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 rule relocates every point, including vertices and endpoints, and moves asymptotes. Images of old intercepts need not remain on the axes; find the new intercepts by setting $x=0$ or $y=0$.

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.