Plot the vertical asymptote

Oct 13, 2025 by magicalceylontours.com 122 views

Introduction

In mathematics, graphing functions is a crucial concept that helps us visualize and understand the behavior of different types of functions. One of the most common types of functions is the radical function, which involves the use of roots or radicals. In this article, we will explore the graph of the function $$y=\sqrt[3]{x-5}$$ and determine which graph represents it.

Understanding the Function

The given function is $$y=\sqrt[3]{x-5}$$. This is a radical function with a cube root, which means that the function will have a different behavior compared to a square root function. The cube root function is defined as the inverse of the cube of a number, i.e., $$\sqrt[3]{x} = x^{1/3}$$.

Graphing the Function

To graph the function $$y=\sqrt[3]{x-5}$$, we need to consider the following key points:

Domain: The domain of the function is all real numbers, i.e., $$x \in (-\infty, \infty)$$.
Range: The range of the function is all real numbers greater than or equal to 0, i.e., $$y \geq 0$$.
Asymptote: The function has a vertical asymptote at $$x=5$$, which means that the function will approach infinity as $$x$$ approaches 5 from the left.
Intercepts: The function has a y-intercept at $$(0, -1)$$, which means that the function will intersect the y-axis at the point $$(0, -1)$$.

Graphing the Function on a Coordinate Plane

To graph the function $$y=\sqrt[3]{x-5}$$ on a coordinate plane, we can use the following steps:

Plot the vertical asymptote: Plot the vertical line $$x=5$$ on the coordinate plane.
Plot the y-intercept: Plot the point $$(0, -1)$$ on the coordinate plane.
Plot the function: Plot the function $$y=\sqrt[3]{x-5}$$ on the coordinate plane by using the key points mentioned above.

Determining the Correct Graph

Based on the graphing steps mentioned above, we can determine which graph represents the function $$y=\sqrt[3]{x-5}$$. The correct graph will have the following characteristics:

Vertical asymptote: The graph will have a vertical asymptote at $$x=5$$.
Y-intercept: The graph will intersect the y-axis at the point $$(0, -1)$$.
Shape: The graph will have a shape that is similar to a cube root function, with a positive slope and a horizontal asymptote at $$y=0$$.

Conclusion

In conclusion, the graph that represents the function $$y=\sqrt[3]{x-5}$$ is the graph that has a vertical asymptote at $$x=5$$, a y-intercept at $$(0, -1)$$, and a shape that is similar to a cube root function. This graph provides a visual representation of the function and helps us understand its behavior.

Frequently Asked Questions

What is the domain of the function $$y=\sqrt[3]{x-5}$$? The domain of the function is all real numbers, i.e., $$x \in (-\infty, \infty)$$.
What is the range of the function $$y=\sqrt[3]{x-5}$$? The range of the function is all real numbers greater than or equal to 0, i.e., $$y \geq 0$$.
What is the vertical asymptote of the function $$y=\sqrt[3]{x-5}$$? The vertical asymptote of the function is $$x=5$$.

References

Graphing Functions: A comprehensive guide to graphing functions, including radical functions.
Cube Root Function: A detailed explanation of the cube root function and its properties.
Asymptotes: A guide to understanding and graphing asymptotes.

Final Thoughts

Graphing functions is an essential concept in mathematics that helps us visualize and understand the behavior of different types of functions. The graph of the function $$y=\sqrt[3]{x-5}$$ provides a visual representation of the function and helps us understand its behavior. By understanding the key points of the function, we can determine which graph represents it and gain a deeper understanding of the function's behavior.

Q&A Article

In this article, we will answer some of the most frequently asked questions about the function $$y=\sqrt[3]{x-5}$$.

Question 1: What is the domain of the function $$y=\sqrt[3]{x-5}$$?

Answer: The domain of the function is all real numbers, i.e., $$x \in (-\infty, \infty)$$.

Question 2: What is the range of the function $$y=\sqrt[3]{x-5}$$?

Answer: The range of the function is all real numbers greater than or equal to 0, i.e., $$y \geq 0$$.

Question 3: What is the vertical asymptote of the function $$y=\sqrt[3]{x-5}$$?

Answer: The vertical asymptote of the function is $$x=5$$.

Question 4: What is the y-intercept of the function $$y=\sqrt[3]{x-5}$$?

Answer: The y-intercept of the function is $$(0, -1)$$.

Question 5: How do I graph the function $$y=\sqrt[3]{x-5}$$?

Answer: To graph the function, plot the vertical asymptote at $$x=5$$, plot the y-intercept at $$(0, -1)$$, and then plot the function using the key points mentioned above.

Question 6: What is the shape of the graph of the function $$y=\sqrt[3]{x-5}$$?

Answer: The graph of the function has a shape that is similar to a cube root function, with a positive slope and a horizontal asymptote at $$y=0$$.

Question 7: How do I determine which graph represents the function $$y=\sqrt[3]{x-5}$$?

Answer: To determine which graph represents the function, look for the graph that has a vertical asymptote at $$x=5$$, a y-intercept at $$(0, -1)$$, and a shape that is similar to a cube root function.

Question 8: What is the significance of the cube root function in mathematics?

Answer: The cube root function is an important concept in mathematics that is used to solve equations and inequalities involving cube roots. It is also used in various applications such as physics, engineering, and computer science.

Question 9: How do I use the cube root function to solve equations and inequalities?

Answer: To use the cube root function to solve equations and inequalities, first isolate the cube root term, then cube both sides of the equation or inequality to eliminate the cube root.

Question 10: What are some common applications of the cube root function?

Answer: The cube root function has various applications in physics, engineering, and computer science, such as calculating volumes of cubes, solving equations involving cube roots, and modeling real-world phenomena.

Conclusion

In conclusion, the function $$y=\sqrt[3]{x-5}$$ is a radical function with a cube root that has a domain of all real numbers, a range of all real numbers greater than or equal to 0, and a vertical asymptote at $$x=5$$. The graph of the function has a shape that is similar to a cube root function, with a positive slope and a horizontal asymptote at $$y=0$$. By understanding the key points of the function, we can determine which graph represents it and gain a deeper understanding of the function's behavior.

Frequently Asked Questions

What is the domain of the function $$y=\sqrt[3]{x-5}$$? The domain of the function is all real numbers, i.e., $$x \in (-\infty, \infty)$$.
What is the range of the function $$y=\sqrt[3]{x-5}$$? The range of the function is all real numbers greater than or equal to 0, i.e., $$y \geq 0$$.
What is the vertical asymptote of the function $$y=\sqrt[3]{x-5}$$? The vertical asymptote of the function is $$x=5$$.

References

Graphing Functions: A comprehensive guide to graphing functions, including radical functions.
Cube Root Function: A detailed explanation of the cube root function and its properties.
Asymptotes: A guide to understanding and graphing asymptotes.

Final Thoughts

The function $$y=\sqrt[3]{x-5}$$ is a radical function with a cube root that has a domain of all real numbers, a range of all real numbers greater than or equal to 0, and a vertical asymptote at $$x=5$$. By understanding the key points of the function, we can determine which graph represents it and gain a deeper understanding of the function's behavior.