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Reflecting a function involves flipping it across a specific axis, which is commonly the x-axis or the y-axis. In this exercise, we focus on reflecting the function across the y-axis.
When you reflect a graph across the y-axis, you change each point's x-coordinate to its opposite. For instance, if a point on the graph is \( (a, b) \), after reflection, it becomes \( (-a, b) \).
For the square root function specifically, reflecting \( y = \sqrt{x} \) over the y-axis results in \( y = \sqrt{-x} \). This new equation signifies that for every positive x in the original function, a corresponding negative x exists in the new function, effectively mirroring the graph over the y-axis.
Understanding this concept is crucial because reflection changes not just the image visually on a grid, but also affects the domain of the function. With reflection across the y-axis, the domain of \( y = \sqrt{x} \), which is x ≥ 0, is now x ≤ 0 for \( y = \sqrt{-x} \).

Translation of a function involves shifting the entire graph horizontally or vertically. It doesn't alter the shape of the graph, only its position.
In this exercise, we performed a horizontal translation. Translating a graph horizontally to the left consists of replacing \(x\) with \(x + c\) in the function's equation. The value of \(c\) determines how far the graph shifts. A positive \(c\) moves the graph left, while a negative \(c\) shifts it right.
For our reflected square root function \( y = \sqrt{-x} \), translating two units left involves revising the x part to \( x + 2 \). This gives us the new equation: \( y = \sqrt{-(x + 2)} \). This operation shifts every point on the graph two units to the left, moving the whole graph without changing its orientation.
Thus, horizontal translations help us to move the graph of a function along the x-axis efficiently, forming an essential part of graph transformations in mathematical analysis.

The square root function, denoted as \( y = \sqrt{x} \), is familiar due to its distinctive half-parabola shape opening to the right. This characteristic shape stems from how the function operates: as x increases, \(y\) increases at a decreasing rate.
Key features of square root functions include:

• The domain is x ≥ 0, meaning it only includes non-negative numbers since you can't calculate the square root of a negative number without involving complex numbers.
• The range is y ≥ 0, which includes non-negative numbers.

In transformations, square root functions follow similar rules as other types of functions. The primary operations include reflections, translations, and adjustments to the coefficient inside the square root, altering how the graph appears.
Understanding transformations applied to the base function \( y = \sqrt{x} \) is crucial for solving more complex problems involving graph manipulations. They unveil how various mathematical operations visually affect the function and how it can be represented differently on a coordinate plane.