How Does The Graph Of $g(x)=\frac{1}{x+3}+6$ Compare To The Graph Of $f(x)=\frac{1}{x}$?A. The Graph Of $ G ( X ) G(x) G ( X ) [/tex] Is The Graph Of $f(x)$ Shifted Right 3 Units And Up 6 Units.B. The Graph Of
Understanding the Functions
When comparing the graphs of two functions, it's essential to understand the characteristics of each function. In this case, we have two rational functions: $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$. The function $f(x)$ has a vertical asymptote at $x=0$, which means that the graph of $f(x)$ approaches positive or negative infinity as $x$ approaches 0 from the left or right. The function $g(x)$, on the other hand, has a vertical asymptote at $x=-3$, which means that the graph of $g(x)$ approaches positive or negative infinity as $x$ approaches -3 from the left or right.
Horizontal Shift
To compare the graphs of $g(x)$ and $f(x)$, let's consider the horizontal shift. The function $g(x)$ can be rewritten as $g(x)=\frac{1}{x+3}+6=\frac{1}{(x+3)}+6$. This means that the graph of $g(x)$ is the graph of $f(x)$ shifted right 3 units. The reason for this shift is that the denominator of the function $g(x)$ is $x+3$, which is 3 units larger than the denominator of the function $f(x)$, which is $x$.
Vertical Shift
In addition to the horizontal shift, the graph of $g(x)$ is also shifted up 6 units. This is because the function $g(x)$ has a constant term of 6, which is added to the function $f(x)$. This means that the graph of $g(x)$ is 6 units above the graph of $f(x)$.
Comparing the Graphs
Now that we have analyzed the horizontal and vertical shifts, let's compare the graphs of $g(x)$ and $f(x)$. The graph of $g(x)$ is the graph of $f(x)$ shifted right 3 units and up 6 units. This means that the graph of $g(x)$ has the same shape as the graph of $f(x)$, but it is shifted to the right and up.
Graphing the Functions
To visualize the graphs of $g(x)$ and $f(x)$, we can use a graphing calculator or software. When graphing the functions, we can see that the graph of $g(x)$ is indeed the graph of $f(x)$ shifted right 3 units and up 6 units.
Conclusion
In conclusion, the graph of $g(x)=\frac{1}{x+3}+6$ is the graph of $f(x)=\frac{1}{x}$ shifted right 3 units and up 6 units. This means that the graph of $g(x)$ has the same shape as the graph of $f(x)$, but it is shifted to the right and up. Understanding the characteristics of each function and analyzing the horizontal and vertical shifts are essential in comparing the graphs of two functions.
Key Takeaways
- The graph of $g(x)$ is the graph of $f(x)$ shifted right 3 units and up 6 units.
- The function $g(x)$ has a vertical asymptote at $x=-3$.
- The function $f(x)$ has a vertical asymptote at $x=0$.
- The graph of $g(x)$ has the same shape as the graph of $f(x)$, but it is shifted to the right and up.
Practice Problems
- Graph the functions $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$ using a graphing calculator or software.
- Compare the graphs of $g(x)$ and $f(x)$ and identify the horizontal and vertical shifts.
- Write a short paragraph explaining the characteristics of the function $g(x)$ and how it compares to the function $f(x)$.
Solutions
- Graph the functions $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$ using a graphing calculator or software.
- Compare the graphs of $g(x)$ and $f(x)$ and identify the horizontal and vertical shifts.
- Write a short paragraph explaining the characteristics of the function $g(x)$ and how it compares to the function $f(x)$.
Additional Resources
- Graphing calculators or software
- Online graphing tools
- Math textbooks or online resources
Conclusion
In conclusion, the graph of $g(x)=\frac{1}{x+3}+6$ is the graph of $f(x)=\frac{1}{x}$ shifted right 3 units and up 6 units. This means that the graph of $g(x)$ has the same shape as the graph of $f(x)$, but it is shifted to the right and up. Understanding the characteristics of each function and analyzing the horizontal and vertical shifts are essential in comparing the graphs of two functions.
Frequently Asked Questions
We've received many questions about comparing the graphs of $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$. Here are some of the most frequently asked questions and their answers:
Q: What is the main difference between the graphs of g(x) and f(x)?
A: The main difference between the graphs of $g(x)$ and $f(x)$ is the horizontal shift. The graph of $g(x)$ is shifted right 3 units and up 6 units compared to the graph of $f(x)$.
Q: Why is the graph of g(x) shifted right 3 units?
A: The graph of $g(x)$ is shifted right 3 units because the denominator of the function $g(x)$ is $x+3$, which is 3 units larger than the denominator of the function $f(x)$, which is $x$.
Q: Why is the graph of g(x) shifted up 6 units?
A: The graph of $g(x)$ is shifted up 6 units because the function $g(x)$ has a constant term of 6, which is added to the function $f(x)$.
Q: How can I visualize the graphs of g(x) and f(x)?
A: You can use a graphing calculator or software to visualize the graphs of $g(x)$ and $f(x)$. You can also use online graphing tools or math textbooks to help you understand the graphs.
Q: What are the characteristics of the function g(x)?
A: The function $g(x)$ has a vertical asymptote at $x=-3$ and a horizontal asymptote at $y=6$. The graph of $g(x)$ approaches positive or negative infinity as $x$ approaches -3 from the left or right.
Q: What are the characteristics of the function f(x)?
A: The function $f(x)$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. The graph of $f(x)$ approaches positive or negative infinity as $x$ approaches 0 from the left or right.
Q: How can I compare the graphs of g(x) and f(x)?
A: To compare the graphs of $g(x)$ and $f(x)$, you can use a graphing calculator or software to visualize the graphs. You can also analyze the horizontal and vertical shifts between the two graphs.
Q: What are some real-world applications of comparing the graphs of g(x) and f(x)?
A: Comparing the graphs of $g(x)$ and $f(x)$ can be useful in various real-world applications, such as:
- Modeling population growth and decline
- Analyzing economic data
- Understanding the behavior of complex systems
Conclusion
In conclusion, comparing the graphs of $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$ can be a useful tool in understanding the characteristics of each function and analyzing the horizontal and vertical shifts between the two graphs. We hope this Q&A article has been helpful in answering your questions and providing a better understanding of the topic.
Additional Resources
- Graphing calculators or software
- Online graphing tools
- Math textbooks or online resources
Practice Problems
- Graph the functions $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$ using a graphing calculator or software.
- Compare the graphs of $g(x)$ and $f(x)$ and identify the horizontal and vertical shifts.
- Write a short paragraph explaining the characteristics of the function $g(x)$ and how it compares to the function $f(x)$.
Solutions
- Graph the functions $g(x)=\frac{1}{x+3}+6$ and $f(x)=\frac{1}{x}$ using a graphing calculator or software.
- Compare the graphs of $g(x)$ and $f(x)$ and identify the horizontal and vertical shifts.
- Write a short paragraph explaining the characteristics of the function $g(x)$ and how it compares to the function $f(x)$.