Transformations of Functions: Vertical Translations

A vertical translation (also called a vertical shift) takes the graph of a function and slides it straight up or down, without stretching, flipping, or changing its shape in any way. Every single point on the graph moves the same distance in the same direction.

If you already know the graph of \(f(x)\), a vertical translation is one of the easiest transformations to apply, because it only affects the \(y\)-values of the function, not the \(x\)-values.

Given a function \(f(x)\) and a constant \(k > 0\), a vertical translation is written as:

\( g(x) = f(x) + k \) — shifts the graph of \(f(x)\) up by \(k\) units.

\( g(x) = f(x) - k \) — shifts the graph of \(f(x)\) down by \(k\) units.

In other words, every point \((x, y)\) on the original graph becomes \((x, \; y + k)\) on the translated graph. Notice that the constant is added or subtracted outside the function, after \(f(x)\) has already been evaluated. This is what tells you the shift is vertical, not horizontal. If the constant were added or subtracted inside the parentheses instead, you would be looking at a horizontal translation of the graph left or right.

Start with the parent function \(f(x) = x^2\). Its graph is a parabola with vertex at the origin \((0, 0)\).

Now consider \(g(x) = f(x) + 3 = x^2 + 3\). Every \(y\)-value of \(f(x)\) increases by 3, so the entire parabola moves up 3 units and the vertex lands at \((0, 3)\).

Because the shape of the curve does not change, the domain of \(g(x)\) is still all real numbers, exactly like \(f(x)\). But the range changes: \(f(x) = x^2\) has range \(y \ge 0\), while \(g(x) = x^2 + 3\) has range \(y \ge 3\). If you want a refresher on describing these outputs precisely, see how to write domain and range.

Now take \(f(x) = \sqrt{x}\), which starts at the origin and increases as \(x\) grows. Consider \(h(x) = f(x) - 2 = \sqrt{x} - 2\).

Here, \(k = 2\) is subtracted outside the square root, so the whole curve moves down 2 units. The starting point moves from \((0, 0)\) to \((0, -2)\). The domain stays \(x \ge 0\), since we are still only allowed to take the square root of non-negative numbers, but the range shifts from \(y \ge 0\) to \(y \ge -2\).

Vertical translations only reposition a graph up or down; they never flip it or change its width. This makes them different from reflections, which flip a graph across an axis, such as a reflection across the x axis, or from horizontal translations, which shift a graph left or right. When a function is written with several transformations combined, it helps to identify the vertical shift term (the constant added or subtracted outside the main expression) separately from any stretch, reflection, or horizontal shift, and apply it last, after the shape and left/right position are set.

To find the vertical shift in a transformed function, isolate the constant that sits completely outside the function's main operation.

• \(y = x^2 + 5\): shifted up 5 units (\(k = 5\))
• \(y = |x| - 7\): shifted down 7 units (\(k = -7\))
• \(y = 2^x + 1\): shifted up 1 unit (\(k = 1\))
• \(y = \sqrt{x}\): no vertical shift (\(k = 0\))

In each case, the value of \(k\) tells you exactly how far, and in which direction, the graph has moved compared to its parent function.

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