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.
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 5
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 \(f(x) = -\sqrt{x}\).
Recognize that the negative sign in front of the square root affects the output values (the \(y\)-values) of the function, changing their sign.
Understand that changing the sign of the \(y\)-values corresponds to reflecting the graph across the \(x\)-axis.
Therefore, the graph of \(f(x) = -\sqrt{x}\) is a reflection of the graph of \(y = \sqrt{x}\) across the \(x\)-axis.
Fill in the blank with '\(x\)' to complete the sentence: 'The graph of \(f(x) = -\sqrt{x}\) is a reflection of the graph of \(y = \sqrt{x}\) across the \(x\)-axis.'
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Key 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, produces the principal (non-negative) root of x and is defined for x ≥ 0. Its graph starts at the origin and increases slowly, forming a curve in the first quadrant. Understanding this base function is essential before analyzing transformations.
Recommended video:
2:20
Imaginary Roots with the Square Root Property
Reflection Across an Axis
Reflection across an axis means flipping a graph over that axis, creating a mirror image. Reflecting a function y = f(x) across the x-axis changes it to y = -f(x), inverting all y-values. This concept helps identify how the graph of y = -√x relates to y = √x.
Recommended video:
5:00Reflections of Functions
Function Transformations
Function transformations include shifts, stretches, compressions, and reflections that alter a graph's position or shape. Recognizing how multiplying a function by -1 reflects it across the x-axis is key to completing the sentence about the graph of ƒ(x) = -√x.
Recommended video:
4:22Domain and Range of Function Transformations
Related Practice
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