IB Maths AA SLTopic 2 — FunctionsPaper 1 & 2~9 min read
Reflections of Graphs
A reflection flips a graph in a mirror line — either the x-axis or the y-axis. The shape and size stay the same, but the orientation flips. Here’s exactly when to use a minus sign inside the bracket vs outside.
📘 What you need to know
A reflection flips the graph in a mirror line. The size doesn’t change — only the orientation does.
The two reflections you need are in the x-axis (the line y = 0) and the y-axis (the line x = 0).
Reflection in the x-axis:y = −f(x) — the minus sign is outside the function, so the y-coords flip sign.
Reflection in the y-axis:y = f(−x) — the minus is inside the bracket, so the x-coords flip sign.
Any point already on the mirror line doesn’t move — these are your fixed points and they’re huge for sketching.
What is a reflection?
A reflection is exactly what it sounds like — imagine holding the graph up to a mirror. Every point on the curve flips to the other side of the mirror line, but stays the same distance away. The graph keeps its shape and size, but its orientation is now reversed.
For IB Maths AA SL, the mirror line is always one of the two coordinate axes:
The x-axis (the line y = 0) — flipping vertically, like turning a sheet of paper upside-down across a horizontal fold.
The y-axis (the line x = 0) — flipping horizontally, like a mirror standing upright on the y-axis.
A reflection changes orientation but not size. If the curve gets bigger, smaller, or just slides — that’s a stretch or translation, not a reflection.
A curve reflected in both axes
Reflection in the y-axis (sideways flip)
This reflection flips the graph from left-to-right. Whatever was on the right of the y-axis ends up on the left, and vice-versa. The mirror sign goes inside the function bracket.
Reflection in the y-axis
y = f(x)⟶y = f(−x)
🤔 Why does the minus go inside?
Inside the bracket is where you change the input. When you put −x in instead of x, you’re asking the function: “what would you do if I gave you the opposite x-value?” The answer comes out the same height as before, but now placed at the mirrored x-position.
So f(−x) flips the graph horizontally — across the y-axis.
What changes and what stays?
Every point (x, y) becomes (−x, y) — only the x-coordinate flips sign.
A vertical asymptote x = k flips to x = −k.
Horizontal asymptotes don’t move at all.
📍
Fixed points: anything on the y-axis doesn’t move
If the original graph passes through (0, 7), the reflected graph still passes through (0, 7). Points on the mirror line are their own reflection — use them as anchor points when sketching.
Reflection in the x-axis (upside-down flip)
This reflection flips the graph upside-down. Anything that was above the x-axis is now below it, and vice-versa. The mirror sign goes outside the function — it’s just a minus in front.
Reflection in the x-axis
y = f(x)⟶y = −f(x)
🤔 Why does the minus go outside?
Outside the function is where you change the output. Multiplying the whole output by −1 takes every y-value and flips its sign. A point that was at height 5 is now at height −5; a point at −2 jumps to +2.
So −f(x) flips the graph vertically — across the x-axis.
What changes and what stays?
Every point (x, y) becomes (x, −y) — only the y-coordinate flips sign.
A horizontal asymptote y = k flips to y = −k.
Vertical asymptotes don’t move at all.
📍
Fixed points: anything on the x-axis doesn’t move
If the original graph crosses the x-axis at (4, 0), the reflected graph still crosses at (4, 0). The x-intercepts of the original become the x-intercepts of the reflection. Use them to anchor your sketch.
The two reflections side by side
Reflect in y-axis (sideways)
y = f(−x)
Minus inside the bracket
x-coords flip · y-coords stay
Reflect in x-axis (upside-down)
y = −f(x)
Minus outside the function
y-coords flip · x-coords stay
↔ Reflection in y-axis
x-coordinateflips sign
y-coordinatestays the same
Vertical asymptoteflips
Horizontal asymptoteunchanged
Points on y-axisdon’t move
↕ Reflection in x-axis
x-coordinatestays the same
y-coordinateflips sign
Vertical asymptoteunchanged
Horizontal asymptoteflips
Points on x-axisdon’t move
A simple memory hook: negative inside = flip the inputs (x’s) → mirror in the y-axis. Negative outside = flip the outputs (y’s) → mirror in the x-axis.
