Problem 52 - Algebra 2

Question

Reflection in which line takes the figure with vertices \(A(0,0), B(-2,4)\) , \(c(-4,2),\) and \(D(-3,0)\) to \(A^{\prime}(0,0), B^{\prime}(-2,-4), C^{\prime}(-4,-2),\) and \(D^{\prime}(-3,0) ?\) F. \(x\) -axis \(\quad\) G. \(y\) -axis \(\quad\) H. \(y=x \quad\) J. \(y=-x\)

Step-by-Step Solution

Answer

The figure is reflected over the F. x-axis.

Step 1: Identify the Vertex Changes

We compare the coordinates of the original vertices with the coordinates of the reflected vertices. If the reflection is over the x-axis, the y-coordinates change signs while the x-coordinates remain the same. If the reflection is over the y-axis, the x-coordinates change signs while the y-coordinates remain the same. Reflections over the line y=x or y=-x will involve swapping x- and y-coordinates, with or without sign changes.

Step 2: Analyze the Coordinates for Reflection Options

We notice that for all the points reflected, the x-coordinates remain the same while the y-coordinates are negated. This observation eliminates the y-axis, the line y=x, and the line y=-x as the line of reflection, all of which would alter the x-coordinates or swap x- and y-coordinates in a certain manner.

Step 3: Determine the Line of Reflection

Since only the y-coordinates are negated, and the x-coordinates remain unchanged, the reflection must occur over the x-axis. This results in a 'flipping' of the figure over the x-axis where every point's y-coordinate is negated.

Key Concepts

Coordinate Plane Reflections

Reflections in geometry are akin to looking into a mirror: every point of a given figure on a coordinate plane is 'flipped' across a line, known as the line of reflection, creating a mirror image. This type of transformation maintains the size and shape of the figure, but alters its orientation. In reflections on a two-dimensional coordinate plane, this flipping action occurs across a designated line, such as the x-axis or y-axis.

For example, if a point \( (x, y) \) is reflected over the x-axis, its image will be at \( (x, -y) \), since reflection over the x-axis inverts the sign of the y-coordinate. Understanding coordinate plane reflections is crucial for visualizing how objects are mapped one-to-one onto their mirror images during this transformation.

Transformation of Points

During a reflection, each point of a shape is transformed to a new location in a systematic way. This movement, referred to as a point transformation, is dependent on the rule established by the line of reflection. Following set guidelines, the coordinates of each original point, designated as \( (x, y) \), are manipulated algebraically to yield the coordinates of the reflected point.

For instance, when reflecting across the y-axis, the transformation rule is \( (x, y) \rightarrow (-x, y) \), meaning only the x-coordinate changes sign. Visualizing the transformation of points and grasping these rules is essential to understanding how geometric reflections manipulate figures in the coordinate plane.

Line of Reflection

The line of reflection acts as the mirror in a geometric reflection. It's the fixed line that every point of the original figure is reflected across to create the mirrored figure. The position and orientation of this line are crucial because they determine how the points are transformed during reflection. Common lines of reflection include the x-axis, the y-axis, the line y=x, and the line y=-x.

The line of reflection determines whether the coordinates exchange places, change signs, or both. For example, in the given exercise, because only the y-coordinates change and become their opposites, the x-axis serves as the line of reflection. Understanding the characteristics and effects of different lines of reflection is fundamental in predicting the outcome of reflecting a figure in the coordinate plane.

Algebraic Representation of Reflections

Reflections can be represented algebraically by defining specific rules that describe how the coordinates of points change during the reflection process. These rules are concise mathematical expressions that indicate how to calculate the coordinates of the reflected image given the original coordinates of a figure.

For instance, reflecting over the x-axis can be described by the rule \( (x, y) \rightarrow (x, -y) \), meaning that the x-coordinate stays the same while the y-coordinate is multiplied by -1. Similarly, reflecting a point across the y-axis is represented by the rule \( (x, y) \rightarrow (-x, y) \). These algebraic rules offer a clear and straightforward method to carry out geometric reflections, allowing for a procedural solution to problems involving transformations.