Problem 16 - Algebra 2

Question

Find the coordinates of each image after reflection in the given line. \(\left[\begin{array}{rrrrr}{-1} & {-2} & {-4} & {-6} & {-2} \\ {-4} & {0} & {0} & {-3} & {-4}\end{array}\right] ; x\) -axis

Step-by-Step Solution

Answer

The reflected coordinates are \[\begin{array}{rrrrr} {-1} & {-2} & {-4} & {-6} & {-2} \ {4} & {0} & {0} & {3} & {4} \end{array}\].

Step 1: Understand the reflection in the x-axis

To reflect a point across the x-axis, invert the sign of the y-coordinate while keeping the x-coordinate unchanged. The general rule for a point \(P(x, y)\) after reflection across the x-axis is \(P'(x, -y)\).

Step 2: Apply reflection to each point

For each point given by its coordinates in the matrix, apply the reflection rule by keeping the x-values the same and inverting the sign of the y-values.

Step 3: Write down the reflected coordinates

The new set of points after reflection will form a matrix where each column represents the reflected coordinates of the original points, maintaining the x-coordinates and having the negated y-coordinates.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric figures through the use of coordinates. Points on a plane are typically defined by their position along the x (horizontal) and y (vertical) axes, forming a coordinate grid.

When working with reflections in coordinate geometry, it's important to understand how to manipulate these coordinates to accurately represent transformations—such as flipping a shape over a line. For instance, when reflecting points over the x-axis, each point's x-coordinate remains the same, while the y-coordinate changes sign. This is because the x-axis serves as a mirror line, and flipping over this axis inverts the position above or below it.

Here's a graphic example: If we have a point A with coordinates \(3, 4\), after reflecting A over the x-axis, it would become A' with coordinates \(3, -4\). The point has the same distance from the x-axis, but on the opposite side.

Transformation of Points

A transformation in geometry is an operation that moves or changes a shape in some way, while still preserving its size and proportion. One common transformation is reflection, a 'flip' over a line—like a mirror image. Reflecting points over the x-axis is a specific type of transformation that alters each point's position according to a set rule.

In the reflection of points, we start with the original coordinates. Take a point \(x, y\) for example. To reflect this point over the x-axis, we generate a new point \(x', y'\), where \(x' = x\) and \(y' = -y\). This effectively 'flips' the point over the x-axis, creating a symmetrical image of the point as though the x-axis was a mirror. This reflection rule is paramount as it maintains the 'mirror image' relationship between the original figure and its reflected counterpart.

Reflecting a series of points results in a new figure that retains the original shape's size and orientation, but reverses its position above or below the x-axis. Understanding this fundamental transformation is key to manipulating and studying geometric shapes in various applications, including computer graphics and advanced mathematics.

Matrix Operations

Matrix operations are fundamental tools in various branches of mathematics, and they have practical applications in areas such as computer graphics, physics, and engineering. In the context of coordinate geometry and transformations, matrices can be used to represent points and apply transformations in a streamlined manner.

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. When representing points for reflection over the x-axis, we use a two-row matrix where the first row corresponds to the x-coordinates and the second row to the y-coordinates of each point. To reflect these points, we perform a matrix operation that multiplies the second row by -1, keeping the first row unchanged—a process that can be visualized as flipping all the y-values over the x-axis.

For example, if we have a matrix \[\begin{array}{ccc}x_1 & x_2 & x_3\y_1 & y_2 & y_3\end{array}\], reflecting over the x-axis would result in a new matrix \[\begin{array}{ccc}x_1 & x_2 & x_3\-y_1 & -y_2 & -y_3\end{array}\]. This principle of matrix operations allows for a convenient method to perform transformations on a set of points simultaneously, making it a powerful technique in computational applications.