Equation decoder
You’ll see reflections written in different ways. Here’s how to spot which axis is being used:
Equation
→
Transformation
y = f(−x)
→
Reflection in the y-axis
y = −f(x)
→
Reflection in the x-axis
y = (−x)2 + 3(−x)
→
Reflection in the y-axis (every x replaced by −x)
y = −(x2 − 4x + 1)
→
Reflection in the x-axis (whole function negated)
In the exam, name the axis you reflected in — examiners want “reflection in the x-axis”, not just “reflection”.
Worked examples
WE 1
Find the equation after a y-axis reflection
The graph of y = x3 − 2x + 1 is reflected in the y-axis. Write the equation of the new graph.
Step 1: Pick the rule
Reflection in y-axis ⇒ replace every x with −x.
Step 2: Substitute carefully
(−x)3 = −x3, −2(−x) = +2x.
Step 3: Tidy upy = −x3 + 2x + 1always replace EVERY x — not just one!
WE 2
Find the equation after an x-axis reflection
The graph of y = x2 − 6x + 5 is reflected in the x-axis. Write the equation of the new graph.
Step 1: Pick the rule
Reflection in x-axis ⇒ multiply the whole function by −1.
Step 2: Apply ity = −(x2 − 6x + 5)Step 3: Tidy upy = −x2 + 6x − 5don’t forget to flip every term’s sign
WE 3
Reflect 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) (b) y = f(−x)
(a) y = −f(x) → reflect in x-axisy-coords flip sign, x-coords stayA′(−1, −5), B′(3, 3)also: max becomes min, and min becomes max!(b) y = f(−x) → reflect in y-axisx-coords flip sign, y-coords stayA′(1, 5), B′(−3, −3)max stays a max, min stays a min — only swapped sideways
WE 4
Reflect a graph with asymptotes
The graph of y = f(x) has asymptotes x = 4 and y = −2. State the equations of the asymptotes after reflecting in:
(a) the y-axis (b) the x-axis
(a) Reflection in y-axis
Vertical asymptote flips: x = 4 → x = −4.
Horizontal asymptote stays: y = −2.
x = −4 & y = −2(b) Reflection in x-axis
Vertical asymptote stays: x = 4.
Horizontal asymptote flips: y = −2 → y = 2.
x = 4 & y = 2
WE 5
Identify the reflection
The function f(x) = 2x + 3 is transformed into g(x) = −2x − 3. Describe the single transformation that maps f onto g.
Step 1: Compare side by sidef(x) = 2x + 3
g(x) = −2x − 3
Step 2: Spot the pattern
Every term has flipped sign → the whole function is multiplied by −1.
g(x) = −f(x)Step 3: Name the transformationReflection in the x-axisif only the x-terms had flipped, it’d be a y-axis reflection instead
💡 Top tips
Inside vs outside still rules. Minus inside the bracket = flip the x‘s = mirror in y-axis. Minus outside the function = flip the y‘s = mirror in x-axis.
Flip the named points first. If the question gives you a max, min, or intercept, flip those points before sketching the new curve through them.
Use fixed points as anchors. Anything on the mirror line stays exactly where it is — these are free anchor points for your sketch.
Reflecting in the x-axis swaps maxima and minima. A peak becomes a trough and vice-versa.
Reflecting in the y-axis keeps maxima as maxima and minima as minima — they just swap sides.
Always name the axis in your written answer: “reflection in the x-axis”, never just “reflection”.
If you’re given a transformed equation, expand it carefully — sign mistakes when distributing the minus are the #1 way to lose marks.
⚠ Common mistakes
Mixing up the axes. Many students reflexively say “minus = flip in x-axis” without checking where the minus is. Always look first: inside or outside the bracket?
Replacing only the first x. For f(−x), every x in the formula must be replaced with −x — not just the first one.
Forgetting to distribute the minus. When writing −f(x) = −(x2 − 6x + 5), the minus must hit every term, giving −x2 + 6x − 5.
Flipping the wrong asymptote. A y-axis reflection only flips vertical asymptotes; an x-axis reflection only flips horizontal ones.
Saying just “reflection”. The mark scheme always wants you to name the axis. “Reflection” alone is not enough.
Confusing reflections with rotations. A reflection flips orientation but preserves it left-right or top-bottom; a 180° rotation flips both. They give the same result only for very symmetric curves.
Errors with even/odd functions. If f(x) = x2, then f(−x) = x2 too — the graph looks unchanged because it’s already symmetric in the y-axis. Don’t think you’ve done something wrong!
Once you’ve got translations and reflections down, the only piece left is stretches — and that’s where scale factors come in next.
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