Welcome to Graph Transformations!
In this chapter, we are going to learn how to pick up a mathematical graph and move it, stretch it, or flip it. Think of a transformation like a filter on a photo app—the original image (the function) stays the same, but you change how it looks or where it sits on the screen.
Understanding transformations is a "superpower" in coordinate geometry. Instead of memorizing a hundred different graphs, you only need to know a few basic shapes and the rules for moving them around. Don't worry if it feels like there is a lot to remember at first; we will break it down into simple patterns that always work the same way.
The Golden Rule: Inside vs. Outside
Before we look at specific moves, there is one trick that will help you for the rest of your A-Level journey. We look at whether the change is happening outside the brackets of the function or inside them.
- Outside the brackets like \(y = f(x) + a\): These affect the y-coordinates. They do exactly what you expect them to do (up is plus, down is minus).
- Inside the brackets like \(y = f(x + a)\): These affect the x-coordinates. They are "counter-intuitive"—they often do the opposite of what you might expect!
Quick Review: If it's outside, it's vertical. If it's inside, it's horizontal.
1. Translations (Sliding the Graph)
A translation is just a fancy word for sliding the graph without changing its shape or orientation.
Vertical Translation: \(y = f(x) + a\)
This moves the graph up or down. Since the change is outside the function, we add \(a\) to every \(y\)-coordinate.
- \(y = f(x) + 3\): Move the graph up 3 units.
- \(y = f(x) - 5\): Move the graph down 5 units.
Notation: We describe this using a translation vector: \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).
Horizontal Translation: \(y = f(x + a)\)
This moves the graph left or right. Because it is inside the brackets, it feels backwards!
- \(y = f(x + 2)\): Move the graph left 2 units. (Think: we reach the x-values 2 units "sooner").
- \(y = f(x - 4)\): Move the graph right 4 units.
Notation: We describe this using a translation vector: \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).
Example: If the original graph has a peak at \((1, 5)\), the graph of \(y = f(x - 3) + 2\) would move that peak to \((4, 7)\).
Key Takeaway: For translations, use the vector \(\begin{pmatrix} x \\ y \end{pmatrix}\). Remember that the \(x\) move inside the bracket uses the opposite sign!
2. Stretches (Pulling the Graph)
A stretch pulls the graph away from an axis or squashes it toward it. Every point is moved by a scale factor.
Vertical Stretch: \(y = a f(x)\)
This pulls the graph vertically. It is outside the brackets, so it affects \(y\).
- Multiply every \(y\)-coordinate by \(a\).
- The \(x\)-coordinates stay the same.
- Points on the \(x\)-axis don't move at all!
How to describe it: "A stretch, scale factor \(a\), parallel to the y-axis."
Horizontal Stretch: \(y = f(ax)\)
This squashes or pulls the graph horizontally. It is inside the brackets, so it's "counter-intuitive" again.
- Multiply every \(x\)-coordinate by \(\frac{1}{a}\).
- If \(a = 2\), the graph actually gets twice as narrow (Scale Factor \(\frac{1}{2}\)).
- Points on the \(y\)-axis don't move.
How to describe it: "A stretch, scale factor \(\frac{1}{a}\), parallel to the x-axis."
Common Mistake: Students often forget to flip the fraction for horizontal stretches. If you see \(y = f(3x)\), the scale factor is \(\frac{1}{3}\), not \(3\)!
Key Takeaway: Vertical stretches use the number "as is." Horizontal stretches use \(1\) divided by the number.
3. Reflections (Flipping the Graph)
Reflections happen when the scale factor is negative. For your AS Level, you specifically need to know these two:
Reflection in the x-axis: \(y = -f(x)\)
The minus sign is outside. All positive \(y\)-values become negative, and negative \(y\)-values become positive. The graph flips upside down over the \(x\)-axis.
Reflection in the y-axis: \(y = f(-x)\)
The minus sign is inside. All positive \(x\)-values swap with negative \(x\)-values. The graph flips left-to-right over the \(y\)-axis.
Did you know? Some graphs look exactly the same after a reflection! For example, if you reflect \(y = x^2\) in the \(y\)-axis, it doesn't change because it is symmetrical.
Key Takeaway: \(y = -f(x)\) flips it vertically (over the \(x\)-axis). \(y = f(-x)\) flips it horizontally (over the \(y\)-axis).
Step-by-Step: How to Form an Equation
If the exam gives you a graph and says it has been transformed, follow these steps to find the new equation:
- Identify the type: Did it slide (Translation), pull (Stretch), or flip (Reflection)?
- Check the direction: Is it moving up/down (Vertical/Outside) or left/right (Horizontal/Inside)?
- Find the value: How many units did it move? Or what is the scale factor?
- Write the function:
- Moved right by 5? Replace \(x\) with \((x - 5)\).
- Stretched vertically by 2? Put a \(2\) in front: \(2f(x)\).
- Moved up by 1? Add \(+ 1\) at the very end.
Quick Summary Table
Use this as a checklist when practicing questions:
- \(f(x) + a\): Translation by \(\begin{pmatrix} 0 \\ a \end{pmatrix}\) (Vertical shift)
- \(f(x + a)\): Translation by \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\) (Horizontal shift - watch the sign!)
- \(a f(x)\): Stretch, scale factor \(a\), parallel to \(y\)-axis
- \(f(ax)\): Stretch, scale factor \(\frac{1}{a}\), parallel to \(x\)-axis
- \(-f(x)\): Reflection in the \(x\)-axis
- \(f(-x)\): Reflection in the \(y\)-axis
Don't worry if this seems tricky at first! The best way to master transformations is to grab a piece of graph paper (or a graphing calculator) and try sketching \(y = x^2\), then \(y = (x-2)^2\), then \(y = 3(x-2)^2\). Seeing it move with your own eyes makes the rules much easier to remember!