Grades Geometry & Functions Functions — AdvancedTransformations of Functions

Transformations of Functions

📊 Functions — Advanced

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Transformations of Functions

Shift, stretch, and reflect parent functions systematically.

Given a parent function f(x), we can produce a family of related functions through transformations. Vertical and horizontal shifts, vertical and horizontal stretches/compressions, and reflections all have predictable effects on the graph.

g(x) = a\cdot f(b(x - h)) + k

Transformation Parameters

k: vertical shift up by k (down if k < 0)h: horizontal shift right by h (left if h < 0)a: vertical stretch by |a|; reflects over x-axis if a < 0b: horizontal compression by factor b; reflects over y-axis if b < 0

Vertical Changes (outside f)

f(x) + k → shift up kf(x) − k → shift down ka·f(x) → stretch/compress vertically−f(x) → reflect over x-axis

Horizontal Changes (inside f)

f(x − h) → shift right hf(x + h) → shift left hf(bx) → compress horizontallyf(−x) → reflect over y-axis

Horizontal transformations always work in the opposite direction of what you might expect.

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Remember This!

Apply transformations in order: horizontal shifts and stretches first (inside the function), then vertical stretches, then vertical shifts.

✏️ Try It!

How does the graph of g(x) = √(x − 3) + 2 differ from f(x) = √x?

Practice with CalcVerse

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