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.

검증된 단계별 안내

Identify the original function and the transformed function. The original function is \(y = \sqrt{x}\), and the transformed function is \(f(x) = \sqrt{-x}\).

Understand the effect of the negative sign inside the square root on the input variable \(x\). Replacing \(x\) with \(-x\) reflects the graph across the vertical axis because it changes the sign of the input values.

Recall that reflecting a graph across the \(y\)-axis means that every point \((x, y)\) on the original graph is mapped to \((-x, y)\) on the transformed graph.

Since \(f(x) = \sqrt{-x}\) replaces \(x\) with \(-x\), the graph of \(f(x)\) is a reflection of \(y = \sqrt{x}\) across the \(y\)-axis.

Therefore, the blank should be filled with '\(y\)' to complete the sentence: 'The graph of \(f(x) = \sqrt{-x}\) is a reflection of the graph of \(y = \sqrt{x}\) across the \(y\)-axis.'

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주요 개념

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

Square Root Function

The square root function, y = √x, is defined for x ≥ 0 and produces non-negative outputs. Its graph starts at the origin and increases slowly, forming a curve in the first quadrant. Understanding its domain and shape is essential for analyzing transformations.

Reflection Across an Axis

Reflection is a transformation that flips a graph over a specific axis. Reflecting a graph across the x-axis changes the sign of the y-values, while reflecting across the y-axis changes the sign of the x-values. Recognizing which axis causes which effect helps identify the transformed graph.

Effect of Negative Input Inside a Function

Replacing x with -x inside a function, as in f(x) = √-x, reflects the graph horizontally across the y-axis. This changes the domain to x ≤ 0, flipping the graph leftwards. Understanding this substitution clarifies how the graph shifts or reflects.

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.