Question

Which point is the reflection of \((-4,7)\) in the \(x\)-axis? A. \((-4,-7)\) B. \((4,-7)\) C. \((7,-4)\) D. \((-7,4)\)

Short answer

A. (-4, -7)

Step-by-step solution

- Understand Reflection in the x-axis

When reflecting a point in the x-axis, we change the sign of the y-coordinate while keeping the x-coordinate the same.

- Apply Reflection to Given Point

Given the point (-4, 7), reflect it in the x-axis by changing the sign of the y-coordinate. The new coordinates are (-4, -7)n.

- Select the Correct Answer

Compare the reflected point (-4,-7) with the answer choices. The correct choice is A.

Key concepts

coordinate geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. In coordinate geometry, we represent geometric figures using points on a graph.
This allows us to analyze shapes, sizes, and positions using algebra and geometry together.

• Each point on the graph is represented by an ordered pair, \((x, y)\), where \('x'\) is the position along the horizontal x-axis and \('y'\) is the position along the vertical y-axis.
• The x-axis and y-axis intersect at the origin, \( (0,0) \).
• Moving right or left along the x-axis, or up and down the y-axis, we can pinpoint any location in this coordinate system.

Coordinate geometry is quite useful in solving various geometric problems, including reflections, translations, rotations, and more.

reflection

Reflection is one of the fundamental transformations in geometry. It flips a point or shape over a specified line, called the mirror line.
When a point is reflected, it creates a mirror image on the opposite side of the line.

• This process displays symmetry, where each point and its reflection are at the same distance from the mirror line.
• In our coordinate system, this mirror line can be the x-axis, y-axis, or any other line.

Reflection is important as it maintains the original size and shape of the figure, only changing its position.

x-axis reflection

Reflecting a point over the x-axis means inverting its y-coordinate while leaving the x-coordinate unchanged.
For example, reflecting the point \((a, b)\) across the x-axis will change its coordinates to \((a, -b)\).

• If you have the point \((-4, 7)\), reflecting it across the x-axis means changing the sign of the y-coordinate.
• The new coordinate after reflection will be \((-4, -7)\).

This is very useful for graphically determining positions and visualizing symmetry in coordinate geometry.