Function Translation and the Coordinate Grid

Oct 13, 2025 by windailys.com 41 views

Introduction to the Coordinate Grid and Function Translation

The coordinate grid is a fundamental concept in mathematics, used to represent points and functions in a two-dimensional space. When dealing with functions, it's essential to understand how they are translated and how this affects their domain. In this article, we will explore the concept of function translation, specifically focusing on the graph of $$y=\sqrt[3]{x-1}+3$$ and its relationship to the function $$y=\sqrt[3]{x}$$.

Function Translation and the Coordinate Grid

Function translation refers to the process of shifting a function's graph up, down, left, or right by a certain value. In the case of the function $$y=\sqrt[3]{x-1}+3$$, we can see that it is a translation of the function $$y=\sqrt[3]{x}$$. The translation occurs in two parts: a horizontal translation and a vertical translation.

Horizontal Translation

The horizontal translation is represented by the value inside the parentheses, which is $$x-1$$. This means that the graph of $$y=\sqrt[3]{x-1}+3$$ is shifted one unit to the right compared to the graph of $$y=\sqrt[3]{x}$$.

Vertical Translation

The vertical translation is represented by the value added to the function, which is $$+3$$. This means that the graph of $$y=\sqrt[3]{x-1}+3$$ is shifted three units up compared to the graph of $$y=\sqrt[3]{x}$$.

Domain of the Graphed Function

Now that we have understood the concept of function translation and its relationship to the coordinate grid, let's focus on finding the domain of the graphed function. The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in an undefined or imaginary output.

Domain of the Original Function

The original function is $$y=\sqrt[3]{x}$$. Since the cube root of any real number is defined, the domain of this function is all real numbers, which can be represented as $$(-\infty, \infty)$$.

Domain of the Translated Function

The translated function is $$y=\sqrt[3]{x-1}+3$$. To find the domain of this function, we need to consider the restrictions imposed by the horizontal translation. Since the graph is shifted one unit to the right, the domain of the translated function is all real numbers greater than 1, which can be represented as $$(1, \infty)$$.

Conclusion

In conclusion, the domain of the graphed function $$y=\sqrt[3]{x-1}+3$$ is all real numbers greater than 1, which can be represented as $$(1, \infty)$$. This is because the horizontal translation shifts the graph one unit to the right, resulting in a domain that starts at 1 and extends to infinity.

Final Answer

The final answer is: $\boxed{(1, \infty)}$

Introduction to the Coordinate Grid and Function Translation

In our previous article, we explored the concept of function translation and its relationship to the coordinate grid. We specifically focused on the graph of $$y=\sqrt[3]{x-1}+3$$ and its relationship to the function $$y=\sqrt[3]{x}$$. In this article, we will answer some frequently asked questions related to the coordinate grid and domain of a function.

Q&A

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in an undefined or imaginary output. The range of a function is the set of all possible output values (y-values) that the function can produce.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to consider the restrictions imposed by the function itself. For example, if a function has a square root, the domain will be restricted to non-negative values. If a function has a cube root, the domain will be all real numbers.

Q: What is the significance of the horizontal translation in a function?

A: The horizontal translation in a function represents a shift of the graph to the left or right. This can affect the domain of the function, as we saw in the example of $$y=\sqrt[3]{x-1}+3$$.

Q: Can a function have a domain that is not a set of real numbers?

A: Yes, a function can have a domain that is not a set of real numbers. For example, a function that involves complex numbers or imaginary numbers may have a domain that includes these values.

Q: How do I graph a function on the coordinate grid?

A: To graph a function on the coordinate grid, you need to identify the x-values and corresponding y-values of the function. You can then plot these points on the grid and connect them to form the graph of the function.

Q: What is the difference between a function and a relation?

A: A function is a relation where each input value (x-value) corresponds to exactly one output value (y-value). A relation, on the other hand, is a set of ordered pairs where each input value may correspond to multiple output values.

Conclusion

In conclusion, understanding the coordinate grid and domain of a function is essential in mathematics. By answering these frequently asked questions, we hope to have provided a better understanding of these concepts and how they relate to each other.

Final Answer

The final answer is: $\boxed{(1, \infty)}$