IB Maths AA SLTopic 2 — FunctionsPaper 1 & 2~9 min read
Translations of Graphs
A translation slides a graph across the plane without changing its shape, size, or orientation. In this note you’ll learn how to read a translation vector, write the new equation, and sketch the result confidently.
📘 What you need to know
A translation just slides a graph — its shape and size never change.
The translation is described by a vector with two parts: how far across, and how far up or down.
For a horizontal shift, the change happens inside the bracket: y = f(x − a) moves the graph a units to the right.
For a vertical shift, the change happens outside the bracket: y = f(x) + b moves the graph b units up.
Imagine you’ve drawn a curve on a piece of paper. Now you slide that paper — keeping it flat, not rotating it, not stretching it — and tape it down somewhere else on your desk. The curve hasn’t changed at all; only its position has.
That is exactly what a translation does to a graph. Every single point moves the same distance, in the same direction. The shape, size, and orientation are all preserved. The only thing that changes is where the graph sits on the axes.
To describe the slide precisely, we use a translation vector. It has two numbers stacked on top of each other:
abTranslation vector
→
a — how far across (left or right)
b — how far up or down
Reading the signs
The sign of each number tells you which direction the graph moves:
→
Positive a
Move right
←
Negative a
Move left
↑
Positive b
Move up
↓
Negative b
Move down
A translation only changes where the graph sits — it never tilts, flips, or resizes it. If the curve gets taller, narrower, or upside-down, that’s a stretch or a reflection, not a translation.
A curve translated in 4 directions
Horizontal translations (left and right)
A horizontal translation slides the graph sideways. The change always happens inside the function brackets — and here’s where most students go wrong, so read carefully.
Horizontal translation
y = f(x − a)⟶translation bya0
🤔 Why does “minus a” mean “move RIGHT by a”?
This trips up almost every student the first time. The trick I always teach: the new graph copies whatever the original did at x = 0, but now it does it at x = a instead.
Take f(x − 3). For the inside to equal 0, we need x = 3. So the new graph is doing at x = 3 what the old graph did at x = 0 — meaning the curve has shifted 3 units to the right. Once you see this, you’ll never get the direction wrong again.
Move RIGHT by a
y = f(x − a)
Vector a0 (a > 0)
Move LEFT by a
y = f(x + a)
Vector −a0
What happens to coordinates and asymptotes?
For a horizontal translation by a units to the right:
Every point (x, y) becomes (x + a, y) — only the x-coordinate changes.
A vertical asymptote x = k moves with the graph: it becomes x = k + a.
Horizontal asymptotes don’t move at all — they’re parallel to the direction of the slide.
Vertical translations (up and down)
Vertical translations are friendlier — the rule actually matches your intuition. Adding a positive number outside the function lifts the whole graph up; subtracting drops it down.
Vertical translation
y = f(x) + b⟶translation by0b
Move UP by b
y = f(x) + b
Vector 0b (b > 0)
Move DOWN by b
y = f(x) − b
Vector 0−b
For a vertical translation by b units up:
Every point (x, y) becomes (x, y + b) — only the y-coordinate changes.
A horizontal asymptote y = k rides up with the graph: it becomes y = k + b.
Vertical asymptotes stay exactly where they were.
Notice the pattern: the asymptote that’s parallel to the direction of the slide is the one that doesn’t move. A horizontal slide pushes vertical lines along with it; a vertical slide pushes horizontal lines along with it.
↔ Horizontal translation
x-coordinatechanges
y-coordinatestays the same
Vertical asymptotemoves
Horizontal asymptoteunchanged
↕ Vertical translation
x-coordinatestays the same
y-coordinatechanges
Vertical asymptoteunchanged
Horizontal asymptotemoves
Equation decoder — read the shift in seconds
Once you’ve practised a few, reading translations becomes automatic. Here’s a quick reference to lock it in:
Equation
→
Translation vector
y = f(x − 5)
→
50 5 units right
y = f(x + 2)
→
−20 2 units left
y = f(x) + 4
→
04 4 units up
y = f(x) − 7
→
0−7 7 units down
y = f(x − 3) + 1
→
31 3 right & 1 up
In the exam, write the answer using proper vector notation, e.g. “translate by 2−4“. Sloppy phrasing like “moved 2 across and 4 down” can lose easy marks.
