Cambridge IGCSE International Maths · Extended
Transformations of Graphs
The Big Idea: When you change the equation of a graph in a specific way, the whole graph
slides — up, down, left, or right — without changing its shape, size, or orientation at all.
Summary — What This Chapter Covers
- A translation is a graph transformation that shifts a graph without changing its shape, size or orientation.
- Translations are described using a column vector like
(0, a) or (−a, 0).
- Vertical translation:
y = f(x) + a shifts the graph up/down — only y-coordinates change.
- Horizontal translation:
y = f(x + a) shifts the graph left/right — only x-coordinates change, and the direction is the opposite of what you'd expect.
- Asymptotes move with the graph — unless they're parallel to the direction of the shift.
- Combined translations use vector
(−p, q) and the equation y = f(x − p) + q.
- For
y = a(x − p)² + q, the vertex is always at (p, q) — no matter what a is.
1. What Is a Translation?
Imagine your graph is drawn on a clear sheet of glass. A translation is what happens when you
pick that whole sheet up and slide it — left, right, up, or down — without rotating it, flipping it, or
stretching it in any way. Every single point on the curve moves by the exact same amount in the
exact same direction.
That's the key distinguishing feature of a translation compared to other graph transformations (like
stretches or reflections): the shape, size, and orientation of the graph stay completely unchanged.
Only its position on the coordinate grid changes.
↑ (shift up)
┌────┐ ┌────┐ ┌────┐
│ orig│ ←── │orig│ ──→ │orig │
└────┘ └────┘ └────┘
↓ (shift down)
Same wiggly shape, just relocated on the page.
We describe exactly how far and in which direction a graph moves using a translation vector
— the same column vector notation you use in vector geometry:
Translation Vector
(x, y) → (x + h, y + k) is written as the vector (h, k)
The top number tells you the horizontal shift; the bottom number tells you the vertical shift.
Quick Check
If a graph translates by the vector (3, −2), every point moves 3 units right and 2 units down. That's it — nothing about the curve's shape changes.
2. Vertical Translations: y = f(x) + a
Let y = f(x) be the equation of your original graph. If you add a constant a to
the whole equation — outside the function, applied to the final y-value — the graph shifts
vertically.
Think of it this way: for every x-value, the new graph's height is simply the old graph's height
plus a. Since every single output gets nudged by the same amount, the entire graph
rises or falls uniformly — like an elevator moving the whole curve up or down a floor.
Rule: Vertical Translation
y = f(x) + a is a vertical translation by the vector (0, a)
• Positive a → graph moves up
• Negative a → graph moves down
• x-coordinates stay the same — only y-coordinates change
Worked Example. Take y = x² − 3x + 7, which has a minimum point at (2, −3)
and passes through (0, 7) and (4, 3). Translate it 1 unit up:
- New equation:
y = f(x) + 1 = x² − 3x + 8
- The minimum point (2, −3) becomes (2, −2) — the y-coordinate increased by 1, x stayed the same.
- Every other point shifts up by exactly 1 unit too.
Why does "+a" mean UP?
Because you're literally adding a to every y-value the function produces. Bigger y-value = higher up on the graph. It's the most intuitive of the two translation types — the sign matches the direction you'd guess.
Practice Question 1
The graph of y = f(x) passes through the point (5, −4). The graph is translated by the vector (0, 7). What is the new coordinate of this point, and what is the new equation in terms of f?
Practice Question 2
The graph y = 4x² + 2x + 1 is translated 5 units down. Find the new equation, fully simplified.