In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = √x + 1

Key Concepts

Square Root Function

The square root function, f(x) = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph is a curve that starts at the origin (0,0) and increases gradually, reflecting the relationship between x and its square root. Understanding this function is crucial as it serves as the foundation for applying transformations.

Graph Transformations

Graph transformations involve shifting, stretching, compressing, or reflecting the graph of a function. In this case, the transformation applied to f(x) = √x to obtain g(x) = √x + 1 is a vertical shift upwards by 1 unit. Recognizing how these transformations affect the graph is essential for accurately sketching the new function.

Vertical Shift

A vertical shift occurs when a constant is added to or subtracted from a function's output. For g(x) = √x + 1, the '+1' indicates that every point on the graph of f(x) = √x is moved up by one unit. This concept is vital for understanding how the original function's graph is altered to create the new function.