Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Assignment: Write an Equation for the Function Graphed Below**
The image displays a graph with a function, and the task is to formulate an equation that represents this function.
### Graph Description:
- **Axes and Asymptotes:**
- The graph is centered with x-axis ranging from -7 to 7 and the y-axis spanning from -5 to 5.
- There are red dashed lines indicating asymptotes at x = -1, x = 3, and y = 2. These asymptotes indicate points where the graph approaches but never touches or crosses these lines.
- **Function Behavior:**
- The function has two separate parts:
- A portion on the left side is approaching the asymptotes at x = -1 and y = 2.
- A portion on the right side is approaching the asymptotes at x = 3 and y = 2.
- In the middle, between x = -1 and x = 3, there is a smooth curve, resembling a parabola opening downwards, stretching from approximately the point (-1, 2) to (3, 2) and reaching a minimum point at (1, -4).
### Mathematical Analysis:
1. **Asymptotes and Symmetry:**
- The vertical asymptotes at x = -1 and x = 3 suggest that the function has undefined points (likely due to division by zero in a rational function).
- The horizontal asymptote at y = 2 suggests that as x approaches ±∞, y approaches 2.
2. **Equation Insight:**
- The parabola-like behavior and asymptotes imply that the function might be a rational function of the form \( y = \frac{ax^2 + bx + c}{dx^2 + ex + f} + g \).
- Given vertical asymptotes, likely factors are in the denominator such as \( (x + 1)(x - 3) \).
### Conclusion:
Based on the given information, an equation for this function likely involves a rational expression considering the asymptotes and the shape, such as:
\[ y = \frac{A}{(x + 1)(x - 3)} + 2 \]
Students are encouraged to utilize critical points and properties such as asymptotes and intervals to further refine the equation. Solutions will vary depending on specific values determined by](https://content.bartleby.com/qna-images/question/83ee5966-93c8-4294-9553-af3c0a4b3c47/eb9d496a-61a2-4f10-82ac-7d840908f1e3/f5kni4_processed.jpeg)
Transcribed Image Text:**Assignment: Write an Equation for the Function Graphed Below**
The image displays a graph with a function, and the task is to formulate an equation that represents this function.
### Graph Description:
- **Axes and Asymptotes:**
- The graph is centered with x-axis ranging from -7 to 7 and the y-axis spanning from -5 to 5.
- There are red dashed lines indicating asymptotes at x = -1, x = 3, and y = 2. These asymptotes indicate points where the graph approaches but never touches or crosses these lines.
- **Function Behavior:**
- The function has two separate parts:
- A portion on the left side is approaching the asymptotes at x = -1 and y = 2.
- A portion on the right side is approaching the asymptotes at x = 3 and y = 2.
- In the middle, between x = -1 and x = 3, there is a smooth curve, resembling a parabola opening downwards, stretching from approximately the point (-1, 2) to (3, 2) and reaching a minimum point at (1, -4).
### Mathematical Analysis:
1. **Asymptotes and Symmetry:**
- The vertical asymptotes at x = -1 and x = 3 suggest that the function has undefined points (likely due to division by zero in a rational function).
- The horizontal asymptote at y = 2 suggests that as x approaches ±∞, y approaches 2.
2. **Equation Insight:**
- The parabola-like behavior and asymptotes imply that the function might be a rational function of the form \( y = \frac{ax^2 + bx + c}{dx^2 + ex + f} + g \).
- Given vertical asymptotes, likely factors are in the denominator such as \( (x + 1)(x - 3) \).
### Conclusion:
Based on the given information, an equation for this function likely involves a rational expression considering the asymptotes and the shape, such as:
\[ y = \frac{A}{(x + 1)(x - 3)} + 2 \]
Students are encouraged to utilize critical points and properties such as asymptotes and intervals to further refine the equation. Solutions will vary depending on specific values determined by
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