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Chapter 2: Problem 31

Find the point that is symmetric to the given point with respect to the \(x\) -axis, the \(y\) -axis, and the origin. $$(0,-4)$$

Short Answer

Expert verified
Symmetric to the x-axis: (0, 4); Symmetric to the y-axis: (0, -4); Symmetric to the origin: (0, 4)

Step by step solution

01

Understanding Symmetry with Respect to the x-axis

To find the symmetric point of \( (0, -4) \) with respect to the \( x \)-axis, you need to keep the \( x \)-coordinate the same and change the sign of the \( y \)-coordinate. The new point will be \( (0, 4) \).
02

Understanding Symmetry with Respect to the y-axis

To find the symmetric point of \( (0, -4) \) with respect to the \( y \)-axis, you need to keep the \( y \)-coordinate the same and change the sign of the \( x \)-coordinate. The new point will be \( (0, -4) \).
03

Understanding Symmetry with Respect to the Origin

To find the symmetric point of \( (0, -4) \) with respect to the origin, you need to change the signs of both the \( x \)-coordinate and \( y \)-coordinate. The new point will be \( (0, 4) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-axis symmetry
Symmetry with respect to the x-axis means that the point is mirrored across the x-axis. When you reflect a point over the x-axis, the x-coordinate remains unchanged, but the y-coordinate changes its sign. For example, if you have the point \(0, -4\), its symmetric point with respect to the x-axis will be \(0, 4\). This is because we keep the x-coordinate (which is 0) the same and take the negative of the y-coordinate, changing -4 to 4. In general, for any point \(x, y\), its symmetric point across the x-axis will be \(x, -y\).
y-axis symmetry
When dealing with y-axis symmetry, the reflection occurs over the y-axis. This means the y-coordinate stays the same, and the x-coordinate changes its sign. Given the point \(0, -4\), its symmetric point with respect to the y-axis is \(0, -4\). Since the original x-coordinate is 0, flipping its sign will still result in 0. Therefore, the point remains the same. In a more general sense, if you have a point \(x, y\), its symmetric point across the y-axis will be \(-x, y\).
origin symmetry
Origin symmetry involves reflecting the point through the origin, effectively flipping both the x and y coordinates. For the point \(0, -4\), reflecting it through the origin will change both signs, resulting in the point \(0, 4\). This is because the original coordinates \(0, -4\) are transformed to \(-0, -(-4) = 0, 4\). Similarly, for any general point \(x, y\), its symmetric point with respect to the origin will be \(-x, -y\). This type of symmetry essentially places the point in the exact opposite quadrant of the coordinate plane.

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