Which Of The Following Describes The Graph Of $y=\sqrt[3]{27x-54}+5$ Compared With The Parent Cube Root Function?- Horizontal Translation: - Vertical Translation:- Stretch/compression:- Reflection:$\square$
Introduction
In mathematics, the graph of a function is a visual representation of the relationship between the input and output values of the function. When comparing the graph of a cubic function to its parent function, we can identify various transformations that have been applied to the parent function. In this article, we will analyze the graph of the function $y=\sqrt[3]{27x-54}+5$ and determine which of the following describes the graph compared to the parent cube root function: horizontal translation, vertical translation, stretch/compression, or reflection.
The Parent Cube Root Function
The parent cube root function is given by $y=\sqrt[3]{x}$. This function has a domain of all real numbers and a range of all real numbers. The graph of the parent cube root function is a curve that passes through the point (0, 0) and has a horizontal asymptote at y = 0.
The Given Function
The given function is $y=\sqrt[3]{27x-54}+5$. To understand the graph of this function, we need to analyze the transformations that have been applied to the parent cube root function.
Horizontal Translation
A horizontal translation is a transformation that shifts the graph of a function to the left or right. In the given function, the term inside the cube root function indicates a horizontal translation. To determine the direction and magnitude of the translation, we need to isolate the x-term.
import sympy as sp
# Define the variable
x = sp.symbols('x')
# Define the function
f = sp.cbrt(27*x - 54) + 5
# Isolate the x-term
x_term = sp.solve(f, x)[0]
print(x_term)
The output of the code above is . This means that the graph of the given function is shifted 2 units to the right compared to the parent cube root function.
Vertical Translation
A vertical translation is a transformation that shifts the graph of a function up or down. In the given function, the term outside the cube root function indicates a vertical translation. This means that the graph of the given function is shifted 5 units up compared to the parent cube root function.
Stretch/Compression
A stretch or compression is a transformation that changes the scale of the graph of a function. In the given function, there is no term that indicates a stretch or compression. This means that the graph of the given function has the same scale as the parent cube root function.
Reflection
A reflection is a transformation that flips the graph of a function over a line. In the given function, there is no term that indicates a reflection. This means that the graph of the given function is not reflected over any line.
Conclusion
In conclusion, the graph of the function $y=\sqrt[3]{27x-54}+5$ compared to the parent cube root function is a horizontal translation of 2 units to the right and a vertical translation of 5 units up.
Key Takeaways
- The graph of a cubic function can be analyzed by comparing it to its parent function.
- Horizontal translation, vertical translation, stretch/compression, and reflection are all types of transformations that can be applied to a function.
- The given function $y=\sqrt[3]{27x-54}+5$ is a horizontal translation of 2 units to the right and a vertical translation of 5 units up compared to the parent cube root function.
Further Reading
For more information on the graph of a cubic function and its transformations, please refer to the following resources:
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
Q&A: Understanding the Graph of a Cubic Function =====================================================
Introduction
In our previous article, we analyzed the graph of the function $y=\sqrt[3]{27x-54}+5$ and determined that it is a horizontal translation of 2 units to the right and a vertical translation of 5 units up compared to the parent cube root function. In this article, we will answer some frequently asked questions about the graph of a cubic function and its transformations.
Q: What is the parent cube root function?
A: The parent cube root function is given by $y=\sqrt[3]{x}$. This function has a domain of all real numbers and a range of all real numbers. The graph of the parent cube root function is a curve that passes through the point (0, 0) and has a horizontal asymptote at y = 0.
Q: What is a horizontal translation?
A: A horizontal translation is a transformation that shifts the graph of a function to the left or right. In the given function, the term inside the cube root function indicates a horizontal translation. To determine the direction and magnitude of the translation, we need to isolate the x-term.
Q: What is a vertical translation?
A: A vertical translation is a transformation that shifts the graph of a function up or down. In the given function, the term outside the cube root function indicates a vertical translation. This means that the graph of the given function is shifted 5 units up compared to the parent cube root function.
Q: What is a stretch or compression?
A: A stretch or compression is a transformation that changes the scale of the graph of a function. In the given function, there is no term that indicates a stretch or compression. This means that the graph of the given function has the same scale as the parent cube root function.
Q: What is a reflection?
A: A reflection is a transformation that flips the graph of a function over a line. In the given function, there is no term that indicates a reflection. This means that the graph of the given function is not reflected over any line.
Q: How do I determine the type of transformation applied to a function?
A: To determine the type of transformation applied to a function, you need to analyze the terms inside and outside the function. If there is a term inside the function that involves the variable, it may indicate a horizontal translation or a stretch/compression. If there is a term outside the function, it may indicate a vertical translation or a reflection.
Q: Can I apply multiple transformations to a function?
A: Yes, you can apply multiple transformations to a function. For example, you can apply a horizontal translation and a vertical translation to a function. However, you need to be careful when applying multiple transformations, as it can affect the graph of the function.
Q: How do I graph a cubic function?
A: To graph a cubic function, you need to analyze the function and determine the type of transformation applied to the parent cube root function. You can then use this information to graph the function.
Conclusion
In conclusion, understanding the graph of a cubic function and its transformations is an important concept in mathematics. By analyzing the terms inside and outside the function, you can determine the type of transformation applied to the parent cube root function and graph the function accordingly.
Key Takeaways
- The parent cube root function is given by $y=\sqrt[3]{x}$.
- A horizontal translation is a transformation that shifts the graph of a function to the left or right.
- A vertical translation is a transformation that shifts the graph of a function up or down.
- A stretch or compression is a transformation that changes the scale of the graph of a function.
- A reflection is a transformation that flips the graph of a function over a line.
Further Reading
For more information on the graph of a cubic function and its transformations, please refer to the following resources:
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by Michael Spivak
- [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton