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-8 -7 -6 -5 -4 -3 -2 -1 -4 3 2 -1- . -1- -2- -3- & -4 -5- -6- -7 2 3 4 5 6 7

-8 -7 -6 -5 -4 -3 -2 -1 -4 3 2 -1- . -1- -2- -3- & -4 -5- -6- -7 2 3 4 5 6 7

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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  Make three (3) observations about the graph shown below. 

 

The image displays a graph of the mathematical function \( y = \frac{1}{x} \). ### Graph Details: - **Axes**: The x-axis and y-axis are labeled with evenly spaced grid lines. The x-axis ranges from approximately -8 to 8, and the y-axis ranges from approximately -8 to 8. - **Function**: This is a hyperbolic curve illustrating the function \( y = \frac{1}{x} \). It has two branches: - One branch occupies the first quadrant, where as x increases from zero to positive infinity, y decreases from positive infinity to zero. - The second branch occupies the third quadrant, where as x decreases from zero to negative infinity, y increases from negative infinity to zero. - **Asymptotes**: - **Vertical Asymptote**: The function approaches but never touches the y-axis (x = 0). - **Horizontal Asymptote**: The function approaches but never touches the x-axis (y = 0). - **Open Circle**: There is an open circle on the graph at \( (3, -\frac{1}{3}) \). This indicates a removed discontinuity or a point where the function is not defined on this graph. ### Educational Context: This graph is an example of how rational functions behave, illustrating the concepts of asymptotes and discontinuities. It showcases how values of x near zero result in large, undefined values for y, and emphasizes the nature of rational functions with a division by zero. The open circle is also a teaching point for understanding removable discontinuities.
Transcribed Image Text:The image displays a graph of the mathematical function \( y = \frac{1}{x} \). ### Graph Details: - **Axes**: The x-axis and y-axis are labeled with evenly spaced grid lines. The x-axis ranges from approximately -8 to 8, and the y-axis ranges from approximately -8 to 8. - **Function**: This is a hyperbolic curve illustrating the function \( y = \frac{1}{x} \). It has two branches: - One branch occupies the first quadrant, where as x increases from zero to positive infinity, y decreases from positive infinity to zero. - The second branch occupies the third quadrant, where as x decreases from zero to negative infinity, y increases from negative infinity to zero. - **Asymptotes**: - **Vertical Asymptote**: The function approaches but never touches the y-axis (x = 0). - **Horizontal Asymptote**: The function approaches but never touches the x-axis (y = 0). - **Open Circle**: There is an open circle on the graph at \( (3, -\frac{1}{3}) \). This indicates a removed discontinuity or a point where the function is not defined on this graph. ### Educational Context: This graph is an example of how rational functions behave, illustrating the concepts of asymptotes and discontinuities. It showcases how values of x near zero result in large, undefined values for y, and emphasizes the nature of rational functions with a division by zero. The open circle is also a teaching point for understanding removable discontinuities.
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