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)-1

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Start by graphing the parent function f(x) = √x. This is the basic square root function, which starts at the origin (0, 0) and increases gradually as x increases. The domain is x ≥ 0, and the range is y ≥ 0.

Next, analyze the transformation g(x) = 2√(x+1) - 1. 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.

The coefficient 2 in front of the square root represents a vertical stretch. This means that the y-values of the graph will be multiplied by 2, making the graph steeper compared to the parent function.

The -1 outside the square root represents a vertical shift downward by 1 unit. This means that the entire graph will be shifted 1 unit lower along the y-axis.

Combine all these transformations: Start with the graph of f(x) = √x, shift it 1 unit to the left, stretch it vertically by a factor of 2, and then shift it 1 unit downward. Plot the resulting graph to visualize g(x) = 2√(x+1) - 1.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 in the given problem.

Transformations of Functions

Transformations of functions involve shifting, stretching, compressing, or reflecting the graph of a function. For example, adding a constant inside the function shifts the graph horizontally, while adding outside shifts it vertically. In the case of g(x) = 2√(x+1)-1, the transformations include a horizontal shift left by 1 unit, a vertical stretch by a factor of 2, and a downward shift by 1 unit.

Graphing Techniques

Graphing techniques involve plotting points and understanding the behavior of functions to create accurate representations. For the function g(x), one must first graph f(x) = √x, then apply the identified transformations step-by-step. This process helps visualize how the original function is altered, leading to a correct graph of the transformed function.