Which Point Would Map Onto Itself After A Reflection Across The Line $y = -x$?A. $(-4, -4$\] B. $(-4, 0$\] C. $(0, -4$\] D. $(4, -4$\]

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Reflection Across the Line y=−xy = -x: Understanding the Concept

Reflection across a line is a fundamental concept in geometry and mathematics. It involves flipping a point or a shape over a given line, resulting in a new position or shape. In this article, we will explore the concept of reflection across the line y=−xy = -x and determine which point would map onto itself after such a reflection.

What is Reflection Across a Line?

Reflection across a line is a transformation that flips a point or a shape over a given line. This transformation can be thought of as a mirror image of the original point or shape, with respect to the line of reflection. The line of reflection acts as a mirror, and the point or shape is reflected over it to produce a new position or shape.

Reflection Across the Line y=−xy = -x

The line y=−xy = -x is a diagonal line that passes through the origin (0, 0) and has a slope of -1. To reflect a point across this line, we need to find the point that is symmetric to the original point with respect to the line y=−xy = -x.

How to Reflect a Point Across the Line y=−xy = -x

To reflect a point (x,y)(x, y) across the line y=−xy = -x, we need to find the point that is symmetric to the original point with respect to the line. This can be done by using the following formula:

(x′,y′)=(−y,−x)(x', y') = (-y, -x)

where (x′,y′)(x', y') is the reflected point.

Applying the Formula to the Given Points

Now, let's apply the formula to the given points to determine which point would map onto itself after a reflection across the line y=−xy = -x.

A. (−4,−4)(-4, -4)

Using the formula, we get:

(−4′,−4′)=(−(−4),−(−4))=(4,4)(-4', -4') = (-(-4), -(-4)) = (4, 4)

This is not the original point, so option A is incorrect.

B. (−4,0)(-4, 0)

Using the formula, we get:

(−4′,0′)=(−0,−(−4))=(0,4)(-4', 0') = (-0, -(-4)) = (0, 4)

This is not the original point, so option B is incorrect.

C. (0,−4)(0, -4)

Using the formula, we get:

(0′,−4′)=(−(−4),−0)=(4,0)(0', -4') = (-(-4), -0) = (4, 0)

This is not the original point, so option C is incorrect.

D. (4,−4)(4, -4)

Using the formula, we get:

(4′,−4′)=(−(−4),−4)=(4,−4)(4', -4') = (-(-4), -4) = (4, -4)

This is the original point, so option D is correct.

Conclusion

In conclusion, the point that would map onto itself after a reflection across the line y=−xy = -x is (4,−4)(4, -4). This is because when we apply the formula for reflection across the line y=−xy = -x to this point, we get the original point as the result.

Reflection Across the Line y=−xy = -x: Key Takeaways

  • Reflection across a line is a transformation that flips a point or a shape over a given line.
  • The line of reflection acts as a mirror, and the point or shape is reflected over it to produce a new position or shape.
  • To reflect a point across the line y=−xy = -x, we need to find the point that is symmetric to the original point with respect to the line.
  • The formula for reflection across the line y=−xy = -x is (x′,y′)=(−y,−x)(x', y') = (-y, -x).
  • The point that would map onto itself after a reflection across the line y=−xy = -x is (4,−4)(4, -4).
    Reflection Across the Line y=−xy = -x: Q&A

In this article, we will continue to explore the concept of reflection across the line y=−xy = -x and answer some frequently asked questions related to this topic.

Q: What is the purpose of reflecting a point across the line y=−xy = -x?

A: The purpose of reflecting a point across the line y=−xy = -x is to find the point that is symmetric to the original point with respect to the line. This can be useful in various mathematical and real-world applications, such as geometry, trigonometry, and computer graphics.

Q: How do I determine if a point is reflected across the line y=−xy = -x?

A: To determine if a point is reflected across the line y=−xy = -x, you can use the formula (x′,y′)=(−y,−x)(x', y') = (-y, -x), where (x′,y′)(x', y') is the reflected point. If the reflected point is the same as the original point, then the point is reflected across the line y=−xy = -x.

Q: What is the relationship between the line y=−xy = -x and the point (4,−4)(4, -4)?

A: The point (4,−4)(4, -4) is reflected across the line y=−xy = -x. This means that if you reflect the point (4,−4)(4, -4) across the line y=−xy = -x, you will get the same point back.

Q: Can I reflect a point across the line y=−xy = -x using a graphing calculator?

A: Yes, you can reflect a point across the line y=−xy = -x using a graphing calculator. Most graphing calculators have a built-in function to reflect a point across a line. You can also use the formula (x′,y′)=(−y,−x)(x', y') = (-y, -x) to reflect a point manually.

Q: How do I reflect a point across the line y=−xy = -x in three dimensions?

A: To reflect a point (x,y,z)(x, y, z) across the line y=−xy = -x in three dimensions, you can use the formula (x′,y′,z′)=(−y,−x,z)(x', y', z') = (-y, -x, z). This will give you the reflected point in three dimensions.

Q: Can I reflect a point across the line y=−xy = -x using a computer program?

A: Yes, you can reflect a point across the line y=−xy = -x using a computer program. Most programming languages, such as Python and MATLAB, have built-in functions to reflect a point across a line. You can also use the formula (x′,y′)=(−y,−x)(x', y') = (-y, -x) to reflect a point manually.

Q: What are some real-world applications of reflecting a point across the line y=−xy = -x?

A: Some real-world applications of reflecting a point across the line y=−xy = -x include:

  • Computer graphics: Reflecting a point across the line y=−xy = -x can be used to create mirror-like effects in computer graphics.
  • Geometry: Reflecting a point across the line y=−xy = -x can be used to find the midpoint of a line segment.
  • Trigonometry: Reflecting a point across the line y=−xy = -x can be used to find the sine and cosine of an angle.

Conclusion

In conclusion, reflecting a point across the line y=−xy = -x is a fundamental concept in geometry and mathematics. It can be used to find the point that is symmetric to the original point with respect to the line, and has various real-world applications. We hope this article has helped you understand the concept of reflection across the line y=−xy = -x and answer some frequently asked questions related to this topic.

Reflection Across the Line y=−xy = -x: Key Takeaways

  • Reflection across a line is a transformation that flips a point or a shape over a given line.
  • The line of reflection acts as a mirror, and the point or shape is reflected over it to produce a new position or shape.
  • To reflect a point across the line y=−xy = -x, you can use the formula (x′,y′)=(−y,−x)(x', y') = (-y, -x).
  • The point that would map onto itself after a reflection across the line y=−xy = -x is (4,−4)(4, -4).
  • Reflection across the line y=−xy = -x has various real-world applications, including computer graphics, geometry, and trigonometry.