Transformation of Curves
In Maths, understanding how graphs of functions can be transformed is essential. Transformations of curves include translations (shifting graphs) and reflections (flipping graphs). These transformations affect the equations of functions in predictable ways.
This guide covers:
- Translating graphs (moving graphs up/down or left/right)
- Reflecting graphs (flipping graphs across the axes)
Translating Graphs
A translation moves a graph without changing its shape. There are two types of translations:
- Vertical Translation (Moves the graph up or down)
- Horizontal Translation (Moves the graph left or right)
Vertical Translation: $y = f(x) + k$
- Adding $k$ moves the graph up by $k$ units.
- Subtracting $k$ moves the graph down by $k$ units.
Example:
If $y=f(x)$ is the original graph, then:
- $y=f(x)+3$ moves the graph up by 3.
- $y=f(x)−5$ moves the graph down by 5.
Key Point: The $x$-values stay the same; only the $y$-values change.
Horizontal Translation: $y=f(x−a)$
- $y=f(x−a)$ moves the graph right by $a$ units.
- $y=f(x+a)$ moves the graph left by $a$ units.
Example:
If $y=f(x)$ is the original graph, then:
- $y=f(x−4)$ moves the graph right by 4.
- $y=f(x+2)$ moves the graph left by 2.
Key Point: This transformation affects the $x$-values, not the $y$-values.
Reflections of Graphs
A reflection flips a graph over a specific axis. There are two types of reflections:
- Reflection in the $x$-axis
- Reflection in the $y$-axis
Reflection in the $x$-axis: $y=−f(x)$
- Every $y$-value becomes its opposite.
- The graph is flipped vertically across the $x$-axis.
Example:
If the original graph is $y=f(x)$, then:
- $y=−f(x)$ reflects it in the $x$-axis.
For example, if $y = x^2$, then:
$y = -x^2$
flips the parabola downwards.
Reflection in the $y$-axis: $y=f(−x)$
- Every $x$-value becomes its opposite.
- The graph is flipped horizontally across the y-axis.
Example:
If the original graph is $y=f(x)$, then:
- $y=f(−x)$ reflects it in the y-axis.
For example, if $y = x^3$, then:
$y = (-x)^3 = -x^3$
mirrors the cubic graph from left to right.
Summary Table of Transformations
| Transformation | Equation | Effect |
|---|---|---|
| Move up by $k$ | $y=f(x)+k$ | Add $k$ to all $y$-values |
| Move down by $k$ | $y=f(x)−k$ | Subtract $k$ from all $y$-values |
| Move right by $a$ | $y=f(x−a)$ | Replace $x$ with $x−a$ |
| Move left by $a$ | $y=f(x+a)$ | Replace $x$ with $x+a$ |
| Reflect in $x$-axis | $y=−f(x)$ | Flip vertically |
| Reflect in $y$-axis | $y=f(−x)$ | Flip horizontally |
This video takes a look at Transformations including: Reflections, Rotations and Enlargements.
Exam Tips
✔ Always check the sign in transformations—negative values indicate reflections.
✔ For horizontal translations, remember that ($x−a$) means moving right, and ($x+a$) means moving left.
✔ Draw key points before and after transformations to check accuracy.
✔ Know the difference between vertical and horizontal changes—they affect different parts of the equation.
This guide covers the essential transformations of curves for GCSE Maths. Keep practising and soon, these will become second nature.
