Given The Functions $f(x)=\frac{1}{x}$ And $g(x)=\frac{1}{x+5}+9$, Which Statement Describes The Transformation Of The Graph Of Function $f$ Into The Graph Of Function $g$?A. The Graph Shifts 5 Units Left And 9 Units

Introduction

In mathematics, graph transformation is a crucial concept that helps us understand how functions change when certain operations are applied to them. Given two functions, $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x+5}+9$, we are tasked with determining the transformation of the graph of function $f$ into the graph of function $g$. This article will delve into the world of graph transformation, exploring the key concepts and principles that govern this process.

Understanding the Functions

Before we dive into the transformation, let's take a closer look at the two functions in question.

Function $f(x)$

The function $f(x)=\frac{1}{x}$ is a simple reciprocal function. It has a single asymptote at $x=0$ and a vertical tangent at this point. The graph of $f(x)$ is a hyperbola that approaches the asymptote as $x$ approaches infinity or negative infinity.

Function $g(x)$

The function $g(x)=\frac{1}{x+5}+9$ is a more complex function that involves a reciprocal term and a constant term. The reciprocal term $\frac{1}{x+5}$ has a vertical asymptote at $x=-5$, while the constant term $9$ shifts the graph upward by $9$ units. The graph of $g(x)$ is also a hyperbola, but it is shifted and scaled compared to the graph of $f(x)$.

Transformation of Graphs

Now that we have a good understanding of the two functions, let's explore the transformation of the graph of function $f$ into the graph of function $g$. To do this, we need to identify the key transformations that occur when we move from $f(x)$ to $g(x)$.

Horizontal Shift

The first transformation we notice is a horizontal shift of $5$ units to the left. This means that the graph of $g(x)$ is shifted $5$ units to the left compared to the graph of $f(x)$. This shift occurs because the reciprocal term in $g(x)$ has a horizontal asymptote at $x=-5$, which is $5$ units to the left of the vertical asymptote of $f(x)$.

Vertical Shift

In addition to the horizontal shift, we also notice a vertical shift of $9$ units upward. This means that the graph of $g(x)$ is shifted $9$ units upward compared to the graph of $f(x)$. This shift occurs because the constant term in $g(x)$ is $9$, which shifts the graph upward by $9$ units.

Scaling

Finally, we notice that the graph of $g(x)$ is scaled compared to the graph of $f(x)$. The reciprocal term in $g(x)$ has a smaller denominator than the reciprocal term in $f(x)$, which means that the graph of $g(x)$ is stretched horizontally compared to the graph of $f(x)$.

Conclusion

In conclusion, the transformation of the graph of function $f$ into the graph of function $g$ involves a horizontal shift of $5$ units to the left, a vertical shift of $9$ units upward, and a scaling of the graph. These transformations occur because of the changes in the reciprocal term and the constant term in the function $g(x)$. By understanding these transformations, we can gain a deeper insight into the behavior of functions and their graphs.

Answer

Based on our analysis, the correct statement that describes the transformation of the graph of function $f$ into the graph of function $g$ is:

A. The graph shifts 5 units left and 9 units upward.

Introduction

In our previous article, we explored the transformation of the graph of function $f(x)=\frac{1}{x}$ into the graph of function $g(x)=\frac{1}{x+5}+9$. We identified the key transformations that occur when we move from $f(x)$ to $g(x)$, including a horizontal shift of $5$ units to the left, a vertical shift of $9$ units upward, and a scaling of the graph. In this article, we will continue to explore graph transformation and function analysis, answering some of the most frequently asked questions in this field.

Q1: What is graph transformation?

A: Graph transformation is the process of changing the graph of a function by applying certain operations to it. These operations can include horizontal and vertical shifts, scaling, and other transformations that change the shape and position of the graph.

Q2: What are the different types of graph transformations?

A: There are several types of graph transformations, including:

• Horizontal shifts: These occur when the graph is moved to the left or right.
• Vertical shifts: These occur when the graph is moved up or down.
• Scaling: This occurs when the graph is stretched or compressed horizontally or vertically.
• Reflections: This occurs when the graph is reflected across a line or axis.
• Rotations: This occurs when the graph is rotated around a point or axis.

Q3: How do I determine the type of graph transformation that occurs when a function is changed?

A: To determine the type of graph transformation that occurs when a function is changed, you need to analyze the changes made to the function. Look for changes in the coefficients, variables, and constants in the function. These changes can indicate the type of transformation that occurs.

Q4: What is the difference between a horizontal shift and a vertical shift?

A: A horizontal shift occurs when the graph is moved to the left or right, while a vertical shift occurs when the graph is moved up or down. Horizontal shifts change the x-coordinate of the graph, while vertical shifts change the y-coordinate.

Q5: How do I graph a function that has undergone a transformation?

A: To graph a function that has undergone a transformation, you need to apply the transformation to the original graph. This can involve shifting, scaling, or reflecting the graph. You can use graphing software or a calculator to help you graph the function.

Q6: What is the significance of graph transformation in real-world applications?

A: Graph transformation has many real-world applications, including:

• Physics: Graph transformation is used to model the motion of objects and predict their behavior.
• Engineering: Graph transformation is used to design and optimize systems, such as bridges and buildings.
• Economics: Graph transformation is used to model economic systems and predict economic trends.
• Computer Science: Graph transformation is used to develop algorithms and data structures.

Q7: How do I determine the type of transformation that occurs when a function is changed?

A: To determine the type of transformation that occurs when a function is changed, you need to analyze the changes made to the function. Look for changes in the coefficients, variables, and constants in the function. These changes can indicate the type of transformation that occurs.

Q8: What is the difference between a linear transformation and a non-linear transformation?

A: A linear transformation is a transformation that preserves the straightness of the graph, while a non-linear transformation is a transformation that changes the straightness of the graph.

Conclusion

In conclusion, graph transformation and function analysis are essential concepts in mathematics and have many real-world applications. By understanding the different types of graph transformations and how to determine the type of transformation that occurs when a function is changed, you can gain a deeper insight into the behavior of functions and their graphs.