M
hscscience Maths Adv · Y11
0/100daily goal
0
0
0 due
0
L1 · 0 XP
KJ
Your weak spots
Insights load after your first practice round.
Module 1 · L10 of 15 ~35 min ⚡ +50 XP in Learn · +25 to complete

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.

Today's hook, The point $(3, 4)$ lies on $y = f(x)$. If you negate the right-hand side to get $y = -f(x)$, what happens to the point? What if you replace $x$ with $-x$? Can you predict the new coordinates in each case?
0/5QUESTS
1
You’re here

Orient and recall

Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

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.

auto-saved
02
Formula reference · this lesson
core notation
$y = -f(x)$  , Reflection in the $x$-axis · $(x, y) \to (x, -y)$
$y = f(-x)$  , Reflection in the $y$-axis · $(x, y) \to (-x, y)$
$y = -f(-x)$ Reflection in both axes · equivalent to $180^\circ$ rotation about the origin

Key insight: Reflection in the $x$-axis changes $y$ to $-y$. Reflection in the $y$-axis changes $x$ to $-x$.

03
What you'll master
Know

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
Understand

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
Can do

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
04
Key terms
Function
A relation where each input has exactly one output.
Domain
The set of all possible input values for a function.
Range
The set of all possible output values for a function.
Reflection in x-axis
The transformation $y = -f(x)$; every point $(x, y)$ becomes $(x, -y)$.
Reflection in y-axis
The transformation $y = f(-x)$; every point $(x, y)$ becomes $(-x, y)$.
Point Mapping
Describing how coordinates change under a transformation.
2
You’re here

Establish it yourself with a graphing application

Work through the core explanation before applying it.

04b
Establish it yourself with a graphing application
+10 XP

The syllabus dot-point for this lesson begins "Establish using graphing applications", so this is the activity that earns it rather than being told the rule. Use Desmos, GeoGebra or any graphing calculator.

Task. Graph $y = x^3 - 2x$, then $y = -(x^3 - 2x)$, then $y = (-x)^3 - 2(-x)$.

Predict before you plot. Before you look: predict which axis each version is reflected in. Then check. One of these two reflections leaves the graph looking unchanged for this particular function. Which, and why?
auto-saved
05
Reflections in the $x$-axis and $y$-axis
core concept · +3 XP at end

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 $(-x, -y)$.

Connection to symmetry. If $f(-x) = f(x)$, reflecting in the $y$-axis leaves the graph unchanged. The function is even. If $-f(-x) = f(x)$, rotating $180^\circ$ about the origin leaves the graph unchanged. The function is odd. Reflections and symmetry are two sides of the same coin.
Reflecting y = (x−1)² in the x-axis gives y = −(x−1)²; in the y-axis gives y = (x+1)².

$x$-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

3
You’re here

Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

1

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.

2

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 $x$.

3

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.

4

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

4
You’re here

Work examples end to end

Follow the reasoning through complete worked solutions.

Worked example 1 · identifying a reflection +5 XP on full reveal

Describe the transformation that maps $y = f(x)$ to $y = -f(x)$.

1
Identify where the negative sign is
The negative sign is outside the function, multiplying the entire output.
2
Determine the effect on coordinates
Every point $(x, y)$ becomes $(x, -y)$. This flips the graph vertically.
3
Reflection in the $x$-axis ✓
A vertical flip is a reflection in the $x$-axis.
Step 1 of 3
Worked example 2 · finding reflected coordinates +5 XP on full reveal

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 $y = f(-x)$.

1
Identify the transformation
$f(-x)$ reflects the graph in the $y$-axis. This negates the $x$-coordinate of every point.
2
$(1, 3) \to (-1, 3)$
Negate the $x$-coordinate only.
3
$(2, -1) \to (-2, -1)$
Negate the $x$-coordinate only.
4
$(0, 4) \to (0, 4)$ ✓
The $y$-axis ($x = 0$) maps to itself, points on the axis don't move.
Step 1 of 4
Worked example 3 · equation of a reflected graph +5 XP on full reveal

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.

1
Reflection in the $x$-axis
Multiply the entire function by $-1$.
2
$$y = -f(x) = -(x^3 - 2x) = -x^3 + 2x$$
Distribute the negative sign across all terms.
3
Reflection in the $y$-axis
Replace every $x$ with $-x$.
4
$$y = f(-x) = (-x)^3 - 2(-x) = -x^3 + 2x$$ ✓
Both reflections give the same result because $f$ is an odd function: $-f(-x) = f(x)$.
Step 1 of 4

Fill the blanks: drag each token into the matching blank.

x-coordinate y-coordinate x-axis y-axis

$y = -f(x)$ 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.

5
You’re here

Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

1

$y = -f(x)$

auto-saved
2

$y = f(-x)$

auto-saved
3

$y = -f(-x)$

auto-saved
4

$y = f(x) + 2$

auto-saved

Odd one out: Which of these does NOT represent a reflection of $y = f(x)$?

Quick-fire practice · 5 reps +2 XP per reveal

6
You’re here

Drill it, then lock it in

Run the quick drill and copy the summary into your book.

1

If $(3, -2)$ lies on $y = f(x)$, what point lies on $y = -f(x)$?

2

If $(3, -2)$ lies on $y = f(x)$, what point lies on $y = f(-x)$?

3

The graph $y = f(x)$ has a $y$-intercept at $(0, 5)$. After reflection in the $x$-axis, where is the new $y$-intercept?

4

Is $f(x) = x^4 - 3x^2$ an even function? What happens when you reflect it in the $y$-axis?

5

A student says "$y = -f(-x)$ means you reflect in the $x$-axis only." Are they correct?

09
Revisit your thinking

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.

auto-saved
1
You’re here

Multiple choice

Answer the drill bank and rate your confidence.

01
Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

2
You’re here

Short answer

Write full responses, then check them against the model answers.

02
Short answer
ApplyBand 33 marks

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) $y = -f(-x)$

auto-saved
ApplyBand 44 marks

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.

auto-saved
EvaluateBand 53 marks

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.

auto-saved
📖 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.

Reflection in the $x$-axis. $-f(x)$ reflects in the $x$-axis.

Reflection in the $y$-axis. $f(-x)$ reflects in the $y$-axis.

Effect on the $y$-coordinate. $-f(x)$ negates the $y$-coordinate.

Effect on the $x$-coordinate. $f(-x)$ negates the $x$-coordinate.

Reflection in both axes. $-f(-x)$ 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 $-f(-x) = f(x)$.

Module overview · Maths Advanced