Combining Transformations

Every transformed function y = a f(b(x - h)) + k is built from four moves on the parent graph y = f(x). The number a stretches or compresses vertically (and flips the graph over the x-axis if a is negative). The number b stretches or compresses horizontally (and flips over the y-axis if negative). h shifts the graph left or right, and k shifts it up or down. The order you apply them matters: do the stretches and reflections tied to a and b first, since they act directly on f(x), then do the shifts h and k last, since those move the whole already-stretched graph to its final position.

The reason order changes the outcome is that a stretch multiplies distances from the origin. If you shift first and then stretch, the stretch also grabs the shift amount and scales it, moving the graph further than you meant. If you stretch first and shift second, the shift amount stays exactly what you intended, since nothing acts on it afterward. That is why the standard form keeps h and k on the outside: it bakes in the correct order.

This topic ties together everything from earlier in the unit, since every polynomial and rational parent function, from x² to 1/x to x³, gets graphed using this same four-step recipe. Later, when you compare end behavior or build new functions from these pieces, you will keep reading a, b, h, and k off an equation the same way, so getting the order locked in now pays off for the rest of the unit.