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 6
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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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, 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.
Recommended video:
2:20
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.
Recommended video:
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.
Recommended video:
5:20Introduction to Relations and Functions
Related Practice
Textbook Question
CONCEPT PREVIEW Which one is not a linear equation? A. 5x + 7 (x - 1) = -3x B. 9x² - 4x + 3 = 0 C. 7x + 8x = 13 D. 0.04x - 0.08x = 0.40
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Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The numerical coefficient in the term -7yz² is ___________.
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Textbook Question
CONCEPT PREVIEW Perform the indicated operation, and write each answer in lowest terms (2x/5) • (10/x²)
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Textbook Question
Solve each quadratic equation using the square root property. See Example 6. x² = 16
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7.
-3x² + 6x + 5 = 0