IB Maths AA HL
Topic 2 — Functions
Paper 1 & 2
HL only
~8 min read
Square Transformations
The square transformation y = [f(x)]2 keeps every x-coordinate the same and squares each y-value. Negatives flip up (like with |f|), but with a key difference: the result is smooth at the x-intercepts, not a cusp. Squaring also stretches values with |y| > 1 away from the axis and pulls values with |y| < 1 toward it. The whole transformation reduces to another feature-mapping exercise.
📘 What you need to know
- Definition: y = [f(x)]2 replaces each y on the original graph with y2. x-coordinates are unchanged.
- Range is always non-negative: y ≥ 0 everywhere — no parts below the x-axis.
- Magnitude effect: |y| > 1 → squared value larger; |y| < 1 → squared value smaller.
- Fixed points: any point with y = 0 or y = 1 stays exactly the same.
- Roots become smooth minima: an x-intercept of f at (a, 0) becomes a smooth minimum touching the x-axis at (a, 0) — not a cusp.
- Maxima ↔ Minima depend on sign of y1: a max with y1 > 0 stays a max; a max with y1 < 0 becomes a min (because squaring flips negatives up).
- Vertical asymptotes are unchanged: a VA at x = a on f stays a VA at x = a on [f]2.
- Horizontal asymptote at y = k becomes HA at y = k2.
The basic idea — square the y-coordinate
Square transformation
y = f(x) → y = [f(x)]2
Every point (x, y) on the original transforms to (x, y2). Since y2 is always non-negative, the resulting graph never dips below the x-axis — similar to y = |f(x)|, but with an important geometric difference at the x-intercepts.
Two regions matter: where the original |y| > 1, squaring pushes the height further from the axis (e.g., y = 3 → y2 = 9). Where |y| < 1, squaring pulls the height closer to the axis (e.g., y = 0.5 → y2 = 0.25). At y = 0 and y = 1, nothing changes.
How key features transform
| On y = f(x) | becomes on y = [f(x)]2 |
|---|
| y-intercept (0, c) | y-intercept at (0, c2) |
| x-intercept (root) at (a, 0) | smooth minimum at (a, 0) — touches the x-axis |
| vertical asymptote at x = a | vertical asymptote at x = a (unchanged) |
| local max at (x1, y1), y1 > 0 | local max at (x1, y12) — type preserved |
| local max at (x1, y1), y1 ≤ 0 | local min at (x1, y12) — flipped up |
| local min at (x1, y1), y1 ≥ 0 | local min at (x1, y12) — type preserved |
| local min at (x1, y1), y1 < 0 | local max at (x1, y12) — flipped up |
| horizontal asymptote y = k | horizontal asymptote y = k2 |
The single rule for extrema: a max or min with negative y1 flips to the opposite type when squared (just like with |f|). With positive or zero y1, the type stays the same. Think of the squaring as “reflect-then-rescale” — but only for negative values.
Square vs absolute value — the key difference
Both y = [f(x)]2 and y = |f(x)| produce graphs that never go below the x-axis. But they differ in two important ways:
[f(x)]2
smooth minimum at x-intercepts
curve touches axis tangentially — no cusp
|f(x)|
cusp at x-intercepts
sharp corner — gradient changes sign abruptly
Squaring also stretches and compresses heights non-uniformly — heights with |y| > 1 get pushed further from the axis (often dramatically), while heights with |y| < 1 get pulled in. The absolute value just reflects without rescaling.
🧭 Recipe — sketch y = [f(x)]2 from y = f(x)
- Mark every key feature on the original — intercepts, asymptotes, extrema (with their signs).
- Square each y-coordinate (keep x-coordinates unchanged).
- Decide max-or-min type for each extremum based on the sign of y1: positive ⇒ same type, negative ⇒ swap.
- Convert x-intercepts to smooth minima on the x-axis. Draw smoothly, not as cusps.
- Keep VAs in place; convert any HA at y = k to HA at y = k2.
- Sketch confirming the range is y ≥ 0 everywhere.
Worked examples
WE 1Square of a quadratic
Sketch the graph of y = (x2 − 4)2, identifying all key features.
Step 1: Identify features of f(x) = x² − 4
parabola; roots ±2; min at (0, −4); f → +∞ as x → ±∞
Step 2: Apply feature map
roots ±2 → smooth minima at (±2, 0)
min (0, −4) with y₁ = −4 < 0 → local max at (0, 16)
f → +∞ → [f]² → +∞ (range y ≥ 0)
Step 3: Y-intercept
y(0) = (0 − 4)² = 16 → (0, 16)
smooth minima at (±2, 0); local max (0, 16); symmetric about y-axis
curve touches x-axis tangentially at ±2 — no cusps, unlike |x² − 4| which would have sharp corners there
WE 2Square of a linear function
Sketch the graph of y = (2x − 6)2, identifying the vertex and y-intercept.
