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.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
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.
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Wrong: √(a + b) = √a + √b.
Right: The square root of a sum is not the sum of square roots; √(a+b) cannot be simplified this way.
📚 Core Content
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.
Multiplying the entire function by $-1$ flips the graph upside down. Every point $(x, y)$ on the original graph moves to $(x, -y)$.
Replacing $x$ with $-x$ flips the graph left-to-right. Every point $(x, y)$ on the original graph moves to $(-x, y)$.
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 $(-x, -y)$.
🧮 Worked Examples
🧪 Activities
1 $y = -f(x)$
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2 $y = f(-x)$
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3 $y = -f(-x)$
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4 $y = f(x) + 2$
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1 $y = -f(x)$
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2 $y = f(-x)$
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3 $y = -f(-x)$
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4 $y = f(x - 1)$
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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.
Now revisit your initial response. What did you get right? What has changed in your thinking?
Look back at your initial response in your book. Annotate it with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
5 random questions from a replayable lesson bank — feedback shown immediately
✍️ Short Answer
8. 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) $y = -f(-x)$ 3 MARKS
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9. 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. 4 MARKS
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10. 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. 3 MARKS
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Answer in your workbook.
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
1. $(3, 2)$ — $y$-coordinate sign flips
2. $(-3, -2)$ — $x$-coordinate sign flips
3. $(-3, 2)$ — both coordinates flip
4. $(4, -2)$ — horizontal translation 1 unit right
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.
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(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 $-f(-x) = f(x)$.
Challenge the boss using your knowledge of function reflections and transformations. Pool: lessons 1–10.
Tick when you've finished all activities and checked your answers.