y 11 x 3x² -15x+18 x-intercepts: Vertical Asymptotes: Horizontal or Slant Asymptotes: Holes: y-Intercept(s): Domain: Range
y 11 x 3x² -15x+18 x-intercepts: Vertical Asymptotes: Horizontal or Slant Asymptotes: Holes: y-Intercept(s): Domain: Range
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Graph the function and label the following information.
![The image presents a rational function given by the equation:
\[ y = \frac{x^2 - 4}{3x^2 - 15x + 18} \]
This function is plotted on a grid with both x and y axes marked with arrows. The grid is intended for graphing the rational function. To the right of the graph, there is a checklist meant to analyze the function's characteristics. The components of the checklist include:
- **x-intercepts**: The points where the graph of the function crosses the x-axis.
- **Vertical Asymptotes**: Vertical lines that the graph approaches but never crosses or touches.
- **Horizontal or Slant Asymptotes**: Lines that the graph approaches as x approaches infinity or negative infinity.
- **Asymptotes**: A general term for lines that the graph approaches.
- **Holes**: Specific points on the graph where the function is not defined, typically due to factors cancelled in the rational expression.
- **y-Intercept(s)**: The point(s) where the graph crosses the y-axis.
- **Domain**: The set of all possible x-values for which the function is defined.
- **Range**: The set of all possible y-values that the function can take.
This structure guides a comprehensive analysis of the function's graphical behavior.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd86db678-bc86-44fd-bd72-7c276c312ad6%2F96da5d4a-5b2f-4fb7-9bc9-ea533cece106%2Fq1b94u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image presents a rational function given by the equation:
\[ y = \frac{x^2 - 4}{3x^2 - 15x + 18} \]
This function is plotted on a grid with both x and y axes marked with arrows. The grid is intended for graphing the rational function. To the right of the graph, there is a checklist meant to analyze the function's characteristics. The components of the checklist include:
- **x-intercepts**: The points where the graph of the function crosses the x-axis.
- **Vertical Asymptotes**: Vertical lines that the graph approaches but never crosses or touches.
- **Horizontal or Slant Asymptotes**: Lines that the graph approaches as x approaches infinity or negative infinity.
- **Asymptotes**: A general term for lines that the graph approaches.
- **Holes**: Specific points on the graph where the function is not defined, typically due to factors cancelled in the rational expression.
- **y-Intercept(s)**: The point(s) where the graph crosses the y-axis.
- **Domain**: The set of all possible x-values for which the function is defined.
- **Range**: The set of all possible y-values that the function can take.
This structure guides a comprehensive analysis of the function's graphical behavior.
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