IB Maths AA SLTopic 2 — FunctionsPaper 1 & 2~10 min read
Stretches of Graphs
A stretch resizes a graph either vertically or horizontally by a scale factor. The shape stays the same, but the curve gets taller, shorter, wider, or narrower. The trickiest part — and where most marks are lost — is the f(ax) gotcha. We’ll cover it head-on.
📘 What you need to know
A stretch resizes a graph — taller/shorter or wider/narrower — by a scale factor a.
Vertical stretch:y = af(x) — the multiplier is outside the function. The y-coordinates get multiplied by a.
Horizontal stretch:y = f(xa) — the multiplier is inside the bracket as a denominator. The x-coordinates get multiplied by a.
Big trap:y = f(ax) is a horizontal stretch by scale factor 1a — not a!
Scale factor > 1 stretches outwards from the axis; 0 < scale factor < 1 squashes inwards. Points on the axis don’t move.
What is a stretch?
Imagine your graph is printed on a stretchy rubber sheet. If you grab the top and bottom and pull, the curve gets taller — that’s a vertical stretch. If you grab the sides and pull, the curve gets wider — that’s a horizontal stretch.
The shape of the curve doesn’t change in any meaningful way — peaks are still peaks, troughs are still troughs, intercepts on the stretching axis stay put. What changes is the size of the graph in one direction.
Stretches always act parallel to one axis. A vertical stretch pulls things parallel to the y-axis (up or down). A horizontal stretch pulls things parallel to the x-axis (left or right).
A stretch changes the size; a translation slides it; a reflection flips it. Always check what’s actually happening to the curve before you name the transformation.
Vertical stretch (taller) vs horizontal stretch (wider)
Vertical stretches (taller or shorter)
A vertical stretch resizes the graph up and down. The multiplier sits outside the function — and just like translations, “outside” means the change is intuitive: multiply by 2 and the graph doubles in height.
Vertical stretch
y = af(x)⟶vertical stretch, scale factor a, parallel to y-axis
🤔 Why does multiplying outside stretch vertically?
Multiplying outside the function changes the output. Every y-value gets multiplied by a. A point at height 3 becomes a point at height 3a; a point at height −2 becomes 3a… wait, it becomes −2a.
So if a = 2, every height doubles. If a = 12, every height halves. That’s a vertical stretch.
Scale factor a > 1
y = af(x)
Graph stretches AWAY from the x-axis
Taller / steeper
Scale factor 0 < a < 1
y = af(x)
Graph squashes TOWARDS the x-axis
Shorter / flatter
In your written answer, never say “compress” or “squash” — use proper terminology: “a stretch by scale factor 12“. Even when the curve gets shorter, it’s still called a stretch.
What changes and what stays?
Every point (x, y) becomes (x, ay) — only the y-coordinate changes.
A horizontal asymptote y = k moves to y = ak.
Vertical asymptotes don’t move at all.
📍
Fixed points: anything on the x-axis doesn’t move
If a point has y = 0, then a × 0 = 0 — it stays exactly where it was. So all x-intercepts of the original graph are also x-intercepts of the stretched graph. Use these to anchor your sketch.
Horizontal stretches (wider or narrower)
A horizontal stretch resizes the graph left and right. The multiplier sits inside the bracket — and just like translations, “inside” means the behaviour is the opposite of what you’d guess. This is where students lose the most marks.
Horizontal stretch (clean form)
y = f(xa)⟶horizontal stretch, scale factor a, parallel to x-axis
🤔 Why does dividing inside stretch horizontally by a?
Inside the bracket changes the input. For f(x2) to give the same output as the original f(2), we need x/2 = 2 — so x = 4.
The new graph does at x = 4 what the old one did at x = 2 — every x-coordinate has doubled. So y = f(x/2) is a horizontal stretch by factor 2.
⚠ The #1 trap: f(ax) is NOT a stretch by factor a
Almost every student gets this wrong the first time. When the multiplier is multiplying the x instead of dividing it, the scale factor is the reciprocal:
Wrong instinct
y = f(2x) → factor 2
This is what most students assume — but it’s wrong.
Actual rule
y = f(2x) → factor 12
The graph gets narrower, not wider.
The trick: if you see f(ax), immediately rewrite it as f(x1/a). Then the scale factor is whatever’s underneath — namely 1a.
Scale factor a > 1
y = f(xa)
Graph stretches AWAY from the y-axis
Wider
Scale factor 0 < a < 1
y = f(xa)
Graph squashes TOWARDS the y-axis
Narrower
What changes and what stays?
Every point (x, y) becomes (ax, y) — only the x-coordinate changes.
A vertical asymptote x = k moves to x = ak.
Horizontal asymptotes don’t move at all.
📍
Fixed points: anything on the y-axis doesn’t move
If a point has x = 0, then a × 0 = 0 — it stays put. So the y-intercept is unchanged by any horizontal stretch. Anchor your sketch with this point.
