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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?

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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.

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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 $-x$.

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

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.

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.

$x$-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

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

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

Identify where the negative sign is

The negative sign is outside the function, multiplying the entire output.

Determine the effect on coordinates

Every point $(x, y)$ becomes $(x, -y)$. This flips the graph vertically.

Reflection in the $x$-axis ✓

A vertical flip is a reflection in the $x$-axis.

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 $y = f(-x)$.

Identify the transformation

$f(-x)$ reflects the graph in the $y$-axis. This negates the $x$-coordinate of every point.

$(1, 3) \to (-1, 3)$

Negate the $x$-coordinate only.

$(2, -1) \to (-2, -1)$

Negate the $x$-coordinate only.

$(0, 4) \to (0, 4)$ ✓

The $y$-axis ($x = 0$) maps to itself, points on the axis don't move.

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.

Reflection in the $x$-axis

Multiply the entire function by $-1$.

$$y = -f(x) = -(x^3 - 2x) = -x^3 + 2x$$

Distribute the negative sign across all terms.

Reflection in the $y$-axis

Replace every $x$ with $-x$.

$$y = f(-x) = (-x)^3 - 2(-x) = -x^3 + 2x$$ ✓

Both reflections give the same result because $f$ is an odd function$-f(-x) = f(x)$.

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.

$y = -f(x)$

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$y = f(-x)$

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$y = -f(-x)$

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$y = f(x) + 2$

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

Work mode · how are you completing this lesson?

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?

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.

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Multiple choice

+5 XP per correct · +25 XP all-correct

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

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) $y = -f(-x)$

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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.

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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.

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📖 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]

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 $-f(-x) = f(x)$.

Boss battle

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Five timed questions on reflections of functions. Beat the boss to bank a tier, gold (perfect + fast), silver (80%+), or bronze (cleared).

⚔ Enter the arena

Science Jump · reflections showdown

arcade practice

Challenge the boss using your knowledge of function reflections and transformations. Pool: lessons 1–10.

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Tick when you've finished the practice and review.

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