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Function transformations modify a parent function's graph by shifting, stretching, compressing, or reflecting it, with rules like f(x) + k for vertical shifts, f(x + h) for horizontal shifts, a times a function of x for vertical stretches or compressions, and -f(x) for reflections. "Strict conditions" likely refers to applying transformations in a specific order, usually from left to right, to accurately represent the final transformed function.

Types of Transformations

Translation (Shifting): Moves the graph up, down, left, or right.
f(x) + k: Shifts the graph of f(x) up by k units.
f(x) - k: Shifts the graph of f(x) down by k units.
f(x + h): Shifts the graph of f(x) left by h units.
f(x - h): Shifts the graph of f(x) right by h units.

Dilation (Stretching or Compressing): Stretches or compresses the graph vertically or horizontally.
a f(x): Stretches or compresses the graph vertically. If the absolute value of a is greater than 1, it's a stretch. If 0 is less than the absolute value of a and less than 1, it's a compression.
f(ax): Stretches or compresses the graph horizontally.

Reflection: Mirrors the graph across the x-axis or y-axis.
-f(x): Reflects the graph of f(x) across the x-axis.
f(-x): Reflects the graph of f(x) across the y-axis.

Applying Transformations in Order
To avoid confusion, apply transformations in a specific order, typically from left to right, to ensure the correct final transformation:

Horizontal shifts: f(x - h) or f(x + h).

Horizontal stretches or compressions: f(ax).

Reflections across the x-axis: -f(x).

Vertical stretches or compressions: a f(x).

Vertical shifts: f(x) + k.

Example of Strict Application
Consider transforming the function g(x) into y = -2f(x - 3) + 1:

Horizontal shift: The (x - 3) inside f() shifts the graph of f(x) three units to the right.

Vertical stretch and reflection: The -2 outside the function stretches the graph vertically by a factor of 2 and reflects it across the x-axis.

Vertical shift: The +1 shifts the graph of -2f(x - 3) up by one unit.