What is the primary difference between the transformations $y = -f(x)$ and $y = f(-x)$?

The transformation $y = -f(x)$ is a vertical reflection across the x-axis, whereas $y = f(-x)$ is a horizontal reflection across the y-axis. The former affects the output values (y-coordinates), while the latter affects the input values (x-coordinates).

How do the invariant points of a reflection across the x-axis differ from those of a reflection across the y-axis?

For a reflection across the x-axis ($y = -f(x)$), invariant points are the x-intercepts where $y = 0$. For a reflection across the y-axis ($y = f(-x)$), the invariant points are the y-intercepts where $x = 0$.

Compare the effect on the range of a function when applying $y = -f(x)$ versus $y = f(-x)$.

A vertical reflection $y = -f(x)$ flips the range values across the x-axis, potentially changing an interval like $[0, \infty)$ to $(-\infty, 0]$. In contrast, a horizontal reflection $y = f(-x)$ does not change the range at all, as it only alters the horizontal distribution of values.

What is a common mistake students make when identifying the axis of reflection for $y = -f(x)$?

Students often mistakenly associate the negative 'outside' the function with the vertical y-axis. In reality, a change in the $y$ value ($y \rightarrow -y$) means the graph moves up or down to the opposite side of the horizontal x-axis.

Why is it incorrect to assume that a reflection in the x-axis moves an asymptote $x = 3$?

An asymptote $x = 3$ is a vertical line. A reflection in the x-axis only changes y-coordinates, so vertical lines (including vertical asymptotes) remain unaffected unless the reflection is across the y-axis.

Explain the error in mapping the point $(4, 7)$ to $(-4, -7)$ when performing a reflection in the y-axis.

A reflection in the y-axis only negates the x-coordinate, mapping $(x, y)$ to $(-x, y)$. Negating both coordinates would represent a rotation of $180^\circ$ or a double reflection, which is not what a single y-axis reflection entails.

Define the algebraic transformation $y = -f(x)$ and its geometric effect.

Algebraically, $y = -f(x)$ represents the negation of the entire function's output. Geometrically, this results in a reflection across the x-axis, where every point $(x, y)$ is mapped to $(x, -y)$.

Define the algebraic transformation $y = f(-x)$ and its geometric effect.

Algebraically, $y = f(-x)$ represents substituting $-x$ for every $x$ in the function. Geometrically, this results in a reflection across the y-axis, where every point $(x, y)$ is mapped to $(-x, y)$.

What is an invariant point in the context of graph transformations?

An invariant point is a coordinate on a graph that remains unchanged after a transformation. In reflections, these points occur where the original graph intersects the line of reflection (the mirror line).

Why does $y = f(-x)$ flip the graph horizontally rather than vertically?

Since the negative sign is applied to the input $x$, it changes the 'direction' of the domain. Positive x-values now generate the outputs previously associated with negative x-values, and vice versa, creating a left-to-right mirror image.

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International A-Level

Reflections

Summary

Reflections are geometric transformations that flip a function's graph over a specified axis, altering the signs of coordinates without changing the shape's dimensions. Understanding the algebraic distinction between input negation and output negation is key to identifying whether a reflection occurs vertically across the x-axis or horizontally across the y-axis.

1. Definition & Core Concepts

Geometrical Nature: A reflection is a transformation where every point on the original graph is mapped to a corresponding point on the opposite side of a mirror line, which is usually a coordinate axis. The distance of each point from the mirror line remains constant, but its relative position is inverted.
Invariant Points: Points that lie directly on the line of reflection do not move during the transformation. These are called invariant points and are essential markers when sketching the reflected function because they provide anchor points where the two graphs intersect.
Preservation of Shape: Unlike stretches, reflections do not change the scale or size of the graph; they only change its orientation. The resulting graph is congruent to the original, meaning all lengths and angles between curves remain identical.

Diagram showing the original function f(x) and its reflection across the x-axis, -f(x), highlighting the invariant point where the curve crosses the axis.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions