Compare The Graph Of G ( X ) = 1 X − 5 − 2 G(x)=\frac{1}{x-5}-2 G ( X ) = X − 5 1 ​ − 2 To The Parent Function F ( X ) = 1 X F(x)=\frac{1}{x} F ( X ) = X 1 ​ . G ( X G(x G ( X ] Is Shifted Right 5 Units And Down 2 Units Compared To F ( X F(x F ( X ].

Introduction

In mathematics, understanding the behavior of functions and their graphs is crucial for solving problems and making predictions. One way to analyze functions is by comparing them to their parent functions, which are the simplest forms of a function. In this article, we will compare the graph of $g(x)=\frac{1}{x-5}-2$ to the parent function $f(x)=\frac{1}{x}$. We will explore how $g(x)$ is shifted compared to $f(x)$ and discuss the implications of these shifts.

The Parent Function $f(x)$

The parent function $f(x)=\frac{1}{x}$ is a rational function that has a single vertical asymptote at $x=0$. This function is also known as the reciprocal function. The graph of $f(x)$ is a hyperbola that approaches the x-axis as $x$ approaches infinity and negative infinity.

The Function $g(x)$

The function $g(x)=\frac{1}{x-5}-2$ is a shifted version of the parent function $f(x)$. To understand how $g(x)$ is shifted compared to $f(x)$, let's break down the function into its components.

Horizontal Shift

The function $g(x)$ has a horizontal shift of 5 units to the right. This means that the graph of $g(x)$ is shifted 5 units to the right compared to the graph of $f(x)$. In other words, if the graph of $f(x)$ has an x-intercept at $x=a$, then the graph of $g(x)$ will have an x-intercept at $x=a+5$.

Vertical Shift

The function $g(x)$ also has a vertical shift of 2 units down. This means that the graph of $g(x)$ is shifted 2 units down compared to the graph of $f(x)$. In other words, if the graph of $f(x)$ has a y-intercept at $y=b$, then the graph of $g(x)$ will have a y-intercept at $y=b-2$.

Comparing the Graphs

Now that we have analyzed the shifts of $g(x)$ compared to $f(x)$, let's compare the graphs of the two functions.

Graph of $f(x)$

The graph of $f(x)=\frac{1}{x}$ is a hyperbola that approaches the x-axis as $x$ approaches infinity and negative infinity. The graph has a single vertical asymptote at $x=0$.

Graph of $g(x)$

The graph of $g(x)=\frac{1}{x-5}-2$ is a shifted version of the graph of $f(x)$. The graph has a horizontal shift of 5 units to the right and a vertical shift of 2 units down. The graph still has a single vertical asymptote, but it is now located at $x=5$.

Implications of the Shifts

The shifts of $g(x)$ compared to $f(x)$ have several implications. For example, if we want to find the x-intercept of the graph of $g(x)$, we need to add 5 to the x-intercept of the graph of $f(x)$. Similarly, if we want to find the y-intercept of the graph of $g(x)$, we need to subtract 2 from the y-intercept of the graph of $f(x)$.

Conclusion

In conclusion, the graph of $g(x)=\frac{1}{x-5}-2$ is a shifted version of the parent function $f(x)=\frac{1}{x}$. The graph of $g(x)$ has a horizontal shift of 5 units to the right and a vertical shift of 2 units down compared to the graph of $f(x)$. Understanding these shifts is crucial for analyzing and solving problems involving rational functions.

Further Exploration

For further exploration, we can analyze other types of shifts, such as vertical stretches and compressions, or horizontal compressions and stretches. We can also explore how these shifts affect the behavior of the function, such as its asymptotes and intercepts.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by James Stewart
• [3] "Precalculus" by Michael Sullivan

Glossary

• Parent function: The simplest form of a function.
• Horizontal shift: A shift of a function to the left or right.
• Vertical shift: A shift of a function up or down.
• Asymptote: A line that a function approaches as $x$ approaches infinity or negative infinity.
• Intercept: A point where a function intersects the x-axis or y-axis.
  Q&A: Understanding the Graphs of $g(x)$ and $f(x)$ =====================================================

Introduction

In our previous article, we compared the graphs of $g(x)=\frac{1}{x-5}-2$ and $f(x)=\frac{1}{x}$. We discussed how $g(x)$ is shifted compared to $f(x)$ and explored the implications of these shifts. In this article, we will answer some frequently asked questions about the graphs of $g(x)$ and $f(x)$.

Q: What is the horizontal shift of $g(x)$ compared to $f(x)$?

A: The horizontal shift of $g(x)$ compared to $f(x)$ is 5 units to the right. This means that the graph of $g(x)$ is shifted 5 units to the right compared to the graph of $f(x)$.

Q: What is the vertical shift of $g(x)$ compared to $f(x)$?

A: The vertical shift of $g(x)$ compared to $f(x)$ is 2 units down. This means that the graph of $g(x)$ is shifted 2 units down compared to the graph of $f(x)$.

Q: How do the asymptotes of $g(x)$ and $f(x)$ compare?

A: The asymptotes of $g(x)$ and $f(x)$ are the same, but they are located at different points. The asymptote of $g(x)$ is located at $x=5$, while the asymptote of $f(x)$ is located at $x=0$.

Q: How do the intercepts of $g(x)$ and $f(x)$ compare?

A: The intercepts of $g(x)$ and $f(x)$ are different due to the shifts. The x-intercept of $g(x)$ is located at $x=5$, while the x-intercept of $f(x)$ is located at $x=0$. The y-intercept of $g(x)$ is located at $y=-2$, while the y-intercept of $f(x)$ is located at $y=1$.

Q: Can you give an example of how to find the x-intercept of $g(x)$?

A: Yes, to find the x-intercept of $g(x)$, we need to set $y=0$ and solve for $x$. This gives us the equation:

$$\frac{1}{x-5}-2=0$$

Solving for $x$, we get:

$$x=5$$

So, the x-intercept of $g(x)$ is located at $x=5$.

Q: Can you give an example of how to find the y-intercept of $g(x)$?

A: Yes, to find the y-intercept of $g(x)$, we need to set $x=0$ and solve for $y$. This gives us the equation:

$$\frac{1}{0-5}-2=-2$$

So, the y-intercept of $g(x)$ is located at $y=-2$.

Q: What is the significance of the shifts in the graphs of $g(x)$ and $f(x)$?

A: The shifts in the graphs of $g(x)$ and $f(x)$ have several implications. For example, if we want to find the x-intercept of the graph of $g(x)$, we need to add 5 to the x-intercept of the graph of $f(x)$. Similarly, if we want to find the y-intercept of the graph of $g(x)$, we need to subtract 2 from the y-intercept of the graph of $f(x)$.

Conclusion

In conclusion, the graphs of $g(x)$ and $f(x)$ are related by a horizontal shift of 5 units to the right and a vertical shift of 2 units down. Understanding these shifts is crucial for analyzing and solving problems involving rational functions. We hope that this Q&A article has helped to clarify any questions you may have had about the graphs of $g(x)$ and $f(x)$.