Fill in the blank(s) to correctly complete each sentence.
The graph of ƒ(x) = -√x is a reflection of the graph of y = √x across the ___-axis.

Verified step by step guidance

1. Step 1: Understand the basic function y = \sqrt{x}. This function represents the principal square root of x, and its graph is a curve that starts at the origin (0,0) and extends to the right in the first quadrant.
2. Step 2: Consider the transformation applied to the function y = \sqrt{x} to obtain f(x) = -\sqrt{x}. The negative sign in front of the square root indicates a reflection.
3. Step 3: Determine which axis the reflection occurs across. A negative sign in front of a function, such as -\sqrt{x}, reflects the graph across the x-axis.
4. Step 4: Visualize the reflection. The original graph of y = \sqrt{x} is above the x-axis. Reflecting it across the x-axis will flip it to be below the x-axis.
5. Step 5: Conclude that the graph of f(x) = -\sqrt{x} is a reflection of the graph of y = \sqrt{x} across the x-axis.

Key Concepts

Reflection in Geometry

Reflection in geometry refers to flipping a shape or graph over a specific line, creating a mirror image. For functions, reflecting across the x-axis means that for every point (x, y) on the graph, there is a corresponding point (x, -y). This concept is crucial for understanding how transformations affect the position and orientation of graphs.

Square Root Function

The square root function, denoted as y = √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. Understanding the properties of this function helps in visualizing how its reflection across the x-axis alters its shape and position.

Transformations of Functions

Transformations of functions involve changing the position or shape of a graph through operations such as translations, reflections, stretches, and compressions. In this case, reflecting the square root function across the x-axis results in the negative square root function, ƒ(x) = -√x, which inverts the y-values while keeping the x-values unchanged.