Function Transformations — SAT Math Explained
Changes to a parent function that shift, reflect, stretch, or compress its graph. These transformations are encoded directly in the equation and follow predictable rules.
The Core Idea
Rather than re-analyzing every modified function from scratch, transformations let you take a function you know and predict exactly how its graph will change based on modifications to its equation.
Parent Functions
f(x) = x — straight line through origin
f(x) = x² — upward parabola with vertex at origin
f(x) = |x| — V-shape with vertex at origin
f(x) = √x — starts at origin, curves upward to the right
Transformation Rules
f(x) + k shifts UP by k; f(x) - k shifts DOWN by k
f(x - h) shifts RIGHT by h; f(x + h) shifts LEFT by h (counterintuitive — the sign inside is opposite!)
-f(x) reflects over the x-axis (flips up-down)
f(-x) reflects over the y-axis (flips left-right)
a·f(x) with |a| > 1 stretches vertically (makes graph taller/narrower)
a·f(x) with 0 < |a| < 1 compresses vertically (makes graph shorter/wider)
Order Of Transformations
Apply in this order: horizontal transformations (inside), stretches/compressions, reflections, then vertical shifts (outside)
Vertex Form Connection
y = a(x - h)² + k is a transformation of y = x²: horizontal shift h, vertical shift k, stretch/compress by a
Common Errors to Avoid
Shifting horizontally in the wrong direction — f(x - 3) shifts RIGHT, not left
Confusing vertical and horizontal reflections
Applying transformations in wrong order
Practice: Function Transformations
5 SAT-style questions. Select your answer and get an instant explanation.
If f(x) = x², what is the graph of g(x) = x² + 3?
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