Horizontal Translations of Functions

A horizontal translation is a transformation that slides every point on the graph of a function to the left or to the right, without stretching, flipping, or reshaping the curve. If you already know how vertical shifts work, horizontal ones follow a similar idea, except the change happens inside the function, not outside it.

For a function \(y = f(x)\), a horizontal translation is written as:

\(y = f(x - h)\)

where \(h\) is a constant. The entire graph of \(f(x)\) moves \(h\) units horizontally, but every \(y\)-value stays exactly the same as it was on the original graph — only its \(x\)-position changes.

This is the part that trips most students up, so it is worth stating plainly:

• \(y = f(x - h)\) with \(h > 0\) shifts the graph right by \(h\) units.
• \(y = f(x + h)\) with \(h > 0\) shifts the graph left by \(h\) units.

Notice the direction is opposite to what the sign looks like. Subtracting inside the function moves the graph to the right, and adding inside the function moves it to the left. A quick way to check this: the graph of \(y = f(x - h)\) reaches the same output as \(f(x)\) when \(x - h\) equals the original input, which means \(x\) has to be \(h\) units larger. That is exactly what "shifting right" means.

Start with the parent function \(f(x) = x^2\). Its graph is the familiar parabola with a vertex at the origin.

Now compare it to \(f(x - 3) = (x - 3)^2\). Every point on the parabola has moved 3 units to the right, so the vertex that used to sit at \(x = 0\) now sits at \(x = 3\).

And \(f(x + 3) = (x + 3)^2\) moves the same parabola 3 units to the left, putting the vertex at \(x = -3\).

Suppose \(f(x) = \sqrt{x}\), and you want to graph \(g(x) = \sqrt{x - 4}\).

1. Compare \(g(x)\) to \(f(x)\): the input is \(x - 4\), so \(h = 4\).
2. Since the sign is subtraction and \(h > 0\), the graph shifts right by 4 units.
3. The starting point of \(f(x) = \sqrt{x}\) is at \((0, 0)\), so the starting point of \(g(x)\) moves to \((4, 0)\).
4. Every other point follows the same shift: a point like \((1, 1)\) on \(f(x)\) becomes \((5, 1)\) on \(g(x)\).

A second example: for \(h(x) = f(x + 2)\) where \(f(x) = \sqrt{x}\), the shift is 2 units to the left, so the starting point moves from \((0,0)\) to \((-2, 0)\).

A horizontal translation only affects the \(x\)-coordinates of a function, so it always changes the domain of a function while leaving the range untouched. For example, shifting \(f(x) = \sqrt{x}\) (domain \(x \ge 0\)) right by 4 units gives a new domain of \(x \ge 4\), but the range stays \(y \ge 0\) in both cases.

Horizontal translations also preserve properties like whether a function is one-to-one, since sliding a graph sideways cannot create or remove repeated outputs. This is different from transformations like a reflection across the x axis, which flips the graph and can change its increasing or decreasing behavior instead of its position.

Horizontal and vertical translations often appear together. The general form

\(y = f(x - h) + k\)

shifts the graph \(h\) units horizontally and \(k\) units vertically. For instance, \(y = (x - 3)^2 + 2\) moves the parabola 3 units right and 2 units up, giving a new vertex at \((3, 2)\). The horizontal and vertical parts act independently, so you can handle each one separately and then combine the results.

• Reading the sign literally: assuming \(f(x - 3)\) shifts left because of the minus sign. It shifts right.
• Forgetting that the shift happens to the input, so it must be applied to every \(x\)-coordinate, not just to specific points you happen to plot.
• Mixing up a horizontal shift with a horizontal stretch or compression, which involves multiplying \(x\) rather than adding or subtracting a constant from it.

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