The two stretches side by side
↕ Vertical stretch (factor a)
Equation formy = af(x)
x-coordinatestays the same
y-coordinatemultiplied by a
Vertical asymptoteunchanged
Horizontal asymptotemultiplied by a
Fixed pointson x-axis
↔ Horizontal stretch (factor a)
Equation formy = f(xa)
x-coordinatemultiplied by a
y-coordinatestays the same
Vertical asymptotemultiplied by a
Horizontal asymptoteunchanged
Fixed pointson y-axis
Equation decoder
Spotting the right scale factor is the most marked-on skill in this topic. Here are the four forms you’ll meet:
In the exam, write your answer in proper form: “horizontal stretch, scale factor 12, parallel to the x-axis”. Naming the axis is worth a mark on its own.
Worked examples
WE 1
Find the equation after a vertical stretch
The graph of y = x2 + 1 is stretched vertically by scale factor 3. Write the equation of the new graph.
Step 1: Pick the rule
Vertical stretch ⇒ multiply the whole function by 3.
Step 2: Apply ity = 3(x2 + 1)Step 3: Tidy upy = 3x2 + 3don’t forget the bracket — multiply EVERY term
WE 2
Find the equation after a horizontal stretch
The graph of y = x2 − 4x is stretched horizontally by scale factor 2. Write the equation of the new graph.
Step 1: Pick the rule
Horizontal stretch by 2 ⇒ replace every x with x2.
Step 2: Substitutey = (x2)2 − 4(x2)Step 3: Simplifyy = x24 − 2xreplace EVERY x — including the one inside the squared term
WE 3
Stretch 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 = 2f(x) (b) y = f(2x)
(a) y = 2f(x) → vertical stretch, factor 2y-coords double, x-coords stayA′(−1, 10), B′(3, −6)(b) y = f(2x) → horizontal stretch, factor 12x-coords HALVED (NOT doubled!), y-coords stayA′(−12, 5), B′(32, −3)f(2x) is the trap form — graph narrows, not widens!
WE 4
Stretch a graph with asymptotes
The graph of y = f(x) has asymptotes x = 5 and y = 4. State the equations of the asymptotes after:
(a) a vertical stretch by scale factor 3 (b) a horizontal stretch by scale factor 12
(a) Vertical stretch, factor 3
Vertical asymptote stays: x = 5.
Horizontal asymptote × 3: y = 4 → y = 12.
x = 5 & y = 12(b) Horizontal stretch, factor 1/2
Vertical asymptote × 12: x = 5 → x = 52.
Horizontal asymptote stays: y = 4.
x = 52 & y = 4
WE 5
Identify the stretch from an equation
The function f(x) = x3 + x is transformed into g(x) = 8x3 + 2x. Describe the single transformation that maps f onto g.
Step 1: Test for a horizontal stretch
Try f(ax) = (ax)3 + (ax) = a3x3 + ax.
Step 2: Match coefficients
Compare with 8x3 + 2x: need a3 = 8 and a = 2.
Both give a = 2 ✓
Step 3: Name itg(x) = f(2x) ⇒ horizontal stretch.
Horizontal stretch, scale factor 12because f(2x) = factor 1/2, not 2!
💡 Top tips
Outside vs inside still rules. Multiplier outside = vertical stretch (intuitive). Multiplier inside = horizontal stretch with reciprocal.
Always rewrite f(ax) as f(x1/a) in your head — then the scale factor is whatever’s in the denominator.
Use fixed points to anchor sketches.x-intercepts are fixed under vertical stretches; the y-intercept is fixed under horizontal stretches.
Always state the scale factor, the direction, and the parallel axis in your written answer.
Never use words like “compress” or “squash” in the exam — even if the curve gets shorter or narrower, it’s still called a “stretch”.
Remember stretches don’t change the type of stationary point. Maxima stay maxima; minima stay minima.
If both axes seem affected, you’ve actually got two transformations — separate them and apply one at a time.
⚠ Common mistakes
The f(ax) trap. Treating f(2x) as a stretch by factor 2 — it’s actually factor 12. Single biggest mistake on this topic.
Stretching only some terms. When applying y = 2f(x), every y-coordinate must be doubled — including negative ones.
Forgetting the bracket. Writing 2x2 + 1 instead of 2(x2 + 1) = 2x2 + 2 misses a sign or term.
Saying “stretch” without naming the direction. Always say “vertical stretch” or “horizontal stretch” — and name the axis it’s parallel to.
Forgetting fixed points. Students sometimes redraw the whole curve from scratch instead of keeping the x-intercepts (vertical stretch) or y-intercept (horizontal stretch) anchored.
Confusing stretches with translations. A stretch resizes; a translation slides. If the shape gets bigger or smaller, it’s a stretch — never a translation.
You now know all three single transformations — translations, reflections, and stretches. Next up: putting them together as composite transformations, where the order really matters.
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