Transformations of Functions
A transformed function is a parent function with four dials: two outside constants that act on outputs exactly as written, and two inside constants that act on inputs in reverse. f(x + 3) moves left, not right; f(2x) compresses, not stretches; and an expression like f(2x + 6) refuses to tell you its shift until you factor it into f(2(x + 3)). Outputs obey the notation; inputs contradict it.
The mistakes here are all direction and order: shifting the wrong way on inside changes, applying an inside change to the output, and adding before stretching when the formula stretches before adding. The lesson drills the one habit that beats all three: set the inside equal to the old input and let the algebra decide.
The work
3 ways in · any order
Lesson
Transformations of Functions
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Outside constants act on outputs as written; inside constants act on inputs in reverse. The lesson builds the four-dial picture of af(b(x-h))+k, shows why the inside runs backward, then closes with ten scenarios: track points, domains, and ranges through shifts, stretches, and reflections.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.12 misconception: horizontal shifts read in the wrong direction, inside dilations applied to outputs, and unfactored insides like f(2x + 6) read as a shift of 6. The bank ships with the Unit 1 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.