Which Of The Following Describes The Graph Of $y=\sqrt{-4x-36}$ Compared To The Parent Square Root Function?A. Stretched By A Factor Of 2, Reflected Over The $x$-axis, And Translated 9 Units Right B. Stretched By A Factor Of 2,

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Introduction

In mathematics, transformations of functions are essential to understand and analyze various types of functions. The square root function is a fundamental function that can be transformed in different ways to create new functions. In this article, we will discuss the transformations of the square root function and compare the graph of $y=\sqrt{-4x-36}$ to the parent square root function.

Parent Square Root Function

The parent square root function is defined as $y=\sqrt{x}$. This function has a domain of $x\geq0$ and a range of $y\geq0$. The graph of the parent square root function is a curve that starts at the origin and increases as $x$ increases.

Transformations of Square Root Functions

There are several types of transformations that can be applied to the square root function, including:

  • Vertical Stretching: This transformation involves multiplying the function by a constant factor. For example, the function $y=2\sqrt{x}$ is a vertical stretch of the parent square root function by a factor of 2.
  • Horizontal Stretching: This transformation involves replacing $x$ with $ax$, where $a$ is a constant factor. For example, the function $y=\sqrt{ax}$ is a horizontal stretch of the parent square root function by a factor of $a$.
  • Reflection: This transformation involves reflecting the function over the $x$-axis or the $y$-axis. For example, the function $y=-\sqrt{x}$ is a reflection of the parent square root function over the $x$-axis.
  • Translation: This transformation involves shifting the function horizontally or vertically. For example, the function $y=\sqrt{x+1}$ is a translation of the parent square root function 1 unit to the left.

Comparing the Graphs

Now that we have discussed the transformations of the square root function, let's compare the graph of $y=\sqrt{-4x-36}$ to the parent square root function.

Vertical Stretching and Reflection

The graph of $y=\sqrt{-4x-36}$ can be obtained by applying a vertical stretch by a factor of 2 and a reflection over the $x$-axis to the parent square root function.

y = \sqrt{-4x-36} = 2\sqrt{-x-9}

This transformation involves multiplying the function by a constant factor of 2 and replacing $x$ with $-x$ to reflect the function over the $x$-axis.

Horizontal Translation

The graph of $y=\sqrt{-4x-36}$ can also be obtained by applying a horizontal translation of 9 units to the left to the parent square root function.

y = \sqrt{-4x-36} = \sqrt{-x-9}

This transformation involves replacing $x$ with $x+9$ to shift the function 9 units to the left.

Conclusion

In conclusion, the graph of $y=\sqrt{-4x-36}$ can be obtained by applying a vertical stretch by a factor of 2 and a reflection over the $x$-axis to the parent square root function. Alternatively, it can be obtained by applying a horizontal translation of 9 units to the left to the parent square root function.

Key Takeaways

  • The parent square root function is defined as $y=\sqrt{x}$.
  • The graph of the parent square root function is a curve that starts at the origin and increases as $x$ increases.
  • The graph of $y=\sqrt{-4x-36}$ can be obtained by applying a vertical stretch by a factor of 2 and a reflection over the $x$-axis to the parent square root function.
  • The graph of $y=\sqrt{-4x-36}$ can also be obtained by applying a horizontal translation of 9 units to the left to the parent square root function.

References

  • [1] "Transformations of Functions" by Khan Academy
  • [2] "Square Root Function" by Math Open Reference

Further Reading

  • "Transformations of Functions" by IXL
  • "Square Root Function" by Purplemath
    Frequently Asked Questions (FAQs) about Transformations of Square Root Functions ====================================================================================

Introduction

In our previous article, we discussed the transformations of square root functions and compared the graph of $y=\sqrt{-4x-36}$ to the parent square root function. In this article, we will answer some frequently asked questions (FAQs) about transformations of square root functions.

Q&A

Q: What is the parent square root function?

A: The parent square root function is defined as $y=\sqrt{x}$. This function has a domain of $x\geq0$ and a range of $y\geq0$. The graph of the parent square root function is a curve that starts at the origin and increases as $x$ increases.

Q: What are the different types of transformations that can be applied to the square root function?

A: There are several types of transformations that can be applied to the square root function, including:

  • Vertical Stretching: This transformation involves multiplying the function by a constant factor. For example, the function $y=2\sqrt{x}$ is a vertical stretch of the parent square root function by a factor of 2.
  • Horizontal Stretching: This transformation involves replacing $x$ with $ax$, where $a$ is a constant factor. For example, the function $y=\sqrt{ax}$ is a horizontal stretch of the parent square root function by a factor of $a$.
  • Reflection: This transformation involves reflecting the function over the $x$-axis or the $y$-axis. For example, the function $y=-\sqrt{x}$ is a reflection of the parent square root function over the $x$-axis.
  • Translation: This transformation involves shifting the function horizontally or vertically. For example, the function $y=\sqrt{x+1}$ is a translation of the parent square root function 1 unit to the left.

Q: How do I determine the type of transformation applied to a square root function?

A: To determine the type of transformation applied to a square root function, you need to analyze the function and identify the following:

  • Vertical Stretching: If the function is multiplied by a constant factor, it is a vertical stretch.
  • Horizontal Stretching: If $x$ is replaced with $ax$, it is a horizontal stretch.
  • Reflection: If the function is multiplied by $-1$, it is a reflection over the $x$-axis.
  • Translation: If the function is shifted horizontally or vertically, it is a translation.

Q: Can I apply multiple transformations to a square root function?

A: Yes, you can apply multiple transformations to a square root function. For example, you can apply a vertical stretch by a factor of 2 and a reflection over the $x$-axis to the parent square root function.

Q: How do I graph a square root function with multiple transformations?

A: To graph a square root function with multiple transformations, you need to apply each transformation in the correct order. For example, if you want to graph the function $y=2\sqrt{-x-9}$, you need to apply a vertical stretch by a factor of 2 and a reflection over the $x$-axis to the parent square root function.

Conclusion

In conclusion, transformations of square root functions are essential to understand and analyze various types of functions. By applying different types of transformations, you can create new functions and graph them accurately. We hope this article has helped you understand the basics of transformations of square root functions and how to apply them.

Key Takeaways

  • The parent square root function is defined as $y=\sqrt{x}$.
  • There are several types of transformations that can be applied to the square root function, including vertical stretching, horizontal stretching, reflection, and translation.
  • To determine the type of transformation applied to a square root function, you need to analyze the function and identify the type of transformation.
  • You can apply multiple transformations to a square root function.
  • To graph a square root function with multiple transformations, you need to apply each transformation in the correct order.

References

  • [1] "Transformations of Functions" by Khan Academy
  • [2] "Square Root Function" by Math Open Reference

Further Reading

  • "Transformations of Functions" by IXL
  • "Square Root Function" by Purplemath