Reflections of Functions
When a video game character turns around, the artist does not redraw the entire sprite — they simply flip the image horizontally. In mathematics, we can flip graphs too: across the $x$-axis, across the $y$-axis, or both. These reflections are powerful tools for sketching and understanding symmetry.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
The point $(3, 4)$ lies on the graph of $y = f(x)$. If you multiply the entire right-hand side by $-1$ to get $y = -f(x)$, what do you think happens to the point? What if instead you replace $x$ with $-x$ to get $y = f(-x)$? Try to predict the new coordinates in each case.
Key insight: Reflection in the $x$-axis changes $y$ to $-y$. Reflection in the $y$-axis changes $x$ to $-x$.
Key facts
- $-f(x)$ reflects the graph in the $x$-axis
- $f(-x)$ reflects the graph in the $y$-axis
- $-f(-x)$ reflects in both axes
Concepts
- How reflections affect the coordinates of key points
- The connection between $f(-x)$ and even functions
- The connection between $-f(-x)$ and odd functions
- Why reflecting in both axes is the same as a $180^\circ$ rotation
Skills
- Sketch reflected graphs from their equations
- Write the equation of a reflected graph
- Determine the new coordinates of points after reflection
- Use reflections to test for odd and even symmetry
Just as you can flip an image in a photo editor, you can reflect the graph of a function across an axis. There are two fundamental reflections you need to know.
Reflection in the $x$-axis: $y = -f(x)$
Multiplying the entire function by $-1$ flips the graph upside down. Every point $(x, y)$ on the original graph moves to $(x, -y)$.
- $x$-intercepts stay the same (where $y = 0$)
- $y$-intercepts change sign
- The range is negated (if the original range was $[a, b]$, the new range is $[-b, -a]$)
Reflection in the $y$-axis: $y = f(-x)$
Replacing $x$ with $-x$ flips the graph left-to-right. Every point $(x, y)$ on the original graph moves to $(-x, y)$.
- $y$-intercepts stay the same (where $x = 0$)
- $x$-intercepts change sign
- The domain is reflected (if the original domain was $[a, b]$, the new domain is $[-b, -a]$)
Reflection in Both Axes: $y = -f(-x)$
When you apply both reflections — multiply by $-1$ outside and replace $x$ with $-x$ inside — the result is equivalent to a $180^\circ$ rotation about the origin. Every point $(x, y)$ becomes $(-x, -y)$.
$x$-axis reflection: $y = -f(x)$ — negates the $y$-coordinate — $(x,y) \to (x,-y)$ — $x$-intercepts stay the same; $y$-axis reflection: $y = f(-x)$ — negates the $x$-coordinate — $(x,y) \to (-x,y)$ — $y$-intercepts stay the same
Pause — copy both reflection rules: $y = -f(x)$ negates $y$-coordinates ($x$-intercepts stay), and $y = f(-x)$ negates $x$-coordinates ($y$-intercept stays) into your book.
Did you get this? True or false: the transformation $y = -f(x)$ reflects the graph in the $y$-axis.
Quick check: The point $(5, -2)$ lies on $y = f(x)$. What is the corresponding point on $y = f(-x)$?
Common mistakes · the 4 traps that cost marks
Confusing $-f(x)$ with $f(-x)$
$-f(x)$ reflects in the $x$-axis (vertical flip). $f(-x)$ reflects in the $y$-axis (horizontal flip). These are completely different transformations, and mixing them up is one of the most common errors in transformation questions.
✓ Fix: Ask yourself: "Where is the negative sign?" Outside = $x$-axis. Inside = $y$-axis.
Changing the wrong coordinate
For $y = -f(x)$, students sometimes change the $x$-coordinate instead of the $y$-coordinate. For $y = f(-x)$, they sometimes change the $y$-coordinate instead of the $x$-coordinate.
✓ Fix: $x$-axis reflection → change $y$. $y$-axis reflection → change $x$.
Forgetting that $f(-x)$ requires substituting $-x$ into every term
When reflecting $f(x) = x^2 + 3x$ in the $y$-axis, some students write $-x^2 + 3x$ instead of $(-x)^2 + 3(-x) = x^2 - 3x$.
✓ Fix: Use brackets. Replace every $x$ with $(-x)$ before simplifying.
Assuming all functions have either $x$-axis or $y$-axis symmetry
Many functions have no reflection symmetry at all. A reflection changes the graph completely, and only special functions (even or odd) map onto themselves.
✓ Fix: If the reflected graph does not match the original, the function simply does not have that symmetry. That is a valid and common conclusion.
Worked examples · reveal as you go
Describe the transformation that maps $y = f(x)$ to $y = -f(x)$.
The graph of $y = f(x)$ passes through the points $(1, 3)$, $(2, -1)$, and $(0, 4)$. Find the corresponding points on the graph of $y = f(-x)$.
