Function graph transformations

Parametric equations

Start from a basic function $f$ . The general transformation is

y = a f (b (x - h)) + k

$h$ and $k$ translate the graph horizontally and vertically.
$a$ scales it vertically. When $a < 0$ , the graph flips upside down.
$b$ scales it horizontally. $∣ b ∣ > 1$ compresses the graph along $x$ , and $0 < ∣ b ∣ < 1$ stretches it. $b < 0$ reflects the input left to right. The graphs of $x$ and $x^{3}$ flip; $x^{2}$ and $∣ x ∣$ look the same.

The figure draws the faint base curve and the transformed curve together, so the same rule can be compared as the shape changes.

Basis f(x): switch among $x$ , $x^{2}$ , $x^{3}$ , and $∣ x ∣$ , and watch the same rule on different shapes
Horizontal shift h and Vertical shift k: drag the sliders, or drag P on the figure, and the graph moves left and right or up and down
Vertical scale a and Horizontal scale b: drag them and watch the opening and the horizontal compression or stretch. $a < 0$ flips the graph upside down. $b < 0$ reflects the input, which shows on $x$ and $x^{3}$

$h$ and $k$ move the graph and leave its shape alone. $a$ and $b$ change the opening and the width, and the sign of $a$ flips the graph upside down.
The same $(a, b, h, k)$ follows one rule on every basis. How bent $f$ is changes what that rule looks like.
The vertex form $a (x - h)^{2} + k$ is this transformation applied to $y = x^{2}$ .