Select All The Correct Answers.Consider Function F F F And Function G G G .${ \begin{align*} f(x) &= \ln X \ g(x) &= -5 \ln X \end{align*} }$How Does The Graph Of Function G G G Compare With The Graph Of Function
Introduction
In mathematics, functions are used to describe the relationship between variables. Two functions, f(x) and g(x), are given as:
In this article, we will compare the graphs of these two functions and discuss their similarities and differences.
Understanding the Functions
The function f(x) = ln x is a natural logarithmic function, which is defined for all positive real numbers. The graph of this function is a curve that increases as x increases.
On the other hand, the function g(x) = -5 ln x is a transformation of the function f(x). The negative sign in front of the function indicates a reflection across the x-axis, while the coefficient -5 indicates a vertical stretch by a factor of 5.
Comparing the Graphs
To compare the graphs of the two functions, we need to understand the effects of the transformation on the graph of f(x).
- Reflection across the x-axis: The graph of g(x) is a reflection of the graph of f(x) across the x-axis. This means that if we have a point (x, y) on the graph of f(x), the corresponding point on the graph of g(x) will be (x, -y).
- Vertical stretch: The graph of g(x) is a vertical stretch of the graph of f(x) by a factor of 5. This means that if we have a point (x, y) on the graph of f(x), the corresponding point on the graph of g(x) will be (x, 5y).
Key Differences
The key differences between the graphs of f(x) and g(x) are:
- Reflection: The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
- Vertical stretch: The graph of g(x) is a vertical stretch of the graph of f(x) by a factor of 5.
- Shape: The graph of g(x) has a steeper slope than the graph of f(x) due to the vertical stretch.
Conclusion
In conclusion, the graph of function g(x) = -5 ln x is a transformation of the graph of function f(x) = ln x. The graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5. The key differences between the graphs of f(x) and g(x) are the reflection, vertical stretch, and shape.
Key Takeaways
- The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
- The graph of g(x) is a vertical stretch of the graph of f(x) by a factor of 5.
- The graph of g(x) has a steeper slope than the graph of f(x) due to the vertical stretch.
Real-World Applications
The comparison of the graphs of f(x) and g(x) has real-world applications in various fields, such as:
- Physics: The transformation of the graph of f(x) to g(x) can be used to model the behavior of physical systems, such as the motion of an object under the influence of gravity.
- Engineering: The comparison of the graphs of f(x) and g(x) can be used to design and optimize systems, such as electronic circuits and mechanical systems.
Final Thoughts
In conclusion, the comparison of the graphs of f(x) and g(x) is an important concept in mathematics that has real-world applications. The key differences between the graphs of f(x) and g(x) are the reflection, vertical stretch, and shape. By understanding these differences, we can apply the concepts of function transformation to solve real-world problems.
References
- [1] "Functions and Graphs" by Michael Sullivan
- [2] "Calculus" by James Stewart
- [3] "Mathematics for Engineers and Scientists" by Donald R. Hill
Glossary
- Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- Graph: A visual representation of a function, showing the relationship between the input and output values.
- Transformation: A change in the graph of a function, such as a reflection, rotation, or vertical stretch.
FAQs
- Q: What is the difference between the graphs of f(x) and g(x)? A: The graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5.
- Q: How does the transformation of the graph of f(x) to g(x) affect the shape of the graph?
A: The transformation of the graph of f(x) to g(x) results in a steeper slope due to the vertical stretch.
Q&A: Comparing the Graphs of Functions f and g =====================================================
Frequently Asked Questions
In this article, we will answer some frequently asked questions about comparing the graphs of functions f(x) and g(x).
Q: What is the difference between the graphs of f(x) and g(x)?
A: The graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5.
Q: How does the transformation of the graph of f(x) to g(x) affect the shape of the graph?
A: The transformation of the graph of f(x) to g(x) results in a steeper slope due to the vertical stretch.
Q: What is the effect of the negative sign in front of the function g(x) = -5 ln x?
A: The negative sign in front of the function g(x) = -5 ln x indicates a reflection across the x-axis.
Q: How does the coefficient -5 in the function g(x) = -5 ln x affect the graph?
A: The coefficient -5 in the function g(x) = -5 ln x indicates a vertical stretch by a factor of 5.
Q: Can you provide an example of how to compare the graphs of f(x) and g(x)?
A: Yes, let's consider the points (1, 0) and (1, -5) on the graphs of f(x) and g(x), respectively. The point (1, 0) is on the graph of f(x), and the point (1, -5) is on the graph of g(x). This shows that the graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5.
Q: How does the comparison of the graphs of f(x) and g(x) have real-world applications?
A: The comparison of the graphs of f(x) and g(x) has real-world applications in various fields, such as physics and engineering. For example, the transformation of the graph of f(x) to g(x) can be used to model the behavior of physical systems, such as the motion of an object under the influence of gravity.
Q: Can you provide more examples of how to compare the graphs of f(x) and g(x)?
A: Yes, let's consider the points (2, 1) and (2, -5) on the graphs of f(x) and g(x), respectively. The point (2, 1) is on the graph of f(x), and the point (2, -5) is on the graph of g(x). This shows that the graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5.
Q: How does the comparison of the graphs of f(x) and g(x) help us understand the behavior of functions?
A: The comparison of the graphs of f(x) and g(x) helps us understand the behavior of functions by showing how transformations affect the shape and position of the graph.
Q: Can you provide a summary of the key differences between the graphs of f(x) and g(x)?
A: Yes, the key differences between the graphs of f(x) and g(x) are:
- The graph of g(x) is a reflection of the graph of f(x) across the x-axis.
- The graph of g(x) is a vertical stretch of the graph of f(x) by a factor of 5.
- The graph of g(x) has a steeper slope than the graph of f(x) due to the vertical stretch.
Conclusion
In conclusion, the comparison of the graphs of functions f(x) and g(x) is an important concept in mathematics that has real-world applications. The key differences between the graphs of f(x) and g(x) are the reflection, vertical stretch, and shape. By understanding these differences, we can apply the concepts of function transformation to solve real-world problems.
References
- [1] "Functions and Graphs" by Michael Sullivan
- [2] "Calculus" by James Stewart
- [3] "Mathematics for Engineers and Scientists" by Donald R. Hill
Glossary
- Function: A relation between a set of inputs (called the domain) and a set of possible outputs (called the range).
- Graph: A visual representation of a function, showing the relationship between the input and output values.
- Transformation: A change in the graph of a function, such as a reflection, rotation, or vertical stretch.
FAQs
- Q: What is the difference between the graphs of f(x) and g(x)? A: The graph of g(x) is a reflection of the graph of f(x) across the x-axis and a vertical stretch by a factor of 5.
- Q: How does the transformation of the graph of f(x) to g(x) affect the shape of the graph? A: The transformation of the graph of f(x) to g(x) results in a steeper slope due to the vertical stretch.