Square Root Function Transformation Analysis Identifying H(x)

In the realm of mathematics, understanding function transformations is crucial for grasping how different functions relate to one another. Specifically, when dealing with the square root parent function, f(x) = √x, transformations can alter its graph in various ways, such as shifting it horizontally, vertically, or reflecting it across an axis. This exploration delves into identifying the function h(x), which represents a transformation of the parent function. The options provided are A. h(x) = √(x-5), B. h(x) = √x + 5, and C. h(x) = √x - 5. To discern the correct transformation, we will analyze each option and its effect on the graph of the parent function.

The square root function, in its most basic form, f(x) = √x, serves as the foundation for understanding more complex transformations. The graph of this function starts at the origin (0,0) and extends to the right, curving upwards. This shape is a direct result of the nature of the square root operation, which only yields real number outputs for non-negative inputs. Transformations applied to this parent function can shift, stretch, compress, or reflect the graph, leading to a variety of related functions. Identifying these transformations requires careful consideration of how the algebraic form of the function changes. For instance, adding or subtracting a constant inside the square root (like in option A) results in a horizontal shift, while adding or subtracting a constant outside the square root (like in options B and C) causes a vertical shift. The ability to recognize these patterns is essential for analyzing and predicting the behavior of transformed functions. Understanding the parent function's characteristics is the cornerstone for accurately interpreting the effects of different transformations, allowing for a deeper appreciation of the relationship between algebraic expressions and their graphical representations.

Option A, h(x) = √(x - 5), represents a horizontal transformation of the square root parent function. The key here is the subtraction of 5 inside the square root. This type of modification affects the x-values before the square root operation is applied. In general, a transformation of the form f(x - c) results in a horizontal shift of c units. If c is positive, the graph shifts to the right; if c is negative, it shifts to the left. In this specific case, subtracting 5 from x before taking the square root means the graph will shift 5 units to the right. This can be visualized by considering the point where the function equals zero. For the parent function, f(x) = √x, this occurs at x = 0. For h(x) = √(x - 5), the function equals zero when x - 5 = 0, which means x = 5. This confirms the horizontal shift of 5 units to the right.

The horizontal shift can be further understood by examining how specific points on the parent function's graph are affected. For example, the point (0, 0) on the parent function's graph corresponds to the point (5, 0) on the transformed graph. Similarly, the point (4, 2) on a scaled version of the parent function (if we were to consider a function like f(x) = √(4x) for demonstration) would correspond to the point (9, 2) on a transformed graph with a similar scaling factor but shifted horizontally. This consistent shift pattern underscores the nature of horizontal transformations. The domain of the function also changes with this transformation. The parent function f(x) = √x has a domain of x ≥ 0. For h(x) = √(x - 5), the expression inside the square root must be non-negative, so x - 5 ≥ 0, which means x ≥ 5. This shift in the domain is another clear indicator of the horizontal transformation. The impact of this shift extends beyond just the starting point of the graph; it influences the entire shape and position of the function on the coordinate plane, demonstrating the profound effect that simple algebraic manipulations can have on the graphical representation of a function.

Options B and C, h(x) = √x + 5 and h(x) = √x - 5, respectively, represent vertical transformations of the square root parent function. Unlike horizontal shifts, vertical shifts involve adding or subtracting a constant outside the square root. This directly affects the y-values of the function. In general, a transformation of the form f(x) + k results in a vertical shift of k units. If k is positive, the graph shifts upwards; if k is negative, it shifts downwards.

For Option B, h(x) = √x + 5, the addition of 5 outside the square root signifies a vertical shift of 5 units upwards. This means that every point on the graph of the parent function f(x) = √x is moved 5 units higher. For instance, the point (0, 0) on the parent function becomes (0, 5) on the transformed function. The shape of the graph remains the same, but its position on the coordinate plane is altered vertically. The vertical shift is evident when comparing the y-intercepts of the two functions. The parent function has a y-intercept of 0, while h(x) = √x + 5 has a y-intercept of 5. This shift affects the range of the function. The parent function f(x) = √x has a range of y ≥ 0, while h(x) = √x + 5 has a range of y ≥ 5. This upward shift in the range is a direct consequence of the vertical transformation.

Conversely, Option C, h(x) = √x - 5, represents a vertical shift of 5 units downwards. The subtraction of 5 outside the square root moves every point on the parent function's graph 5 units lower. The point (0, 0) on the parent function becomes (0, -5) on the transformed function. Again, the shape of the graph remains unchanged, but its vertical position is shifted. The y-intercept of h(x) = √x - 5 is -5, which is 5 units below the y-intercept of the parent function. The range of this function is y ≥ -5, reflecting the downward shift. The distinction between Options B and C highlights the impact of the sign of the constant added or subtracted outside the square root. A positive constant shifts the graph upwards, while a negative constant shifts it downwards. This understanding of vertical shifts is essential for accurately interpreting and manipulating functions.

To determine the correct transformation, it's crucial to distinguish between horizontal and vertical shifts. Option A, h(x) = √(x - 5), represents a horizontal shift, while Options B and C, h(x) = √x + 5 and h(x) = √x - 5, represent vertical shifts. The position of the constant relative to the square root operation is the key differentiator. If the constant is inside the square root, it affects the x-values and results in a horizontal shift. If the constant is outside the square root, it affects the y-values and results in a vertical shift.

The algebraic manipulation of the function is the primary indicator of the type of transformation. For horizontal shifts, the transformation is applied directly to the x variable before any other operations, such as taking the square root. In h(x) = √(x - 5), the subtraction of 5 occurs before the square root is calculated, indicating a horizontal shift. For vertical shifts, the transformation is applied after the square root operation. In h(x) = √x + 5 and h(x) = √x - 5, the addition or subtraction of 5 occurs after the square root is calculated, indicating a vertical shift. This careful consideration of the order of operations is fundamental to understanding function transformations.

The graphical representation of the transformed function provides a visual confirmation of the transformation. By plotting the parent function f(x) = √x and each of the transformed functions, the shifts become apparent. A horizontal shift will move the graph left or right along the x-axis, while a vertical shift will move the graph up or down along the y-axis. This visual analysis complements the algebraic understanding of the transformations. Comparing the key features of the graphs, such as the starting point and the overall shape, helps solidify the understanding of how the transformations affect the function. The ability to connect the algebraic form of a function with its graphical representation is a cornerstone of mathematical literacy.

In conclusion, analyzing transformations of the square root parent function requires a thorough understanding of how algebraic manipulations affect the graph. Horizontal shifts, represented by changes inside the square root, move the graph left or right. Vertical shifts, represented by changes outside the square root, move the graph up or down. By carefully considering the position of constants in the function's equation and how they alter the x- and y-values, we can accurately identify and describe these transformations. This ability to interpret and manipulate function transformations is a fundamental skill in mathematics, allowing for a deeper understanding of the relationships between different functions and their graphical representations.

Therefore, by carefully examining the options and understanding the principles of horizontal and vertical shifts, we can accurately determine the function h(x) that represents a transformation of the square root parent function. Each option presents a distinct transformation, and discerning the correct one relies on a solid grasp of how algebraic changes manifest graphically.