What Is The Effect On The Graph Of $f(x)=\frac{1}{x}$ When It Is Transformed To $g(x)=\frac{1}{x}-14$?A. The Graph Of F ( X F(x F ( X ] Is Shifted 14 Units To The Right.B. The Graph Of F ( X F(x F ( X ] Is Shifted 14 Units Down.C. The

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Understanding the Transformation of a Graph: A Case Study of $f(x)=\frac{1}{x}$ and $g(x)=\frac{1}{x}-14$

Introduction

In mathematics, graph transformation is a crucial concept that helps us understand how functions change when they undergo various transformations. In this article, we will explore the effect of a specific transformation on the graph of the function $f(x)=\frac{1}{x}$. We will examine how the graph of $f(x)$ changes when it is transformed to $g(x)=\frac{1}{x}-14$. Our goal is to understand the nature of this transformation and how it affects the graph of the function.

The Original Function: $f(x)=\frac{1}{x}$

The function $f(x)=\frac{1}{x}$ is a classic example of a rational function. It has a simple form, but its graph is quite complex. The graph of $f(x)$ is a hyperbola that opens in the first and third quadrants. As $x$ approaches 0 from the right, the value of $f(x)$ approaches infinity. Similarly, as $x$ approaches 0 from the left, the value of $f(x)$ approaches negative infinity.

The Transformed Function: $g(x)=\frac{1}{x}-14$

Now, let's consider the transformed function $g(x)=\frac{1}{x}-14$. This function is obtained by subtracting 14 from the original function $f(x)$. To understand the effect of this transformation, we need to analyze the graph of $g(x)$.

Analyzing the Graph of $g(x)$

When we subtract 14 from the original function $f(x)$, we are essentially shifting the graph of $f(x)$ down by 14 units. This is because the constant term -14 is being subtracted from the function, which has the effect of shifting the graph down.

Conclusion

In conclusion, the graph of $f(x)=\frac{1}{x}$ is shifted 14 units down when it is transformed to $g(x)=\frac{1}{x}-14$. This is a classic example of a vertical shift, where the graph of the function is moved up or down by a certain amount. Understanding graph transformations is essential in mathematics, as it helps us analyze and interpret the behavior of functions in different contexts.

Key Takeaways

  • The graph of $f(x)=\frac{1}{x}$ is a hyperbola that opens in the first and third quadrants.
  • The graph of $g(x)=\frac{1}{x}-14$ is obtained by subtracting 14 from the original function $f(x)$.
  • The graph of $g(x)$ is shifted 14 units down compared to the graph of $f(x)$.
  • Understanding graph transformations is essential in mathematics, as it helps us analyze and interpret the behavior of functions in different contexts.

Further Reading

For those interested in learning more about graph transformations, we recommend exploring the following topics:

  • Horizontal shifts: How to shift the graph of a function horizontally by a certain amount.
  • Vertical stretches: How to stretch the graph of a function vertically by a certain amount.
  • Reflections: How to reflect the graph of a function across the x-axis or y-axis.
  • Compositions: How to combine two or more functions to create a new function.

By exploring these topics, you will gain a deeper understanding of graph transformations and how they can be used to analyze and interpret the behavior of functions in different contexts.
Graph Transformation Q&A: Understanding the Effects of Transformations on Functions

Introduction

In our previous article, we explored the effect of a specific transformation on the graph of the function $f(x)=\frac{1}{x}$. We analyzed how the graph of $f(x)$ changes when it is transformed to $g(x)=\frac{1}{x}-14$. In this article, we will continue to explore graph transformations by answering some common questions related to this topic.

Q: What is a graph transformation?

A: A graph transformation is a change made to the graph of a function. This can include shifting the graph horizontally or vertically, stretching or compressing the graph, reflecting the graph across the x-axis or y-axis, or combining two or more functions to create a new function.

Q: What are the different types of graph transformations?

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

  • Horizontal shifts: Shifting the graph of a function horizontally by a certain amount.
  • Vertical shifts: Shifting the graph of a function vertically by a certain amount.
  • Horizontal stretches: Stretching the graph of a function horizontally by a certain amount.
  • Vertical stretches: Stretching the graph of a function vertically by a certain amount.
  • Reflections: Reflecting the graph of a function across the x-axis or y-axis.
  • Compositions: Combining two or more functions to create a new function.

Q: How do I determine the type of graph transformation?

A: To determine the type of graph transformation, you need to analyze the function and its graph. Look for any changes in the function's equation, such as additions or subtractions of constants, or changes in the function's form, such as squaring or cubing the variable.

Q: What is the effect of a horizontal shift on a graph?

A: A horizontal shift is a change made to the graph of a function by shifting it horizontally by a certain amount. This can be represented by adding or subtracting a constant to the variable in the function's equation.

Q: What is the effect of a vertical shift on a graph?

A: A vertical shift is a change made to the graph of a function by shifting it vertically by a certain amount. This can be represented by adding or subtracting a constant to the function's equation.

Q: How do I graph a function with a horizontal or vertical shift?

A: To graph a function with a horizontal or vertical shift, you need to follow these steps:

  • Graph the original function.
  • Identify the type of shift (horizontal or vertical).
  • Determine the amount of the shift (the constant added or subtracted).
  • Apply the shift to the graph of the original function.

Q: What is the effect of a reflection on a graph?

A: A reflection is a change made to the graph of a function by reflecting it across the x-axis or y-axis. This can be represented by multiplying the function by -1 or changing the sign of the variable in the function's equation.

Q: How do I graph a function with a reflection?

A: To graph a function with a reflection, you need to follow these steps:

  • Graph the original function.
  • Identify the type of reflection (across the x-axis or y-axis).
  • Apply the reflection to the graph of the original function.

Q: What is the effect of a composition on a graph?

A: A composition is a change made to the graph of a function by combining two or more functions to create a new function. This can be represented by multiplying or dividing the functions together.

Q: How do I graph a function with a composition?

A: To graph a function with a composition, you need to follow these steps:

  • Graph the original functions.
  • Identify the type of composition (multiplication or division).
  • Apply the composition to the graphs of the original functions.

Conclusion

In conclusion, graph transformations are an essential concept in mathematics that helps us analyze and interpret the behavior of functions in different contexts. By understanding the different types of graph transformations and how to apply them, you can gain a deeper understanding of functions and their graphs.