Problem 14 - Algebra 2

Question

Find the coordinates of each image after reflection in the given line. $$ \left[\begin{array}{llll}{0} & {4} & {8} & {6} \\ {0} & {4} & {4} & {2}\end{array}\right] ; x-a x i s $$

Step-by-Step Solution

Answer

The coordinates of the reflected points are [(0, 0), (4, -4), (8, -4), (6, -2)].

Step 1: Understand the Concept of Reflection over the x-axis

When a point is reflected over the x-axis, the x-coordinate remains the same while the y-coordinate changes sign.

Step 2: Reflect Each Point over the x-axis

Apply the reflection rule to each point. Keep the x-coordinate the same, change the sign of the y-coordinate for each point.

Step 3: Write Down the Coordinates of the Reflected Points

Record the new coordinates after reflection for each point, maintaining the original order.

Key Concepts

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface that we use to plot points, lines, and curves. It is comprised of two number lines: the x-axis (horizontal) and the y-axis (vertical), which intersect at a point called the origin, labeled as (0, 0).

Each point
on this plane is defined by an ordered pair of numbers ���������ного

Transformations in Algebra

In algebra, transformations refer to the movement of figures on the coordinate plane. These can be translations, rotations, reflections, or dilations. A reflection is a type of transformation where a figure is flipped over a line, such as the x-axis or y-axis, resulting in a mirror image.

When we apply a reflection over the x-axis in algebra, we follow specific rules: maintain the x-coordinates and multiply the y-coordinates by -1. This rule is consistent because the x-axis acts as a mirror, leaving positions along the x-axis unchanged while inverting the positions along the y-axis.

Understanding these transformation rules allows students to visualize and perform reflections accurately, which is an essential skill in coordinate geometry and algebra.

Reflection Symmetry

Reflection symmetry, also known as mirror symmetry, occurs when one half of an object or shape is the mirror image of the other half. Imagine a mirror placed along the line of symmetry; the shape appears unchanged. This concept is not limited to two dimensions; three-dimensional shapes can also have reflective symmetry.

In the context of our exercise, when we reflect points over the x-axis, we are essentially finding the mirror image of those points as if there was a mirror placed along the x-axis. The original and reflected points are equidistant from the axis, but on opposite sides, creating symmetrical pairs.

Recognizing reflection symmetry can help in visualizing the effects of reflective transformations on shapes and patterns within the coordinate plane.