Combining Horizontal and Vertical Shifts

Oct 13, 2025 by pynkocean.com 8 views

Introduction

In mathematics, graph transformations are essential concepts that help us understand how functions change and behave under different conditions. Given two functions, $f(\theta) = \frac{1}{x} + 1$ and $g(\theta) = \frac{1}{5x + 4} + 8$ , we are tasked with determining the transformation of the graph of function $f$ onto the graph of function $g$ . In this article, we will delve into the world of graph transformations and explore the statement that describes the transformation of the graph of function $f$ onto the graph of function $g$ .

The Functions $f$ and $g$

Let's start by examining the given functions:

$f(\theta) = \frac{1}{x} + 1$
$g(\theta) = \frac{1}{5x + 4} + 8$

These functions are defined in terms of the variable $x$ , and they both involve fractions. To understand the transformation of the graph of function $f$ onto the graph of function $g$ , we need to analyze the differences between these two functions.

Horizontal Shift

One of the key differences between the two functions is the presence of the constant term in the denominator of function $g$ . Specifically, the denominator of function $g$ is $5x + 4$ , whereas the denominator of function $f$ is simply $x$ . This difference suggests that the graph of function $g$ may be shifted horizontally compared to the graph of function $f$ .

Vertical Shift

Another difference between the two functions is the presence of the constant term in the numerator of function $g$ . Specifically, the numerator of function $g$ is $8$ , whereas the numerator of function $f$ is $1$ . This difference suggests that the graph of function $g$ may be shifted vertically compared to the graph of function $f$ .

Combining Horizontal and Vertical Shifts

Based on the analysis above, we can conclude that the graph of function $g$ is a combination of a horizontal shift and a vertical shift compared to the graph of function $f$ . To determine the exact nature of this transformation, we need to examine the specific values of the constants involved.

Determining the Transformation

Let's examine the constant term in the denominator of function $g$ , which is $4$ . This constant term suggests that the graph of function $g$ is shifted horizontally by $4$ units to the left compared to the graph of function $f$ . Additionally, the constant term in the numerator of function $g$ , which is $8$ , suggests that the graph of function $g$ is shifted vertically by $7$ units upwards compared to the graph of function $f$ .

Conclusion

In conclusion, the graph of function $g$ is a combination of a horizontal shift and a vertical shift compared to the graph of function $f$ . Specifically, the graph of function $g$ is shifted horizontally by $4$ units to the left and vertically by $7$ units upwards compared to the graph of function $f$ . Therefore, the statement that describes the transformation of the graph of function $f$ onto the graph of function $g$ is:

The graph shifts 7 units upwards.

Final Answer

Introduction

In our previous article, we explored the transformation of the graph of function $f$ onto the graph of function $g$ . We analyzed the differences between the two functions and determined that the graph of function $g$ is a combination of a horizontal shift and a vertical shift compared to the graph of function $f$ . In this article, we will answer some frequently asked questions about graph transformations and provide additional insights into this fascinating topic.

Q: What is a horizontal shift in graph transformations?

A: A horizontal shift in graph transformations refers to a change in the position of the graph along the x-axis. When a graph is shifted horizontally, its x-coordinates change, but its y-coordinates remain the same.

Q: What is a vertical shift in graph transformations?

A: A vertical shift in graph transformations refers to a change in the position of the graph along the y-axis. When a graph is shifted vertically, its y-coordinates change, but its x-coordinates remain the same.

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

A: To determine the type of shift in a graph transformation, you need to examine the function that is being transformed. If the function has a constant term in the denominator, it may indicate a horizontal shift. If the function has a constant term in the numerator, it may indicate a vertical shift.

Q: Can a graph be shifted both horizontally and vertically?

A: Yes, a graph can be shifted both horizontally and vertically. In fact, most graph transformations involve a combination of both horizontal and vertical shifts.

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

A: To graph a function with a horizontal shift, you need to shift the graph of the original function by the specified amount along the x-axis. For example, if the original function is $f(x)$ and the shifted function is $f(x - c)$ , you need to shift the graph of $f(x)$ to the right by $c$ units.

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

A: To graph a function with a vertical shift, you need to shift the graph of the original function by the specified amount along the y-axis. For example, if the original function is $f(x)$ and the shifted function is $f(x) + c$ , you need to shift the graph of $f(x)$ upwards by $c$ units.

Q: What are some common types of graph transformations?

A: Some common types of graph transformations include:

Horizontal shifts: $f(x - c)$
Vertical shifts: $f(x) + c$
Reflections: $f(-x)$
Stretches and compressions: $af(x)$
Rotations: $f(\sqrt{x^2 + y^2})$

Conclusion

In conclusion, graph transformations are an essential concept in mathematics that help us understand how functions change and behave under different conditions. By understanding the different types of graph transformations, we can analyze and visualize complex functions and relationships. We hope that this Q&A article has provided you with a deeper understanding of graph transformations and has inspired you to explore this fascinating topic further.

Final Answer

The final answer is: Graph transformations involve a combination of horizontal and vertical shifts, reflections, stretches and compressions, and rotations.