Transformations of functions
y = a·f(b(x − h)) + k: a and k act on y, outside the function; b and h act on x, inside it, and b stretches by its reciprocal.
A worked example
From y = f(x) to y = −2f(3(x + 1)) − 4: a = −2 is a vertical stretch by 2 and a reflection in the x-axis; b = 3 is a horizontal stretch by 1/3; x + 1 is x − (−1), so 1 left; k = −4, so 4 down. The point (6, 5) goes to (6/3 − 1, −2 × 5 − 4) = (1, −14).
The mistake people actually make
Inside the brackets everything is backwards: f(x − 3) moves right, and f(2x) squeezes to half the width. Do the stretches and reflections before the translations.