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)

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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.

Next, analyze the given function, h(x) = -√(x + 1). Notice that the transformation involves two changes: (1) the addition of 1 inside the square root, and (2) the negative sign outside the square root.

The transformation x + 1 inside the square root shifts the graph of f(x) = √x to the left by 1 unit. This is because adding a constant inside the function affects the x-values inversely.

The negative sign outside the square root reflects the graph across the x-axis. This means that all y-values of the graph are multiplied by -1, flipping the graph downward.

Combine these transformations: Start with the graph of f(x) = √x, shift it left by 1 unit, and then reflect it across the x-axis. The resulting graph represents h(x) = -√(x + 1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 manipulation of the base graph to create new functions.

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 all y-values of the square root function are inverted, resulting in a graph that opens downward. This concept is essential for understanding how the shape and position of the graph change with transformations.