vertical translation of 1 unit downward

Oct 13, 2025 by pynkocean.com 100 views

Introduction

In mathematics, transformations are essential concepts that help us understand how functions change and behave under different conditions. When a function is translated, it means that its graph is shifted horizontally or vertically, resulting in a new function. In this article, we will explore the transformations that occur when the graph of the parent function $f(x) = x^3$ is translated to represent the new function $g(x) = (x - 4)^3 - 1$ .

Horizontal Translation

The function $g(x) = (x - 4)^3 - 1$ can be rewritten as $g(x) = (x - 4)^3 - 1 = f(x - 4) - 1$ . This indicates that the graph of $f(x)$ has been shifted 4 units to the right. This is because the term $(x - 4)$ inside the parentheses represents a horizontal shift of 4 units to the right.

Vertical Translation

In addition to the horizontal translation, the function $g(x)$ also undergoes a vertical translation of 1 unit downward. This is evident from the term $-1$ that is subtracted from the function. This means that the graph of $f(x)$ has been shifted 1 unit downward.

Combining Transformations

When we combine the horizontal and vertical translations, we get the new function $g(x) = (x - 4)^3 - 1$ . This function represents the graph of the parent function $f(x) = x^3$ after undergoing both horizontal and vertical translations.

Understanding the Graph

To visualize the graph of $g(x)$ , we can start with the graph of $f(x) = x^3$ and apply the transformations. The graph of $f(x)$ is a cubic function that opens upward. When we shift the graph 4 units to the right, the vertex of the graph moves from $(0, 0)$ to $(4, 0)$ . Then, when we shift the graph 1 unit downward, the vertex moves from $(4, 0)$ to $(4, -1)$ .

Conclusion

In conclusion, the graph of the parent function $f(x) = x^3$ is translated to represent the new function $g(x) = (x - 4)^3 - 1$ . The transformations that occur are a horizontal translation of 4 units to the right and a vertical translation of 1 unit downward. By understanding these transformations, we can visualize the graph of $g(x)$ and appreciate the changes that occur when a function is translated.

Key Takeaways

The function $g(x) = (x - 4)^3 - 1$ represents the graph of the parent function $f(x) = x^3$ after undergoing both horizontal and vertical translations.
The graph of $f(x)$ is shifted 4 units to the right and 1 unit downward to represent the graph of $g(x)$ .
Understanding transformations is essential in mathematics, as it helps us visualize and analyze the behavior of functions.

Real-World Applications

Transformations have numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, transformations are used to design and analyze complex systems, such as bridges and buildings. In physics, transformations are used to describe the motion of objects and predict their behavior under different conditions. In computer science, transformations are used to develop algorithms and data structures that can efficiently process and analyze large datasets.