Welcome to the World of Transformations!
In this chapter, we are going to learn how to take a "parent" graph—like a simple parabola or a straight line—and move it, stretch it, or flip it around. Think of it like using a photo editing app: you can slide a picture across the screen (translation), make it taller or wider (stretch), or flip it to see its mirror image (reflection).
Mastering these transformations is a superpower in A Level Maths because it allows you to sketch complicated-looking equations in seconds without having to plot dozens of points! Don't worry if it feels a bit confusing at first; we will break it down into simple rules that work every single time.
1. The Golden Rule: Inside vs. Outside
Before we look at specific movements, there is one "Secret Trick" that makes this whole chapter much easier to understand. Always ask yourself: Is the change happening inside the brackets or outside?
• Outside the brackets: \(y = f(x) + a\) or \(y = a f(x)\). These affect the y-coordinates. They are "honest" and do exactly what you expect. If you see "+ 2", the graph goes up by 2.
• Inside the brackets: \(y = f(x + a)\) or \(y = f(ax)\). These affect the x-coordinates. These are "liars" and do the opposite of what you expect. If you see "+ 2", the graph actually moves in the negative direction (left) by 2!
Quick Review Box:
Outside = Vertical (Up/Down) = Normal Logic
Inside = Horizontal (Left/Right) = Opposite Logic
2. Translations: Sliding the Graph
A translation moves every point on a graph the same distance in the same direction. We often describe this using a vector: \(\begin{pmatrix} x \\ y \end{pmatrix}\).
Vertical Translation (Outside)
Equation: \(y = f(x) + k\)
Effect: Moves the graph up by \(k\) units. If \(k\) is negative, it moves down.
Example: If \(f(x) = x^2\), then \(y = x^2 + 3\) is the same graph shifted 3 units up.
Horizontal Translation (Inside)
Equation: \(y = f(x + k)\)
Effect: Moves the graph left by \(k\) units. Remember the "Opposite Logic"—a plus sign moves it to the left (negative direction), and a minus sign moves it to the right (positive direction).
Example: \(y = f(x - 5)\) moves the graph 5 units to the right.
Using Vector Notation
In your OCR MEI exam, you might be asked to describe a translation using a vector. For a graph \(y = f(x - a) + b\), the translation vector is \(\begin{pmatrix} a \\ b \end{pmatrix}\).
Common Mistake to Avoid: Many students see \(f(x + 3)\) and want to move the graph right. Always tell yourself: "Inside the brackets is the Opposite World!"
Key Takeaway: Translations slide the graph without changing its shape or orientation.
3. Reflections: The Mirror Effect
Reflections flip the graph over one of the axes. There are only two you need to know.
Reflection in the x-axis (Outside)
Equation: \(y = -f(x)\)
Effect: All the positive \(y\) values become negative, and vice versa. The graph flips upside down over the \(x\)-axis.
Analogy: Think of the x-axis as the surface of a lake; the reflection appears in the water.
Reflection in the y-axis (Inside)
Equation: \(y = f(-x)\)
Effect: All the positive \(x\) values become negative. The graph flips sideways over the \(y\)-axis.
Did you know? If a graph is perfectly symmetrical across the \(y\)-axis (like \(y = x^2\)), the transformation \(y = f(-x)\) won't change its appearance at all!
Key Takeaway: A minus sign outside flips it vertically; a minus sign inside flips it horizontally.
4. Stretches: Pulling and Squashing
Stretches change the shape of the graph by pulling points away from an axis or pushing them toward it. Every stretch has a Scale Factor (SF).
Vertical Stretch (Outside)
Equation: \(y = a f(x)\)
Effect: A stretch parallel to the y-axis with scale factor \(a\).
How to do it: Multiply all your y-coordinates by \(a\). Your \(x\)-coordinates stay exactly the same.
Example: \(y = 3f(x)\) makes the graph 3 times taller.
Horizontal Stretch (Inside)
Equation: \(y = f(ax)\)
Effect: A stretch parallel to the x-axis with scale factor \(\frac{1}{a}\).
How to do it: This is the "Opposite World" again! If you see a 2, you don't multiply; you divide all your \(x\)-coordinates by 2 (multiply by \(\frac{1}{2}\)).
Example: \(y = f(2x)\) actually squashes the graph horizontally, making it half as wide.
Memory Aid: For horizontal stretches, we always use the reciprocal (flip the fraction) of the number next to \(x\).
Key Takeaway: Outside numbers change the height (SF is \(a\)); inside numbers change the width (SF is \(1/a\)).
5. Combined Transformations: Putting it all Together
Sometimes, a graph undergoes more than one change, like \(y = 2f(x + 3)\). When this happens, the order in which you apply the transformations matters!
The Step-by-Step Method:
1. Deal with Horizontal first (Inside): Look inside the brackets. If you have \(f(x + 3)\), move it 3 units left first.
2. Deal with Vertical second (Outside): Look outside the brackets. If you have \(2f(...)\), stretch it vertically by scale factor 2 after you've moved it.
Helpful Tip: If you are performing two vertical transformations (like a stretch and a shift), follow the standard order of operations (BIDMAS/PEMDAS). Multiply by the scale factor first, then add the translation.
Recognising Transformations from a Graph
If you are given a transformed graph and asked for its equation:
• Look at the turning points or intercepts. How far have they moved?
• Look at the distance between points. If the distance between two peaks has doubled, it’s a horizontal stretch with SF 2 (which means \(f(\frac{1}{2}x)\)).
• If the graph is upside down, there is a reflection (\(-f(x)\)).
Key Takeaway: When combining, work from the inside out. Be systematic!
Final Summary Checklist
• \(f(x) + a\): Translation up by \(a\).
• \(f(x + a)\): Translation left by \(a\).
• \(a f(x)\): Vertical stretch, SF \(a\).
• \(f(ax)\): Horizontal stretch, SF \(1/a\).
• \(-f(x)\): Reflection in the x-axis.
• \(f(-x)\): Reflection in the y-axis.
Don't worry if this seems tricky at first—practice sketching a simple curve like \(y = x^2\) with each rule, and you'll soon see the patterns emerge!