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. h(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 base for applying transformations.

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In this case, the function h(x) = -√(x + 1) involves a horizontal shift to the left by 1 unit and a reflection across the x-axis. Mastery of these transformations allows for the accurate graphing of new functions based on known ones.

Reflection Across the X-Axis

Reflection across the x-axis occurs when the output values of a function are negated. For the function h(x) = -√(x + 1), this means that every point on the graph of f(x) = √(x + 1) is mirrored over the x-axis. This transformation changes the direction of the graph, flipping it upside down, which is essential for accurately representing the function.