Step 1: Identify features of f(x) = 2x − 6
linear; root at x = 3; y-int (0, −6)
Step 2: Apply feature map
root x = 3 → smooth minimum at (3, 0)
y-int (0, −6) → y-int (0, 36)
Step 3: Confirm shape
(2x − 6)² = 4(x − 3)² — parabola opening up with vertex (3, 0)
vertex (3, 0); y-int (0, 36); parabola opening upward
the square of a linear function is always a parabola — root becomes the vertex
WE 3Track a local minimum with negative y-value
The function y = f(x) has a local minimum at (4, −3). State the position and type of the corresponding feature on y = [f(x)]2.
Step 1: x-coordinate is unchanged
x = 4
Step 2: y-coordinate becomes y²
y = −3 → y² = 9
Step 3: Sign of y₁ determines type
y₁ = −3 < 0 → min flips to max
local maximum at (4, 9)
a min that sits below the axis becomes a max above it — squaring flips negatives up
WE 4Track both a positive max and a negative min
The function y = f(x) has a local maximum at (1, 5) and a local minimum at (−2, −3). State the position and type of each feature on y = [f(x)]2.
Step 1: (1, 5) — positive max
y₁ = 5 > 0 → max stays a max
new point: (1, 25)
Step 2: (−2, −3) — negative min
y₁ = −3 < 0 → min flips to max
new point: (−2, 9)
local max (1, 25); local max (−2, 9)
both points are now maxima — when y₁ < 0, the squaring inverts the role of extrema
WE 5Square vs absolute value — direct comparison
For f(x) = (x − 1)(x + 2), state the position and type of features on (a) y = [f(x)]2, (b) y = |f(x)|.
Identify features of f(x) = x² + x − 2
roots x = 1 and x = −2; vertex (−1/2, −9/4); y-int (0, −2)
(a) y = [f(x)]²
roots → smooth minima at (1, 0) and (−2, 0)
vertex (−1/2, −9/4) → max at (−1/2, 81/16)
y-int → (0, 4)
(a) smooth minima at (1, 0) and (−2, 0); local max (−1/2, 81/16); y-int (0, 4)
(b) y = |f(x)|
roots → cusps at (1, 0) and (−2, 0)
vertex (−1/2, −9/4) → max at (−1/2, 9/4)
y-int → (0, 2)
(b) cusps at (1, 0) and (−2, 0); local max (−1/2, 9/4); y-int (0, 2)
same range and roots, but [f]² is smooth and stretches |y| > 1 values further (compare 9/4 vs 81/16)
WE 6Comprehensive transformation
The function y = f(x) has:
• local maximum at A(−3, 4)
• y-intercept at B(0, 2)
• x-intercept at C(5, 0)
• vertical asymptote at x = 8
• horizontal asymptote at y = −2.
Describe the position and type of each feature on y = [f(x)]2.
Apply the feature map to each in turn
A(−3, 4): y₁ = 4 > 0 → max stays a max → (−3, 16)
B(0, 2): y-int → y-int (0, 4)
C(5, 0): root → smooth minimum at (5, 0)
VA at x = 8 → still VA at x = 8
HA at y = −2 → HA at y = (−2)² = 4
local max (−3, 16); y-int (0, 4); smooth min (5, 0); VA at x = 8; HA at y = 4
five features, five clean mappings — squaring kept the VA and the max type because y₁ was positive at A
💡 Top tips
- Range is always y ≥ 0. If your sketch dips below the x-axis, something’s gone wrong.
- Smooth, not cusped, at every x-intercept. The curve touches the axis tangentially like a parabola tangent.
- Sign of y1 decides the type swap: y1 < 0 flips max ↔ min; y1 ≥ 0 keeps the type.
- VAs are unchanged. Squaring an infinite value gives infinity — so the asymptote stays.
- HA gets squared: y = k becomes y = k2, always non-negative.
- Points at y = 0 and y = 1 are fixed. Useful as anchor points when sketching.
- Compare with |f|: same shape roughly, but [f]2 is smooth at intercepts and stretches/compresses heights non-uniformly.
⚠ Common mistakes
- Drawing cusps at x-intercepts. The square transformation gives smooth minima — sharp corners are for absolute value, not squaring.
- Squaring the x-coordinate. Only y-values get squared.
- Forgetting to swap max ↔ min when y1 < 0. Negative outputs flip up — the role of the extremum changes accordingly.
- Treating the HA as unchanged. k = −2 becomes k2 = 4, not k = −2 again.
- Missing the magnitude effect: |y| > 1 stretches, |y| < 1 compresses. Easy to overlook for intermediate values.
- Confusing [f]2 with f(x2). The first squares the output; the second squares the input. Different graphs entirely.
- Forgetting that the range is non-negative when listing features. No part of the graph sits below the x-axis.
And that closes Section 2.10 — Modulus & Further Transformations, the final section of Topic 2 — Functions. You now have the full toolkit: linear and quadratic functions, exponentials and logs, polynomials, transformations, inequalities, modulus, reciprocal, and square. Functions are the foundation everything else in the IB AA HL course builds on — calculus, complex numbers, and probability all lean heavily on what you’ve covered here.
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