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

$y = -f(x)$ is a reflection in the x-axis, which flips the graph vertically by changing the sign of the y-coordinates. In contrast, $y = f(-x)$ is a reflection in the y-axis, which flips the graph horizontally by changing the sign of the x-coordinates.

How do invariant points differ between x-axis and y-axis reflections?

In an x-axis reflection, invariant points occur where the graph crosses the x-axis ($y=0$). In a y-axis reflection, invariant points occur where the graph crosses the y-axis ($x=0$).

How does a reflection differ from a translation in terms of graph properties?

A translation shifts the entire graph to a new location without changing its orientation. A reflection flips the graph across an axis, which changes its orientation (handedness) while keeping it congruent to the original.

What is a common error when reflecting a function like $f(x) = x^2 + 3$ across the x-axis?

A common error is only negating the $x^2$ term (resulting in $-x^2 + 3$). To correctly reflect across the x-axis, the entire function must be negated: $-(x^2 + 3) = -x^2 - 3$.

What mistake do students often make regarding asymptotes during a y-axis reflection?

Students often forget to reflect the vertical asymptotes. If a graph has a vertical asymptote at $x = 2$, a reflection in the y-axis moves that asymptote to $x = -2$.

Why might a student incorrectly think a reflection has no effect on a graph?

This happens if the graph possesses line symmetry across the axis of reflection. For example, reflecting $y = x^2$ across the y-axis results in the same graph because the function is symmetric about that axis.

Define an 'invariant point' in the context of graph transformations.

An invariant point is a point on a graph that remains at its original coordinates after a transformation has been applied. For reflections, these points always lie on the axis of reflection.

What is the algebraic rule for a reflection in the x-axis?

The rule is $(x, y) \to (x, -y)$. This is represented by the function transformation $g(x) = -f(x)$.

What is the algebraic rule for a reflection in the y-axis?

The rule is $(x, y) \to (-x, y)$. This is represented by the function transformation $g(x) = f(-x)$.

If a function has a horizontal asymptote at $y = 5$, where is the asymptote after a reflection in the x-axis?

The horizontal asymptote moves to $y = -5$. This is because the transformation $y = -f(x)$ negates all y-values, including the limit the function approaches.

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Reflections

Summary

Reflections are geometric transformations that flip a function's graph across a specific line, known as the axis of reflection. This process changes the orientation of the graph while preserving its shape and size, resulting in a congruent image that is a mirror of the original.

1. Definition & Core Concepts

A reflection is a transformation that maps every point $(x, y)$ on a graph to a new position by 'flipping' it over a line of reflection.
The resulting graph is congruent to the original, meaning the dimensions and shape remain identical, but the orientation is reversed.
In function notation, reflections are achieved by applying a negative sign either to the entire output of the function or to the input variable itself.

Diagram showing a reflection of a curve across the x-axis. The original red curve f(x) is mirrored by the blue curve -f(x), with vertical dashed lines showing the mapping of points.

2. Reflection in the x-axis

3. Reflection in the y-axis

4. Invariant Points

5. Impact on Asymptotes

6. Key Distinctions

7. Exam Strategy & Tips