IB Maths AA HL Topic 2 — Functions Paper 1 & 2 ~6 min read

Reflections of Graphs

A reflection flips a graph across an axis. There are exactly two cases to learn — flip across the x-axis with y = −f(x), or flip across the y-axis with y = f(−x). Each one negates a coordinate; the other coordinate is untouched.

📘 What you need to know

Reflection in the x-axis: y = −f(x). Negates every y-coordinate; x-coordinates stay the same.
Reflection in the y-axis: y = f(−x). Negates every x-coordinate; y-coordinates stay the same.
The minus sign location is the rule: minus outside the bracket → flip up-down (x-axis); minus inside the bracket → flip left-right (y-axis).
Points on the axis of reflection are fixed: points on the x-axis don’t move under y = −f(x); points on the y-axis don’t move under y = f(−x).
Asymptotes flip too: vertical asymptotes flip under f(−x); horizontal asymptotes flip under −f(x).
The shape and size are unchanged — only the orientation flips.

The two reflection rules

Reflection in x-axis

y = −f(x)

(a, b) → (a, −b)
minus outside the bracket

Reflection in y-axis

y = f(−x)

(a, b) → (−a, b)
minus inside the bracket

Both reflections of the same curve

Reflection in the x-axis — y = −f(x)

Negate every y-value Every point (a, b) on y = f(x) maps to (a, −b) on y = −f(x)

Changes

y-coordinates flip sign
horizontal asymptotes flip sign

max ↔ min, above ↔ below

Stays the same

x-coordinates
vertical asymptotes
x-intercepts

anything with y = 0 doesn’t move

Reflection in the y-axis — y = f(−x)

Negate every x-value Every point (a, b) on y = f(x) maps to (−a, b) on y = f(−x)

Changes

x-coordinates flip sign
vertical asymptotes flip sign

left ↔ right

Stays the same

y-coordinates
horizontal asymptotes
y-intercepts

anything with x = 0 doesn’t move

Even and odd functions: if f(−x) = f(x) the graph is its own reflection in the y-axis (even, like x² or cos x). If f(−x) = −f(x), reflecting in both axes gives back the original (odd, like x³ or sin x).

🧭 Recipe — reflecting a graph

Locate the minus sign: outside bracket → x-axis flip; inside bracket → y-axis flip.
For each labelled point (a, b): apply (a, −b) for −f(x), or (−a, b) for f(−x).
Flip the matching asymptotes: HAs flip under −f(x); VAs flip under f(−x).
Sketch the new curve with the same shape, just mirrored. Label all transformed features.

Worked examples

WE 1

Find the equation after a reflection in the x-axis

The graph of f(x) = x² − 4x + 7 is reflected in the x-axis. Find the equation of the new graph.

Multiply the entire function by −1 y = −f(x) = −(x² − 4x + 7) y = −x² + 4x − 7 y = −x² + 4x − 7 every term gets a sign flip — the parabola that opened up now opens down

WE 2

Find the equation after a reflection in the y-axis

The graph of g(x) = 2x³ − 5x + 1 is reflected in the y-axis. Find the equation of the new graph in simplified form.

Replace x with −x y = g(−x) = 2(−x)³ − 5(−x) + 1 Simplify each term 2(−x)³ = −2x³ −5(−x) = 5x y = −2x³ + 5x + 1 y = −2x³ + 5x + 1 odd powers of x flip sign, even powers don’t, constants stay the same

WE 3

Reflect the key points of a graph

The graph of y = f(x) has a maximum at P(−3, 4), a minimum at Q(2, −5), and crosses the y-axis at (0, 1). State the coordinates of the corresponding three points on:
(a) y = −f(x) (b) y = f(−x)

(a) y = −f(x): negate y-coordinates P(−3, 4) → (−3, −4) — now a minimum Q(2, −5) → (2, 5) — now a maximum (0, 1) → (0, −1) (a) min (−3, −4); max (2, 5); y-int (0, −1) (b) y = f(−x): negate x-coordinates P(−3, 4) → (3, 4) — still a maximum Q(2, −5) → (−2, −5) — still a minimum (0, 1) → (0, 1) — unchanged (on the y-axis) (b) max (3, 4); min (−2, −5); y-int (0, 1) x-axis flip swaps max with min; y-axis flip keeps them as max and min — just on the other side

WE 4

Reflection of a rational function

The graph of y = 3x − 1 + 2 is reflected in the x-axis.
(a) Find the equation of the new graph. (b) State its vertical and horizontal asymptotes.

(a) Multiply the whole expression by −1 y = −[3/(x − 1) + 2] y = −3/(x − 1) − 2 (a) y = −3/(x − 1) − 2 (b) Asymptotes — VA stays, HA flips sign original VA: x = 1 → unchanged original HA: y = 2 → flipped to y = −2 (b) VA: x = 1; HA: y = −2 x-axis reflection only touches y-coordinates — including the y-value of the horizontal asymptote

WE 5

Identify the reflection from two equations

The graph of y = e^x is transformed by a single reflection to give the graph of y = e^−x. State the axis of reflection.

Compare the two equations y = e^x vs y = e^(−x) x has been replaced by −x The minus sign is inside the function input that’s the form y = f(−x) reflection in the y-axis classic exam set-up: e^x and e^(−x) are mirror images across the y-axis — same y-intercept (0, 1)

WE 6

Track intercepts under both reflections

The graph of y = f(x) has y-intercept (0, 6) and x-intercepts at (−2, 0) and (5, 0). State the y-intercept and x-intercepts of:
(a) y = −f(x) (b) y = f(−x)

(a) y = −f(x): y-coordinates flip, x-coordinates stay y-int: (0, 6) → (0, −6) x-ints: (−2, 0) and (5, 0) → unchanged (y = 0 stays 0) (a) y-int (0, −6); x-ints (−2, 0) and (5, 0) (b) y = f(−x): x-coordinates flip, y-coordinates stay y-int: (0, 6) → (0, 6) — unchanged (x = 0 stays 0) x-ints: (−2, 0) → (2, 0); (5, 0) → (−5, 0) (b) y-int (0, 6); x-ints (2, 0) and (−5, 0) x-intercepts sit on the x-axis so they’re fixed under x-axis reflection; y-intercept sits on the y-axis so it’s fixed under y-axis reflection

💡 Top tips

Locate the minus sign first: outside the bracket → x-axis reflection; inside the bracket → y-axis reflection.
The “fixed” coordinate trick: under −f(x), x-intercepts don’t move; under f(−x), the y-intercept doesn’t move.
For a polynomial reflection in the y-axis: odd powers flip sign, even powers and constants stay.
For a reflection in the x-axis: every term gets multiplied by −1 — including any constant.
Sanity-check with the GDC: plot original and image on the same axes — they should be true mirror images.
Label both axes of reflection when sketching: it makes the symmetry obvious to the examiner.
Watch for double reflections: y = −f(−x) reflects in both axes — equivalent to a 180° rotation about the origin.

⚠ Common mistakes

Confusing f(−x) with −f(x). The minus sign placement determines the axis — pause and locate it before flipping.
Negating only the leading term when finding −f(x). Every term in the expression must change sign.
Cubing sign errors: (−x)³ = −x³, but (−x)² = x². Even powers don’t flip.
Flipping the wrong asymptote: under −f(x), VAs stay put; under f(−x), HAs stay put.
Forgetting to label the new max/min: under −f(x), maxes become mins and vice versa.
Sketching a curve that loses its shape. Reflections preserve shape — same height, width, and curvature, just mirrored.
Naming the wrong axis: “reflection in x = 0″ is the y-axis. Use the right wording.

Translations slide and reflections flip — both leave the shape intact. The next note tackles stretches, where the size of the graph changes too. The minus sign rules you just learned generalise: outside the bracket affects y; inside the bracket affects x.

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