Overview
Watch
Read
Next Steps
Overview
Grade 12 Advanced Functions, University Preparation (MHF4U)
A. Exponential and Logarithmic Functions
7. 12AF.A2.3
7.4 Reflection across the x-axis: y = -f(x)
Reflection Across the x-axis: y = -f(x)
See exactly how the rule y = -f(x) flips a graph over the x-axis, with clear steps and worked examples for parabolas and absolute value graphs.
What You'll Learn
Reflecting a graph across the x-axis multiplies every output value by negative one, giving the new equation y = -f(x).
Every point (x, y) on the original graph moves to (x, -y) on the reflected graph, so the x-coordinates never change.
Maximum points become minimum points and minimum points become maximum points, while x-intercepts stay exactly where they were.
The domain of the reflected function is the same as the original, but the range flips sign.
A common mistake is negating the input instead of the output; -f(x) is not the same as f(-x), which reflects across the y-axis instead.
What You'll Practice
1
Reflecting graphs across the x-axis by transforming coordinate points
2
Converting positive y-values to negative y-values and vice versa
3
Plotting reflected points and sketching transformed functions
4
Verifying reflections by comparing original and reflected coordinates
Why This Matters
Mastering x-axis reflections is essential for understanding function transformations throughout algebra and precalculus. You'll use this skill to graph absolute value functions, analyze trigonometric functions, and solve real-world problems involving symmetry and inverse relationships.
Before You Start — Make Sure You Can:
This Unit Includes
2 Video lessons
Learning resources
Skills
Reflections
Function Transformations
Coordinate Geometry
Graphing
Invariant Points
Negative Functions
ON Curriculum Aligned