Translations

A transformation changes the position, size, or orientation of a graph without changing which family it belongs to. We always start from a parent (base) function — such as y = x^{2}, y = |x|, y = \sqrt{x}, or y = \dfrac{1}{x} — and then slide, stretch, or flip it. This first section is about translations: sliding a graph rigidly, with no rotation, resizing, or reflection, so that every point moves the same distance in the same direction.

Adding a constant k outside the function raises or lowers the entire graph: y = f(x) + k shifts it up by k when k > 0 and down when k < 0. Every output changes by k, so each point (x, y) moves to (x,\ y + k). This one is intuitive — you are literally adding k to the height of every point.

A constant inside the bracket, y = f(x - h), shifts the graph right by h when h > 0 — the opposite of the sign you might expect. Why? To reproduce the output the base function gave at some input u, you now need x - h = u, i.e. x = u + h, a larger x — so the whole graph slides right. Rule of thumb: replace x with x - h to go right, and with x + h to go left.

Put them together and y = f(x - h) + k shifts the graph right h and up k. The cleanest way to track this is mapping notation: every point transforms as (x,\ y) \to (x + h,\ y + k). This single rule relocates intercepts, vertices, endpoints, and asymptotes all at once — just move the key points and re-draw.

One pattern governs every transformation you will meet this year: changes inside the bracket affect x (horizontal) and act opposite to their sign, while changes outside affect y (vertical) and act with their sign. Translations are the first instance; the stretches and reflections in the next sections obey the very same inside/outside logic, so learning it deeply now pays off repeatedly.

Because a translation moves the graph rigidly, the shape and orientation stay the same. Vertical shifts change the range (and the y-intercept, and the maximum/minimum value); horizontal shifts change the domain (and the x-intercepts, and the position of any vertical asymptote). For y = \sqrt{x - h} + k the domain becomes x \ge h and the range y \ge k; for y = \dfrac{1}{x - h} + k the asymptotes move to x = h and y = k.

Many equations hide a translation in plain sight. Vertex form y = (x - h)^{2} + k is the parabola y = x^{2} translated to have its vertex at (h, k). Given the standard form y = x^{2} + bx + c, complete the square to rewrite it as (x - h)^{2} + k and simply read off the shift. The same completing-the-square trick reveals the centre of a translated circle or the corner of a translated absolute-value graph.