Function Transformations • Topic 2 of 4

Reflections

A reflection flips a graph over an axis. Negating the whole function, −f(x), reflects it across the x-axis: every output changes sign, so the graph flips upside down. Replacing x with −x, giving f(−x), reflects it across the y-axis: the left and right sides swap. For example, y = x² reflected across the x-axis is y = −x², and y = √x reflected across the y-axis is y = √(−x). A function symmetric about the y-axis (like x²) looks unchanged under f(−x). The SAT tests which operation produces a given reflection: negate the output for the x-axis, negate the input for the y-axis.

sat33t2 graphOy = x²y = −x²Reflectiony = −x²: reflected across the x-axis

✅ Solved examples

1. Reflect y = x³ across the x-axis. New equation?
Negate the function: y = −x³.
2. How do you reflect a graph across the y-axis?
Replace x with −x: y = f(−x).
3. Reflect y = √x across the x-axis. New equation?
y = −√x.
4. Reflect y = 2x + 1 across the x-axis. New equation?
y = −(2x + 1).

✏️ Practice — try these, take hints as needed

1. Reflect y = x² + 1 across the x-axis. New equation?
Negate the whole function.
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y = −(x² + 1).
2. To reflect across the y-axis, you replace x with:
Negate the input.
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−x.
3. Reflect y = x³ across the x-axis. New equation?
Negate output.
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y = −x³.
4. Negating the entire function, −f(x), reflects across which axis?
Output flips sign.
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The x-axis.
5. Reflect y = 5x across the x-axis. New equation?
Negate.
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y = −5x.

📝 Topic test — 8 questions

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