Multiple Choice
The green dotted curve below is a graph of the function . Find the domain and range of (the blue solid curve), which is a transformation of .
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0. Review of College Algebra
Transformations
2:12 minutesProblem 6
Textbook Question
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 guidance1
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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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Imaginary Roots with the Square Root Property
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.
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5:00Reflections of Functions
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.
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5:20Introduction to Relations and Functions
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