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1 to graph g(x) = Use transformations of f(x) = (x+6)2" ..... Select the correct graph. O A. OB. OC. D. « Previous

1 to graph g(x) = Use transformations of f(x) = (x+6)2" ..... Select the correct graph. O A. OB. OC. D. « Previous

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Title: Understanding Function Transformations** **Objective:** Use transformations of \( f(x) = \frac{1}{x^2} \) to identify the graph of \( g(x) = \frac{1}{(x + 6)^2} \). **Question:** Select the correct graph. **Options:** - **A.** - Graph depicts the function with a sharp peak centered at \( x = -6 \). The graph appears symmetric around this vertical line, demonstrating a horizontal shift of the basic function. - **B.** - Incorrect positioning, the peak is not at \( x = -6 \). - **C.** - Similar to other options but does not illustrate the correct horizontal shift needed. - **D.** - Shows incorrect placement for the transformation scenario. **Solution:** Option **A** correctly shows the graph of \( g(x) = \frac{1}{(x + 6)^2} \), which is a horizontal shift 6 units to the left from \( f(x) = \frac{1}{x^2} \). **Visual Explanation:** Each graph shows a function with a vertical asymptote at a specific x-value. Option A correctly demonstrates a horizontal shift with the vertex translated to \( x = -6 \). --- **Conclusion:** Understanding the effects of transformations, such as horizontal shifts, on function graphs is crucial in visualizing mathematical concepts.
Transcribed Image Text:**Title: Understanding Function Transformations** **Objective:** Use transformations of \( f(x) = \frac{1}{x^2} \) to identify the graph of \( g(x) = \frac{1}{(x + 6)^2} \). **Question:** Select the correct graph. **Options:** - **A.** - Graph depicts the function with a sharp peak centered at \( x = -6 \). The graph appears symmetric around this vertical line, demonstrating a horizontal shift of the basic function. - **B.** - Incorrect positioning, the peak is not at \( x = -6 \). - **C.** - Similar to other options but does not illustrate the correct horizontal shift needed. - **D.** - Shows incorrect placement for the transformation scenario. **Solution:** Option **A** correctly shows the graph of \( g(x) = \frac{1}{(x + 6)^2} \), which is a horizontal shift 6 units to the left from \( f(x) = \frac{1}{x^2} \). **Visual Explanation:** Each graph shows a function with a vertical asymptote at a specific x-value. Option A correctly demonstrates a horizontal shift with the vertex translated to \( x = -6 \). --- **Conclusion:** Understanding the effects of transformations, such as horizontal shifts, on function graphs is crucial in visualizing mathematical concepts.
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