Vertical shift of a function calculator

Enter a rule such as f(x) = x^2, choose a positive, negative, or zero shift, and receive the transformed rule in standard notation. The result identifies the direction and distance of the movement, explains how every output changes, and can also move supplied key points. It is useful for checking homework, preparing examples, or applying the same transformation reliably in software.

How to write and apply a vertical shift

A vertical shift changes the output of a function while leaving its input untouched. If the original function is written as f(x), adding a constant k produces the new function g(x) = f(x) + k. Enter the original rule in the function field and put the signed constant in the shift field. A positive value moves the graph upward, while a negative value moves it downward. For example, shifting f(x) = x^2 by 5 gives g(x) = (x^2) + 5. Shifting it by -3 gives g(x) = (x^2) - 3. The calculator keeps the original expression inside parentheses so the added constant clearly applies to the entire output, even when the rule contains several terms. You may enter a bare expression, y followed by an equals sign, or a named rule such as h(x) = sin(x). The returned rule uses g(x) to distinguish the transformed function from the original one and also states the transformation in the familiar form g(x) = f(x) + k.

Why the graph moves up or down

Every point on the graph of a function has coordinates (x, f(x)). After a vertical shift by k, the corresponding point is (x, f(x) + k). The x-coordinate stays exactly the same because the input has not changed. Only the y-coordinate changes, and it changes by the same amount at every point. That uniform change is why the graph keeps its shape: peaks, valleys, intercept patterns, and curvature remain arranged in the same way, but the entire picture occupies a higher or lower position. When k is positive, every output is larger and the graph moves up by k units. When k is negative, every output is smaller and the graph moves down by the absolute value of k. A shift of zero causes no movement. If you provide key points, the calculator applies this coordinate rule directly, returning each original point and its new location so the symbolic rule and the geometric movement can be checked together.

Reading the result and avoiding common mistakes

The result separates the signed shift from the nonnegative distance. This matters because a shift of -4 means down 4 units, not down -4 units. It also reports a direction and a sentence describing the movement, making the sign convention visible instead of leaving it implicit. One common mistake is to alter x inside the formula, such as replacing x with x + 4. That creates a horizontal transformation and follows a different direction convention. For a vertical transformation, the constant belongs outside the original function: f(x) + k. Another mistake is to distribute the shift through individual terms. There is no need to modify each term because the constant changes the final output as a whole. This calculator intentionally preserves the entered expression rather than expanding or simplifying it, which keeps the operation transparent and suitable for functions involving powers, absolute values, trigonometric names, or several terms. Use the optional point mapping when you want a numerical cross-check of the new graph position.