Worked examples
WE 1
Find the equation after a horizontal translation
The graph of y = f(x) is translated by the vector 40. Write the equation of the new graph.
Step 1: Read the vector
Top number = 4 → 4 units right.
Bottom = 0 → no vertical shift.
Step 2: Apply the rule
Right by a = 4 ⇒ replace x with (x − 4).
Step 3: Write ity = f(x − 4)minus inside the bracket = move right!
WE 2
Find the equation after a vertical translation
The graph of y = x2 is translated by the vector 0−5. Write the equation of the new graph.
Step 1: Read the vector
Top = 0 → no horizontal shift.
Bottom = −5 → 5 units down.
Step 2: Apply the rule
Down means subtract on the outside.
Step 3: Write ity = x2 − 5
WE 3
Translate named points on a curve
The graph of y = f(x) has a maximum at A(−1, 5) and a minimum at B(3, −3). Find the new coordinates of A and B on:
(a) y = f(x + 3) (b) y = f(x) + 3
(a) y = f(x + 3) → vector −30x-coords drop by 3, y-coords stayA′(−4, 5), B′(0, −3)(b) y = f(x) + 3 → vector 03x-coords stay, y-coords go up by 3A′(−1, 8), B′(3, 0)always shift the named points first, then sketch through them
WE 4
Translate a graph with asymptotes
The graph of y = 1x has asymptotes x = 0 and y = 0. Find the equations of the asymptotes after a translation by the vector 2−1.
Step 1: Move the vertical asymptote
Vertical line, horizontal slide → it moves with the graph.
x = 0 → x = 0 + 2 = 2Step 2: Move the horizontal asymptote
Horizontal line, vertical slide → it moves with the graph.
y = 0 → y = 0 + (−1) = −1x = 2 & y = −1
WE 5
Describe the translation
Describe the single transformation that maps y = f(x) onto y = f(x + 6) − 2.
Step 1: Look inside the bracket
+6 inside → horizontal shift → 6 units LEFT.
Step 2: Look outside the function
−2 outside → vertical shift → 2 units DOWN.
Step 3: Combine into one vectorTranslate by −6−2always answer in column-vector form for full marks
💡 Top tips
Inside vs outside is everything. Inside the bracket = horizontal shift, opposite sign. Outside the function = vertical shift, same sign. This one rule unlocks the whole topic.
Move named points first. If a question gives you a max, min, or intercept, translate those points before drawing anything. The new curve passes through the new points.
Always write the answer as a column vector — IB markschemes reward proper notation over vague phrases.
Translations do not change the shape, gradient, or stationary point types. A maximum stays a maximum; a minimum stays a minimum.
For asymptotes, only the parallel asymptote moves. A sideways shift moves the vertical asymptote (and vice-versa).
If the equation has changes both inside and outside the bracket, just treat them as two separate shifts and combine the vectors at the end.
Sketch lightly in pencil first — translations look obvious once the new key points are plotted.
⚠ Common mistakes
Wrong direction inside the bracket. Students often write y = f(x + 3) for a translation 3 right — but it’s actually 3 left. The sign flips for horizontal shifts.
Confusing horizontal and vertical asymptotes. A horizontal slide moves vertical asymptotes; a vertical slide moves horizontal ones. Easy to mix up under exam pressure.
Translating both coordinates for a single-axis shift. A vector 30 only changes x-coords; the y-coords don’t move at all.
Writing the vector as (a, b). IB markschemes expect the column-vector form with one number above the other.
Saying “shift up” without numbers. Vague descriptions lose marks — always include the exact size of the shift.
Forgetting to redraw the original axes. The translated graph sits on the same set of axes as the original, not on a fresh grid.
Mixing up translation with stretch. If the curve gets taller, narrower, or steeper, that’s a stretch. Translation only slides — it never resizes.
Master this note and the rest of transformations becomes much easier — reflections and stretches all build on the “inside vs outside” pattern you just learned here.
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