Select The Correct Answer.The Parent Square Root Function, { F $}$, Is Transformed To Create Function { G $} . . . { G(x) = \sqrt{x+3} - 4 \} Which Statement Is True?A. The Graph Of { F $}$ Is Translated 3 Units
The parent square root function, f }, is a fundamental concept in mathematics, and its transformation is a crucial aspect of understanding various mathematical functions. In this article, we will explore the transformation of the parent square root function to create function <span class="katex-error" title="ParseError', got 'EOF' at end of input: " style="color$, and determine which statement is true.
The Parent Square Root Function
The parent square root function is defined as { f(x) = \sqrt{x} }$. This function has a domain of { x \geq 0 }$ and a range of { y \geq 0 }$. The graph of the parent square root function is a curve that starts at the origin and increases as x increases.
The Transformed Function
The transformed function, { g(x) = \sqrt{x+3} - 4 }$, is created by applying a series of transformations to the parent square root function. To understand the transformation, let's break it down step by step.
- Horizontal Translation: The function { g(x) = \sqrt{x+3} }$ is a horizontal translation of the parent square root function. This means that the graph of { g(x) }$ is shifted 3 units to the left compared to the graph of { f(x) }$.
- Vertical Translation: The function { g(x) = \sqrt{x+3} - 4 }$ is a vertical translation of the horizontal translation. This means that the graph of { g(x) }$ is shifted 4 units down compared to the graph of { \sqrt{x+3} }$.
Which Statement is True?
Now that we have understood the transformation of the parent square root function, let's examine the given statements and determine which one is true.
A. The graph of { f $}$ is translated 3 units B. The graph of { f $}$ is translated 4 units C. The graph of { f $}$ is translated 3 units down and 4 units left D. The graph of { f $}$ is translated 3 units left and 4 units down
Based on our analysis, the correct answer is:
A. The graph of { f $}$ is translated 3 units
The graph of { f $}$ is translated 3 units to the left, which is represented by the term { x+3 }$ inside the square root function.
Conclusion
In conclusion, the transformation of the parent square root function to create function { g $}$ involves a horizontal translation of 3 units and a vertical translation of 4 units. The correct statement is that the graph of { f $}$ is translated 3 units to the left.
Key Takeaways
- The parent square root function is defined as { f(x) = \sqrt{x} }$.
- The transformed function, { g(x) = \sqrt{x+3} - 4 }$, is created by applying a series of transformations to the parent square root function.
- The graph of { g(x) }$ is shifted 3 units to the left and 4 units down compared to the graph of { f(x) }$.
- The correct statement is that the graph of { f $}$ is translated 3 units to the left.
Frequently Asked Questions
Q: What is the parent square root function?
A: The parent square root function is defined as { f(x) = \sqrt{x} }$.
Q: What is the transformed function?
A: The transformed function is { g(x) = \sqrt{x+3} - 4 }$.
Q: What is the horizontal translation of the parent square root function?
A: The horizontal translation of the parent square root function is 3 units to the left.
Q: What is the vertical translation of the parent square root function?
A: The vertical translation of the parent square root function is 4 units down.
Q: Which statement is true?
A: The graph of { f $}$ is translated 3 units to the left.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by James Stewart
- [3] "Mathematics for the Nonmathematician" by Morris Kline
Q&A: Understanding the Transformation of the Parent Square Root Function ====================================================================
In our previous article, we explored the transformation of the parent square root function to create function { g $}$. We analyzed the horizontal and vertical translations of the parent square root function and determined that the correct statement is that the graph of { f $}$ is translated 3 units to the left.
In this article, we will continue to answer frequently asked questions related to the transformation of the parent square root function.
Q: What is the domain of the parent square root function?
A: The domain of the parent square root function is { x \geq 0 }$.
Q: What is the range of the parent square root function?
A: The range of the parent square root function is { y \geq 0 }$.
Q: How do you determine the horizontal translation of the parent square root function?
A: To determine the horizontal translation of the parent square root function, you need to look at the term inside the square root function. If the term is { x+a }$, then the horizontal translation is { a }$ units to the left.
Q: How do you determine the vertical translation of the parent square root function?
A: To determine the vertical translation of the parent square root function, you need to look at the term outside the square root function. If the term is { b }$, then the vertical translation is { b }$ units down.
Q: What is the difference between a horizontal translation and a vertical translation?
A: A horizontal translation is a shift in the x-direction, while a vertical translation is a shift in the y-direction.
Q: Can you give an example of a horizontal translation?
A: Yes, an example of a horizontal translation is { f(x) = \sqrt{x+2} }$. In this case, the horizontal translation is 2 units to the left.
Q: Can you give an example of a vertical translation?
A: Yes, an example of a vertical translation is { f(x) = \sqrt{x} - 3 }$. In this case, the vertical translation is 3 units down.
Q: How do you graph the parent square root function?
A: To graph the parent square root function, you need to start at the origin and draw a curve that increases as x increases.
Q: How do you graph the transformed function?
A: To graph the transformed function, you need to apply the horizontal and vertical translations to the parent square root function.
Q: What is the significance of the parent square root function?
A: The parent square root function is a fundamental concept in mathematics, and its transformation is a crucial aspect of understanding various mathematical functions.
Q: Can you give an example of a real-world application of the parent square root function?
A: Yes, an example of a real-world application of the parent square root function is in physics, where it is used to model the motion of objects.
Conclusion
In conclusion, the transformation of the parent square root function is a crucial aspect of understanding various mathematical functions. By understanding the horizontal and vertical translations of the parent square root function, you can apply this knowledge to real-world problems.
Key Takeaways
- The domain of the parent square root function is { x \geq 0 }$.
- The range of the parent square root function is { y \geq 0 }$.
- The horizontal translation of the parent square root function is determined by the term inside the square root function.
- The vertical translation of the parent square root function is determined by the term outside the square root function.
- A horizontal translation is a shift in the x-direction, while a vertical translation is a shift in the y-direction.
Frequently Asked Questions
Q: What is the parent square root function?
A: The parent square root function is defined as { f(x) = \sqrt{x} }$.
Q: What is the transformed function?
A: The transformed function is { g(x) = \sqrt{x+3} - 4 }$.
Q: What is the horizontal translation of the parent square root function?
A: The horizontal translation of the parent square root function is 3 units to the left.
Q: What is the vertical translation of the parent square root function?
A: The vertical translation of the parent square root function is 4 units down.
Q: Which statement is true?
A: The graph of { f $}$ is translated 3 units to the left.
References
- [1] "Algebra and Trigonometry" by Michael Sullivan
- [2] "Calculus" by James Stewart
- [3] "Mathematics for the Nonmathematician" by Morris Kline