Which Of The Following Equations Best Describes A Square Root Function That Is Reflected Across The \[$x\$\]-axis And Has A Vertex Of \[$(-4, 2)\$\]?A. \[$y = \sqrt{-(x-4)} + 2\$\]B. \[$y = -\sqrt{x-4} + 2\$\]C. \[$y

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Introduction

In mathematics, a square root function is a type of function that involves the square root of a variable. It is commonly represented as y=xy = \sqrt{x}, where xx is the input and yy is the output. However, square root functions can be modified and transformed in various ways, such as reflecting them across the x-axis. In this article, we will explore which equation best describes a square root function that is reflected across the x-axis and has a vertex of (-4, 2).

Reflection Across the x-axis

When a function is reflected across the x-axis, its graph is flipped upside down. This means that the y-values of the function are negated. In other words, if the original function is y=f(x)y = f(x), then the reflected function is y=βˆ’f(x)y = -f(x).

Vertex Form of a Square Root Function

The vertex form of a square root function is given by y=axβˆ’h+ky = a\sqrt{x-h} + k, where (h,k)(h, k) is the vertex of the function. In this case, the vertex is given as (-4, 2), so we can write the equation as y=axβˆ’(βˆ’4)+2y = a\sqrt{x-(-4)} + 2.

Equation Options

We are given three equation options to choose from:

A. y=βˆ’(xβˆ’4)+2y = \sqrt{-(x-4)} + 2 B. y=βˆ’xβˆ’4+2y = -\sqrt{x-4} + 2 C. y=xβˆ’4+2y = \sqrt{x-4} + 2

Analyzing Equation Option A

Equation option A is y=βˆ’(xβˆ’4)+2y = \sqrt{-(x-4)} + 2. To analyze this equation, we need to understand what the expression βˆ’(xβˆ’4)\sqrt{-(x-4)} represents. The expression inside the square root is βˆ’(xβˆ’4)-(x-4), which is a negative value. This means that the square root of a negative value is an imaginary number. However, in this case, we are not dealing with imaginary numbers, so we need to consider the absolute value of the expression inside the square root.

The absolute value of βˆ’(xβˆ’4)-(x-4) is βˆ£βˆ’(xβˆ’4)∣=xβˆ’4|-(x-4)| = x-4. Therefore, the equation option A can be rewritten as y=xβˆ’4+2y = \sqrt{x-4} + 2. However, this is not the correct equation, as it does not reflect the function across the x-axis.

Analyzing Equation Option B

Equation option B is y=βˆ’xβˆ’4+2y = -\sqrt{x-4} + 2. This equation reflects the function across the x-axis, as the y-values are negated. However, the vertex of the function is given as (-4, 2), which means that the equation should be in the form y=axβˆ’(βˆ’4)+2y = a\sqrt{x-(-4)} + 2. To match this form, we need to rewrite the equation as y=βˆ’x+4+2y = -\sqrt{x+4} + 2. However, this is not the correct equation, as it does not match the given vertex.

Analyzing Equation Option C

Equation option C is y=xβˆ’4+2y = \sqrt{x-4} + 2. This equation does not reflect the function across the x-axis, as the y-values are not negated. However, the vertex of the function is given as (-4, 2), which means that the equation should be in the form y=axβˆ’(βˆ’4)+2y = a\sqrt{x-(-4)} + 2. To match this form, we need to rewrite the equation as y=x+4+2y = \sqrt{x+4} + 2. However, this is not the correct equation, as it does not match the given vertex.

Conclusion

After analyzing all three equation options, we can conclude that none of them match the given vertex of (-4, 2). However, we can rewrite equation option B as y=βˆ’x+4+2y = -\sqrt{x+4} + 2, which reflects the function across the x-axis and matches the given vertex.

Final Answer

Introduction

In our previous article, we explored the concept of square root functions and reflections across the x-axis. We analyzed three equation options to determine which one best describes a square root function that is reflected across the x-axis and has a vertex of (-4, 2). In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on this topic.

Q: What is a square root function?

A: A square root function is a type of function that involves the square root of a variable. It is commonly represented as y=xy = \sqrt{x}, where xx is the input and yy is the output.

Q: What is a reflection across the x-axis?

A: A reflection across the x-axis is a transformation that flips the graph of a function upside down. This means that the y-values of the function are negated.

Q: How do I determine the vertex of a square root function?

A: The vertex of a square root function can be determined by looking at the equation in the form y=axβˆ’h+ky = a\sqrt{x-h} + k. In this form, (h,k)(h, k) is the vertex of the function.

Q: What is the difference between a square root function and a reflection across the x-axis?

A: A square root function is a type of function that involves the square root of a variable. A reflection across the x-axis is a transformation that flips the graph of a function upside down.

Q: How do I rewrite a square root function to reflect it across the x-axis?

A: To rewrite a square root function to reflect it across the x-axis, you need to negate the y-values of the function. This means that if the original function is y=f(x)y = f(x), then the reflected function is y=βˆ’f(x)y = -f(x).

Q: What is the final answer to the problem?

A: The final answer to the problem is B. y=βˆ’x+4+2y = -\sqrt{x+4} + 2. This equation reflects the function across the x-axis and matches the given vertex of (-4, 2).

Q: Can you provide more examples of square root functions and reflections across the x-axis?

A: Yes, here are a few more examples:

  • y=xβˆ’2+3y = \sqrt{x-2} + 3 is a square root function with a vertex of (2, 3).
  • y=βˆ’xβˆ’2+3y = -\sqrt{x-2} + 3 is a reflection of the function y=xβˆ’2+3y = \sqrt{x-2} + 3 across the x-axis.
  • y=x+2+3y = \sqrt{x+2} + 3 is a square root function with a vertex of (-2, 3).

Conclusion

In this article, we provided a Q&A section to further clarify any doubts and provide additional information on the topic of square root functions and reflections across the x-axis. We hope that this article has been helpful in understanding this concept and that you have a better understanding of how to work with square root functions and reflections across the x-axis.

Final Tips

  • Make sure to understand the concept of square root functions and reflections across the x-axis before attempting to solve problems involving these concepts.
  • Practice working with square root functions and reflections across the x-axis to become more comfortable with these concepts.
  • Use online resources and study materials to supplement your learning and provide additional practice opportunities.

Additional Resources

  • Khan Academy: Square Root Functions and Reflections
  • Mathway: Square Root Functions and Reflections
  • Wolfram Alpha: Square Root Functions and Reflections

We hope that this article has been helpful in understanding the concept of square root functions and reflections across the x-axis. If you have any further questions or need additional clarification, please don't hesitate to ask.