Let $f(x) = x^3 - 2x$. Write the equation of the graph after reflection in the $x$-axis, and then after reflection in the $y$-axis.
Fill the blanks: drag each token into the matching blank.
$y = -f(x)$ reflects in the ___ and negates the ___. $y = f(-x)$ reflects in the ___ and negates the ___.
Activity 1 — Identify the reflection
For each equation, state whether it represents a reflection in the $x$-axis, the $y$-axis, both, or neither.
$y = -f(x)$
$y = f(-x)$
$y = -f(-x)$
$y = f(x) + 2$
Odd one out: Which of these does NOT represent a reflection of $y = f(x)$?
Quick-fire practice · 5 reps +2 XP per reveal
If $(3, -2)$ lies on $y = f(x)$, what point lies on $y = -f(x)$?
If $(3, -2)$ lies on $y = f(x)$, what point lies on $y = f(-x)$?
The graph $y = f(x)$ has a $y$-intercept at $(0, 5)$. After reflection in the $x$-axis, where is the new $y$-intercept?
Is $f(x) = x^4 - 3x^2$ an even function? What happens when you reflect it in the $y$-axis?
A student says "$y = -f(-x)$ means you reflect in the $x$-axis only." Are they correct?
Earlier you were asked: If $(3, 4)$ lies on $y = f(x)$, what happens to the point under $y = -f(x)$ and $y = f(-x)$?
For $y = -f(x)$, the negative sign is outside the function, so it flips the $y$-coordinate. The point $(3, 4)$ becomes $(3, -4)$. This is a reflection in the $x$-axis. For $y = f(-x)$, the negative sign is inside the function, so it flips the $x$-coordinate. The point $(3, 4)$ becomes $(-3, 4)$. This is a reflection in the $y$-axis. The key is simple but powerful: outside = vertical flip ($x$-axis), inside = horizontal flip ($y$-axis). Master this distinction and you have mastered one of the most important ideas in graph transformations.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q8. The graph of $y = f(x)$ passes through the points $(-2, 1)$, $(0, 3)$, and $(4, -2)$. Write the coordinates of the corresponding points on: (a) $y = -f(x)$ (b) $y = f(-x)$ (c) $y = -f(-x)$
Q9. Let $f(x) = x^2 - 4x + 3$. (a) Write the equation of the graph after reflection in the $y$-axis. (b) Simplify your answer from part (a) by expanding any brackets. (c) Determine whether the reflected graph is the same as the original graph.
Q10. A student is asked to reflect $y = f(x)$ in the $x$-axis and then in the $y$-axis. They write the final equation as $y = f(x)$, claiming that the two reflections cancel each other out. Evaluate this claim. Is it true for all functions? Provide a specific counterexample or proof to support your answer.
📖 Comprehensive answers (click to reveal)
Multiple choice — drill bank
MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.
1. A — $-f(x)$ reflects in the $x$-axis.
2. B — $f(-x)$ reflects in the $y$-axis.
3. C — $-f(x)$ negates the $y$-coordinate.
4. B — $f(-x)$ negates the $x$-coordinate.
5. C — $-f(-x)$ reflects in both axes.
Activity 1 — Identify the reflection model answers
1. Reflection in the $x$-axis
2. Reflection in the $y$-axis
3. Reflection in both the $x$-axis and the $y$-axis (or $180^\circ$ rotation about the origin)
4. Neither — this is a vertical translation 2 units up
Short answer model answers
Q8 (3 marks):
(a) $(-2, -1)$, $(0, -3)$, $(4, 2)$ [1]
(b) $(2, 1)$, $(0, 3)$, $(-4, -2)$ [1]
(c) $(2, -1)$, $(0, -3)$, $(-4, 2)$ [1]
Q9 (4 marks):
(a) $y = f(-x) = (-x)^2 - 4(-x) + 3 = x^2 + 4x + 3$ [1]
(b) $y = x^2 + 4x + 3$ (already expanded) [1]
(c) The reflected graph is not the same as the original [1]. The original has its vertex at $(2, -1)$, while the reflected graph has its vertex at $(-2, -1)$ [1].
Q10 (3 marks): The student's claim is false in general [1]. For most functions, reflecting in the $x$-axis and then the $y$-axis gives $y = -f(-x)$, which is not the same as $y = f(x)$ [1]. For example, if $f(x) = x + 1$, then $-f(-x) = -(-x + 1) = x - 1 \neq x + 1 = f(x)$ [1]. The claim is only true for odd functions, where $-f(-x) = f(x)$.
Five timed questions on reflections of functions. Beat the boss to bank a tier — gold (perfect + fast), silver (80%+), or bronze (cleared).
⚔ Enter the arenaChallenge the boss using your knowledge of function reflections and transformations. Pool: lessons 1–10.
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