Begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. g(x) = 2√(x+1)

Verified step by step guidance

1. Start by identifying the base function, f(x) = √x. This is the square root function, which has a domain of x ≥ 0 and a range of y ≥ 0. Its graph starts at the origin (0, 0) and curves upward to the right.
2. Next, analyze the given function, g(x) = 2√(x+1). Notice that this function is a transformation of the base function f(x). The transformations include a horizontal shift, a vertical stretch, and a vertical scaling.
3. The term (x+1) inside the square root indicates a horizontal shift. Specifically, the graph of f(x) = √x is shifted 1 unit to the left because of the +1 inside the square root.
4. The coefficient 2 outside the square root represents a vertical stretch. This means that the y-values of the graph are multiplied by 2, making the graph steeper.
5. To graph g(x), start by shifting the graph of f(x) = √x one unit to the left. Then, stretch the graph vertically by multiplying the y-values by 2. Plot key points such as (0, 0), (-1, 0), and others to visualize the transformation.

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. For example, adding a constant inside the function's argument shifts the graph horizontally, while multiplying the function by a constant stretches or compresses it vertically. These transformations allow us to manipulate the basic shape of the square root function to create new functions like g(x) = 2√(x+1).

Horizontal and Vertical Shifts

Horizontal and vertical shifts are specific types of transformations that affect the position of a graph. A horizontal shift occurs when a constant is added or subtracted from the input variable (x), moving the graph left or right. A vertical shift occurs when a constant is added or subtracted from the output of the function, moving the graph up or down. In g(x) = 2√(x+1), the '+1' indicates a leftward shift of the graph by 1 unit.