Reflections Across the X-Axis
Reflecting a function across the x-axis multiplies it by −1: g(x) = −f(x). In Grade 11 enVision Algebra 1 (Chapter 10: Working With Functions), students learn that this transformation changes the sign of every y-coordinate — turning every point (x, y) into (x, −y) — while leaving all x-coordinates unchanged. Positive output values become negative and vice versa, effectively flipping the graph upside down. This is the foundational transformation for understanding negative coefficients in stretched or reflected functions.
Key Concepts
When a function is multiplied by $ 1$, the graph reflects across the x axis: $$g(x) = f(x)$$.
Every point $(x, y)$ on the original graph becomes $(x, y)$ on the reflected graph.
Common Questions
What is the rule for reflecting a function across the x-axis?
Multiply the function by −1: g(x) = −f(x). Every point (x, y) on the original maps to (x, −y) on the reflected graph.
What coordinates change in an x-axis reflection?
Only the y-coordinates change sign. The x-coordinates remain the same.
If f(3) = 5, what is the reflected function's value at x = 3?
For g(x) = −f(x), g(3) = −f(3) = −5. The output changes sign.
What does an x-axis reflection look like visually?
The graph is flipped upside down — peaks become valleys and valleys become peaks, while the horizontal positions of all points are unchanged.
How does x-axis reflection interact with vertical stretch?
When a = −2, the function g(x) = −2f(x) both reflects across the x-axis and stretches vertically by factor 2 simultaneously.
What is the x-axis reflection of f(x) = x²?
The reflection is g(x) = −x², which opens downward instead of upward.