has a turning point of (4, -2) and is reflected in the x-axis, the turning point would become (4, 2).
A function that has been reflected in the y-axis can be written as \(y = j(-x)\). Where the function has been reflected in the y-axis, the y-coordinate of the turning point becomes inverted which the x-coordinate stays the same.
| Reflection in the... | y-coordinate | x-coordinate |
| x-axis | becomes inverted | stays the same |
| y-axis | stays the same | becomes inverted |
Combining graphic transformations
At A-level, you need to be able to work with a combination of graphical transformations within a question. To do so, you need to know the order of operations for graph transformations. Here is a the list of graph transformations, in the correct order:
Stretch
Reflection
Translations
Transformations of Graphs - Key takeaways
- There are three main transformations of graphs: stretches, reflections and translations.
- Translations are a type of graphical transformation where the function is moved. To explain a translation, you use a vector in the form \(\left( \begin{array} {c} x \\ y \end{array} \right) \) where the top part of the vector shows how the function has been translated horizontally, and the bottom part of the vector shows the function has moved vertically.
- A function can be stretched both horizontally and vertically.
- Reflections are where the whole functions are inverted in a line of reflection.
Explanations 
Exams 
Magazine 
