For every point with coordinates (x, y) on y=f(x), the shifted graph has a corresponding point at (x, y+k). The x-coordinate stays fixed, while the output increases or decreases by the same amount k. Applying this rule to all points preserves the graph’s shape and horizontal placement.
The constant k controls both features. Its sign determines direction: a positive value raises every output, whereas a negative value lowers every output. Its absolute value determines the distance moved. Because the same adjustment applies to every output, the transformation does not stretch, compress, or otherwise change the graph’s shape.
A vertical shift changes every value in the range by k, so the new range consists of the original outputs with that constant added. The y-intercept also moves vertically: the original value f(0) becomes f(0)+k. The domain remains unchanged because the transformation does not alter which x-values are available.
First identify the constant k outside the function expression. Select points or key features on the original graph, keep each x-coordinate unchanged, and add k to every y-coordinate. Plot the resulting points and connect them with the same shape as the original. This procedure gives the translated graph without rebuilding the function.
Compare the equations for an unchanged input expression and a constant added outside it. If one graph can be written as y=f(x)+k relative to the other, corresponding points share x-coordinates and differ in output by k. This comparison helps determine both the direction and distance separating the two graphs.
In a mathematical model, adding a constant can represent a fixed increase or decrease in every predicted output. The model’s input values and horizontal position stay the same, while its outputs move consistently by k. Interpreting that constant allows students to connect an equation change with a uniform adjustment in the modeled quantities.