Choosing the Appropriate Answers

Oct 13, 2025 by wbrum.org 61 views

Introduction

In mathematics, function transformations are essential concepts that help us understand how functions change when they are shifted or scaled. In this article, we will explore the transformation of a given model function, $$y=\frac{1}{x}$$, into a translated function, $$f(x)=-\frac{3}{x+2}-4$$. We will analyze the horizontal and vertical shifts involved in this transformation and provide a detailed explanation of the process.

Model Function: $$y=\frac{1}{x}$$

The model function, $$y=\frac{1}{x}$$, is a simple rational function that represents a hyperbola. This function has a vertical asymptote at $$x=0$$ and a horizontal asymptote at $$y=0$$. The graph of this function is a pair of intersecting lines that approach the asymptotes as $$x$$ approaches infinity or negative infinity.

Translated Function: $$f(x)=-\frac{3}{x+2}-4$$

The translated function, $$f(x)=-\frac{3}{x+2}-4$$, is a transformed version of the model function. This function has been shifted horizontally and vertically to create a new graph. The horizontal shift is represented by the term $$x+2$$ in the denominator, while the vertical shift is represented by the constant term $$-4$$.

Horizontal Shift

To determine the horizontal shift, we need to analyze the term $$x+2$$ in the denominator of the translated function. This term represents a horizontal shift of $$2$$ units to the left. In other words, the graph of the translated function is shifted $$2$$ units to the left compared to the graph of the model function.

Vertical Shift

The vertical shift is represented by the constant term $$-4$$ in the translated function. This term indicates that the graph of the translated function is shifted $$4$$ units downward compared to the graph of the model function.

Choosing the Appropriate Answers

Based on our analysis, we can choose the following answers:

a) Horizontal Shift: [Select] Left

b) Vertical Shift: [Select] Downward

Discussion

The transformation of the model function into the translated function involves both horizontal and vertical shifts. The horizontal shift of $$2$$ units to the left is represented by the term $$x+2$$ in the denominator, while the vertical shift of $$4$$ units downward is represented by the constant term $$-4$$.

In conclusion, understanding function transformations is essential in mathematics, as it helps us analyze and interpret the behavior of functions. By analyzing the horizontal and vertical shifts involved in the transformation of the model function, we can gain a deeper understanding of the properties of functions and their graphs.

Conclusion

In this article, we have explored the transformation of a model function, $$y=\frac{1}{x}$$, into a translated function, $$f(x)=-\frac{3}{x+2}-4$$. We have analyzed the horizontal and vertical shifts involved in this transformation and provided a detailed explanation of the process. By understanding function transformations, we can gain a deeper understanding of the properties of functions and their graphs.

References

[1] Algebra.com. (n.d.). Horizontal Shift. Retrieved from https://www.algebra.com/algebra/homework/Horizontal-Shift/
[2] Math Open Reference. (n.d.). Vertical Shift. Retrieved from https://www.mathopenref.com/verticalshift.html

Introduction

In our previous article, we explored the transformation of a model function, $$y=\frac{1}{x}$$, into a translated function, $$f(x)=-\frac{3}{x+2}-4$$. We analyzed the horizontal and vertical shifts involved in this transformation and provided a detailed explanation of the process. In this article, we will answer some frequently asked questions about function transformations and provide additional insights into this topic.

Q&A

Q: What is a horizontal shift in a function?

A: A horizontal shift in a function is a transformation that moves the graph of the function to the left or right. This is represented by a change in the input variable, x, in the function.

Q: What is a vertical shift in a function?

A: A vertical shift in a function is a transformation that moves the graph of the function up or down. This is represented by a change in the output variable, y, in the function.

Q: How do I determine the horizontal shift in a function?

A: To determine the horizontal shift in a function, you need to analyze the term that represents the shift. If the term is in the form of x + c, where c is a constant, then the graph of the function is shifted c units to the left. If the term is in the form of x - c, then the graph of the function is shifted c units to the right.

Q: How do I determine the vertical shift in a function?

A: To determine the vertical shift in a function, you need to analyze the constant term that represents the shift. If the constant term is positive, then the graph of the function is shifted up. If the constant term is negative, then the graph of the function is shifted down.

Q: Can a function have both horizontal and vertical shifts?

A: Yes, a function can have both horizontal and vertical shifts. This is represented by a combination of terms in the function that indicate both shifts.

Q: How do I graph a function with both horizontal and vertical shifts?

A: To graph a function with both horizontal and vertical shifts, you need to first graph the original function. Then, you need to apply the horizontal shift by moving the graph to the left or right. Finally, you need to apply the vertical shift by moving the graph up or down.

Q: What are some common types of function transformations?

A: Some common types of function transformations include:

Horizontal shift: moving the graph of the function to the left or right
Vertical shift: moving the graph of the function up or down
Reflection: flipping the graph of the function over a line or axis
Scaling: stretching or compressing the graph of the function

Q: Why are function transformations important in mathematics?

A: Function transformations are important in mathematics because they help us analyze and interpret the behavior of functions. By understanding how functions can be transformed, we can gain a deeper understanding of the properties of functions and their graphs.

Conclusion

In this article, we have answered some frequently asked questions about function transformations and provided additional insights into this topic. We hope that this article has been helpful in understanding the concept of function transformations and how they can be applied to various mathematical functions.

References

[1] Algebra.com. (n.d.). Horizontal Shift. Retrieved from https://www.algebra.com/algebra/homework/Horizontal-Shift/
[2] Math Open Reference. (n.d.). Vertical Shift. Retrieved from https://www.mathopenref.com/verticalshift.html