2.9.3 Reflections

Reflections are a type of transformation in mathematics where a function is "flipped" across a specific axis. Understanding reflections is crucial for analysing and manipulating functions, especially with graphical representations.

Key Reflections

1. Reflection in the $\ x$-axis:

• The reflection of a function $\ f(x)$in the $x$ -axis is given by $\ y = -f(x)$.
• Effect: Each point $(x, y)$ on the graph of $\ f(x)$ is mapped to $\ (x, -y)$ on the graph of $\ -f(x) .$

Graphical Impact:

• The entire graph flips vertically, making all positive $\ y$-values negative and vice versa.

2. Reflection in the $\ y$-axis:

• The reflection of a function $\ f(x)$ in the $\ y$-axis is given by $\ y = f(-x)$.
• Effect: Each point $\ (x, y)$ on the graph of $\ f(x)$ is mapped to$\ (-x, y)$ on the graph of$\ f(-x) .$

Graphical Impact:

• The entire graph flips horizontally, making all points that were to the right of the $\ y$-axis move to the left, and vice versa.

Combining Reflections:

Sometimes, a function may undergo multiple reflections. For example:

• Reflection in both axes:
  • Reflecting $\ f(x)$ in both the$\ x$-axis and the $\ y$-axis results in $\ y = -f(-x)$.
  • Effect: This is equivalent to rotating the graph 180 degrees around the origin.

Worked Example:

Practice Problem:

Solution:

• Reflection in the $\ x$-axis: $\ y = -\frac{1}{x}$.
• Further reflection in the $\ y$-axis: $\ y = -\frac{1}{-x} = \frac{1}{x}$.
• Interestingly, reflecting $\ \frac{1}{x}$ in both axes brings it back to the original function $\ \frac{1}{x}$. This shows how understanding reflections can help predict and analyse functions' behaviour.

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