Transformation Rules for Functions - Algebra-2

1. Fundamental Concepts

Function transformation rules describe how modifying a parent function’s equation affects its graph. These transformations include shifts (horizontal/vertical), stretches/compressions (horizontal/vertical), and reflections (over x-axis/y-axis). They are summarized by the general form of a transformed function:
where each parameter () and sign changes correspond to specific changes in the graph’s position, size, or orientation.

The general form breaks down into transformations affecting x-coordinates and y-coordinates of the parent function :

Reflection over x-axis: A negative sign in front of a (i.e., ) reflects the graph over the x-axis (flips vertically).
Vertical stretch/compression: The coefficient a determines vertical scaling:
- : Vertical stretch (graph becomes taller).
- : Vertical compression (graph becomes shorter).
Vertical shift: The term $\pm k$ shifts the graph up or down:
- (where ): Shifts up by k units.
- (where ): Shifts down by k units.

Reflection over y-axis: A negative sign in front of b (i.e., ) reflects the graph over the y-axis (flips horizontally).
Horizontal stretch/compression: The coefficient b determines horizontal scaling:
- : Horizontal compression (graph becomes narrower).
- : Horizontal stretch (graph becomes wider).
Horizontal shift: The term $(x - h)$ shifts the graph left or right:
- with : Shifts right by h units.
- (equivalent to ) with : Shifts left by h units.

Given the parent function , what is the equation of the function after a vertical stretch by a factor of 4 and a shift up by 3 units?

Solution:

The parent function is . Describe the transformations applied to get , and identify the new vertex (the vertex of is ).

Solution:

$-2f(x - 1) + 5$ breaks down as:
- : Reflect over the x-axis and vertical stretch by 2.
- : Shift right by 1 unit.
- : Shift up by 5 units.
Original vertex $(0, 0)$ transforms as:
- Right 1: .
- Up 5 and reflected/stretched (y-coordinate: ).
New vertex: .

Decompose the Transformation Equation: Break down the function into individual components ( and signs) to identify each transformation (stretch, shift, reflection).
Reverse Engineering for Points: To find a point on the transformed function, work backwards from the parent function’s point:
- For x-coordinates: Adjust for horizontal shifts first, then stretches/compressions/reflections.
- For y-coordinates: Adjust for stretches/compressions/reflections first, then vertical shifts.
Use the General Form as a Checklist: Verify each parameter against the rules (e.g., = horizontal compression) to avoid mixing up horizontal/vertical transformations.
Test with Key Points: Use critical points of the parent function (e.g., vertices, intercepts) to track how transformations affect the graph, ensuring consistency with the rules.