Reflections: Flipping a Graph Across an Axis

Two minus signs, two completely different results. Where the sign sits — outside the function or inside its bracket — decides whether the graph flips top to bottom or left to right.

Reflection Across the x-Axis

Negate the output and the graph flips top to bottom.

Whatever was above the -axis is now the same distance below it, and vice versa. Points that were on the axis do not move at all — they are their own reflection.

Reflection Across the y-Axis

Negate the input and the graph flips left to right.

This is the inside-outside rule again: a change inside the bracket acts horizontally, a change outside acts vertically.

The Two Side by Side

Using an off-centre curve makes the difference obvious. A parabola with vertex reflects to across the -axis, and to across the -axis — two clearly different graphs.

Try this with a symmetric parent such as and you learn nothing: the -axis reflection leaves it completely unchanged. That is not a failure of the rule, it is symmetry — and it is precisely how symmetry is defined.

What Each Does to a Point

“Flipped” for an interval means negating and swapping the endpoints: a domain of becomes .

Both at Once

Applying both gives , which sends to . That is not a third kind of reflection — it is a 180° rotation about the origin.

Order does not matter here: reflecting vertically then horizontally gives the same result as the other way round, because the two operations touch different coordinates.

Reflections Hiding in Vertex Form

A negative in is a reflection. is a parabola stretched by 2 and reflected, which is why it opens downward.

The same applies to , and . A negative sign in front of any parent function flips it across the -axis.

Careful with the vertex, though. In vertex form the reflection is across the horizontal line through the vertex, not across the -axis itself — because the is applied after the negation. The vertex stays at ; only the arms flip.

Worked Example A: Which Reflection?

Describe .

Reflected across the -axis. Heights negated, domain unchanged.

Worked Example B: The Other One

Describe .

Reflected across the -axis. Horizontal positions negated, range unchanged.

Worked Example C: On a Concrete Function

If , write and describe both reflections.

: the curve falls to the right instead of rising. Domain still ; range becomes .

: the curve runs to the left from the origin. Domain becomes ; range still .

Worked Example D: Domain and Range

has domain and range . Give both for .

A -axis reflection flips the domain: .

The range is unchanged: .

Worked Example E: A Reflection That Does Nothing

What is when ?

Identical to the original. The graph is unchanged because it is already symmetric about the -axis — an even function, covered in symmetry.

Common Mistakes to Avoid

• Swapping the two. Outside the bracket is the -axis; inside is the -axis.
• Writing as . They are different functions with different graphs.
• Flipping the wrong coordinate. negates ; negates .
• Testing on a symmetric parent. hides the -axis reflection entirely.
• Forgetting to swap interval endpoints. flips to , not .
• Thinking is a new reflection. It is a 180° rotation.
• Reflecting the vertex in vertex form. With applied after, the vertex stays put and only the arms flip.

Where Reflections Lead Next

• Symmetry. A graph unchanged by a reflection is symmetric — see symmetry for the even and odd tests.
• Inverse functions. An inverse is a reflection across the line , a third mirror beyond the two axes.
• The full toolkit. Function transformations fits reflections alongside shifts and stretches.
• Downward parabolas. Every negative leading coefficient is a reflection in disguise.
• Absolute value as a reflection. reflects only the part below the axis, leaving the rest alone.

Practice Problems

Work each one before opening the answer.

Problem 1. Which axis does reflect across?

Problem 2. Which axis does reflect across?

Problem 3. Where does go under ?

Problem 4. Where does go under ?

Problem 5. Where does go under ?

Problem 6. Write reflected across the -axis.

Problem 7. has range . What is the range of ?

Problem 8. has domain . What is the domain of ?

Problem 9. What is when ?

Problem 10. What is when ?

Problem 11. Describe .

Problem 12. Is the same as reflecting twice, and what single move does it equal?

Quick Reference

Reflections are the flipping half of function transformations, alongside horizontal and vertical shifts. A graph that survives one unchanged has symmetry, and reflecting across instead of an axis produces an inverse function. More Algebra lessons are available.