Dilations and Reflections

Dilations change scale without changing shape. If you write y = a f(x), every output gets multiplied by a, so the graph stretches away from the x-axis when a is greater than 1, and squeezes toward it when a is between 0 and 1. If instead you write y = f(bx), you're rescaling the input before the function even sees it. Since x is being multiplied by b before plugging in, the graph actually squeezes horizontally when b is greater than 1, and stretches when b is between 0 and 1 — the opposite feel from the vertical case, because you're working backward from the output to find which x produces it.

A negative sign does the same rescaling but by a factor of -1, which flips the graph instead of resizing it. A negative out front, y = -f(x), negates every output, flipping the graph over the x-axis. A negative inside, y = f(-x), negates every input, flipping it over the y-axis. Combine a stretch and a flip, like y = -3x², and you get both effects at once: the 3 pulls points away from the x-axis three times farther, and the negative sign turns the upward parabola into a downward one.

These transformations matter because the leading coefficient of a polynomial is really just a vertical dilation and possible reflection of the parent power function, and it's what determines end behavior far from the origin. Horizontal dilations matter differently: they rescale the domain or range's speed but never move its boundary, which is why √x stretched or squeezed horizontally still starts at x = 0.