Pay close attention to whether the change is *inside* the function (affecting x) or *outside* the function (affecting y). Inside changes affect the x-values, leading to horizontal transformations. A horizontal shift is of the form f(x + a) or f(x - a), while a horizontal stretch/compression is of the form f(kx).
The most common mistake is applying the reflections in the wrong order or forgetting to apply one of them. Remember to apply each reflection sequentially. Reflecting over the x-axis changes y to -y, and reflecting over the y-axis changes x to -x.
Follow the order of operations in reverse when interpreting transformations. Address horizontal shifts first, then stretches/compressions/reflections, and finally vertical shifts. For example, in y = a*f(b(x - h)) + k, the order is: horizontal shift (h), horizontal stretch/compression (b), vertical stretch/compression/reflection (a), and vertical shift (k).
If the function is multiplied by -1 *outside* (i.e., -f(x)), its a reflection across the x-axis. If the x-variable is multiplied by -1 *inside* (i.e., f(-x)), its a reflection across the y-axis.
Vertical stretches/compressions affect the y-values (the output), so they are applied *outside* the function: a*f(x). Horizontal stretches/compressions affect the x-values (the input), so they are applied *inside* the function: f(bx). If |a| > 1, its a vertical stretch; if 0 < |a| < 1, its a vertical compression. If |b| > 1, its a horizontal compression; if 0 < |b| < 1, its a horizontal stretch.
Graph the original function and the transformed function using graphing software or a calculator. Compare key points (e.g., intercepts, maximum/minimum points) on both graphs to see if they have been transformed as expected. Also, substitute a few x-values into both the original and transformed functions and verify that the resulting y-values correspond to the applied transformations.