Function transformations

A transformation of the graph of a function shifts it, stretches or compresses it, or reflects it. With four parameters the transformed function is $g(x) = a\,f(b(x - h)) + k$, where $b \ne 0$. By default this page uses a polyline $f$ with no symmetry, so every transformation shows; seven common parent functions are there too: $x^2$, $x^3$, $\sqrt{x}$, $1/x$, $|x|$, $2^x$ and $\sin x$.

The dashed curve is the parent graph $y = f(x)$, the solid curve is $y = g(x)$. A point P = $(x_0, f(x_0))$ on the parent and its image P′ on $g$ are joined by red arrows, one for each step. Every point of the graph moves by the same rule:

Why the inside acts the opposite way: a point $(x, y)$ lies on the graph of $g$ when $x_0 = b(x - h)$ is an input of $f$ and $y = a\,f(x_0) + k$. Solving for the new x-coordinate gives $x = x_0/b + h$: the shift $h$ is added, but the old coordinate is divided by $b$. So $h = 2$ moves the graph right although the formula shows $x - 2$, and $b = 2$ halves its width. The outside parameters act on the output directly.

Also written $y = f(x + \varphi)$: for $\varphi > 0$ the graph moves $\varphi$ to the left, which is $h = -\varphi$ here. The sinusoid $y = A\sin(\omega x + \varphi) + k$ is a special case of this page too: $a = A$, $b = \omega$, $h = -\varphi/\omega$.

The red button builds $g$ from $f$ step by step: inside the bracket (horizontal) first, then outside (vertical). Inside there are two ways, chosen with the lever beside the button. Scale first: divide the x-coordinates by $b$ (a horizontal stretch or compression relative to the y-axis, with a reflection across it when $b < 0$), then add $h$. Shift first: add $bh$ to the x-coordinates, then divide them by $b$ — which divides the shift too, leaving exactly $h$. Outside: multiply the y-coordinates by $a$ (a vertical stretch or compression relative to the x-axis, with a reflection across it when $a < 0$), then add $k$. Steps whose parameter changes nothing are skipped.

Both ways end on the same $g$. After each run the graphs on the way stay, faint, until a parameter changes; throw the lever and press again, and the two ways lie side by side.

In each direction the stretch comes before the shift when the shift is written as $h$ or $k$: shifting by $h$ first and then dividing the x-coordinates by $b$ divides the shift as well, and the graph ends up shifted by $h/b$. (Shifting first works if the shift is $bh$ — the lever’s Shift side.) Horizontal and vertical steps do not affect each other, so either direction may go first.