Reflections of Functions - Algebra-2

1. Fundamental Concepts

Reflections of functions are an important form of function graph transformation, referring to the symmetric flipping of a function's graph with respect to coordinate axes (or other lines). Common types of reflections include:
- Reflection about the x-axis: When the graph of the function \(y = f(x)\) is reflected about the x-axis, the expression of the new function obtained is \(y=-f(x)\). At this time, each point \((x,y)\) on the graph becomes \((x,-y)\).
- Reflection about the y-axis: When the graph of the function \(y = f(x)\) is reflected about the y-axis, the expression of the new function obtained is \(y = f(-x)\). At this time, each point \((x,y)\) on the graph becomes \((-x,y)\).

Reflection transformations do not change the shape of the function, only the position and direction (up-down or left-right flipping) of the function's graph.
When reflecting about the x-axis, the function value becomes the opposite of the original, that is, the sign of the ordinate of all points changes.
When reflecting about the y-axis, the independent variable becomes the opposite of the original, that is, the sign of the abscissa of all points changes.
Reflections can be combined with other transformations (such as translations). In such cases, the transformation should be carried out in the order of "reflection first, then translation" or "translation first, then reflection", which depends on the function expression.

Easy: If the function \(f(x)=2x + 1\), the expression of the function after reflection about the x-axis is \(y=\underline{\quad\quad}\). (Answer: \(-2x-1\))
Medium: If the function \(f(x)=x^2-3\), the expression of the function after reflection about the y-axis is \(y=\underline{\quad\quad}\). (Answer: \((-x)^2-3\) or \(x^2-3\))
Hard: If the function \(f(x)=3x + 2\) is first reflected about the y-axis and then about the x-axis, the expression of the resulting function is \(y=\underline{\quad\quad}\). (Answer: \(3x-2\))

Clarify the type of reflection: Determine whether it is a reflection about the x-axis or the y-axis, and identify the corresponding function transformation rule (\(y=-f(x)\) or \(y = f(-x)\)).
Handle composite transformations step by step: If reflection is combined with transformations such as translation, first perform the reflection transformation according to the rules, and then carry out other transformations; or derive step by step in accordance with the operation order of the expression.
Substitution verification: For the expression of the transformed function, several special points on the original function (such as intersection points with the coordinate axes) can be selected, corresponding points can be obtained through reflection rules, and substituted into the new function expression to verify whether it holds, so as to ensure the correctness of the result.