Problem 27 - Algebra 2

Question

A quadrilateral is reflected across the \(y\) -axis. The coordinates of the vertices are \(P^{\prime}(-2,2), Q^{\prime}(4,1), R^{\prime}(-1,-5),\) and \(S^{\prime}(-3,-4) .\) What were the coordinates of the quadrilateral in its original position?

Step-by-Step Solution

Answer

Original coordinates: \((2,2), (-4,1), (1,-5), (3,-4)\).

Step 1: Understanding Reflection Across the y-axis

When a point is reflected across the y-axis, the x-coordinate of the point changes sign, while the y-coordinate remains unchanged. For example, if the coordinate of the vertex is \(x, y\), the reflected point would be \((-x, y)\).

Step 2: Reflecting Point P'

The given coordinates of point \(P'\) are \((-2, 2)\). To find the original point \(P\), change the sign of the x-coordinate: \\[P = (2, 2)\]

Step 3: Reflecting Point Q'

For point \(Q'\) with coordinates \(4, 1\), reflect it back to get the original point \(Q\) by changing the sign of the x-coordinate: \((4, 1) \rightarrow (-4, 1)\).\Thus, \[Q = (-4, 1)\]

Step 4: Reflecting Point R'

The coordinates of point \(R'\) are \((-1, -5)\). Reflect it back to find the original point \(R\) by changing the sign of the x-coordinate: \((-1, -5) \rightarrow (1, -5)\).\Thus, \[R = (1, -5)\]

Step 5: Reflecting Point S'

Finally, point \(S'\) has coordinates \((-3, -4)\). Change the sign of the x-coordinate to reflect it back to the original point \(S\): \((-3, -4) \rightarrow (3, -4)\).\Thus, \[S = (3, -4)\]

Step 6: Presenting Original Coordinates

The original coordinates of the quadrilateral vertices before reflection are \[P = (2, 2), Q = (-4, 1), R = (1, -5), S = (3, -4)\]

Key Concepts

Coordinates Transformation

Coordinates transformation is a vital concept in geometry, allowing us to understand how shapes change position or orientation on a plane. When we talk about transforming coordinates, we often refer to applying operations that modify one or more aspects of these coordinates.

In the case of reflection across the y-axis, one of the simplest transformations, only the x-coordinates of a point are affected. Specifically, their signs are changed. The y-coordinates remain unchanged.

This transformation is simple but crucial when we need to determine a shape’s placement before or after certain operations. In practical terms, knowing how to perform such basic transformations helps greatly in solving complex problems involving reflections, rotations, or translations.

Quadrilateral Vertices

Quadrilaterals are one of the most common types of polygons, having four sides and four vertices. Each vertex is a point on the plane defined by its coordinates. Whether you’re dealing with squares or trapezoids, understanding vertices is key to knowing how a shape will behave under various transformations.

Reflecting a quadrilateral across the y-axis involves transforming each vertex individually. For our exercise, we start with the vertices of the reflected quadrilateral:

• Point P': \((-2, 2)\)
• Point Q': \((4, 1)\)
• Point R': \((-1, -5)\)
• Point S': \((-3, -4)\)

By changing the sign of each x-coordinate, we find their respective original positions on the plane. This returns the quadrilateral to how it originally appeared before reflection.

X-coordinate Sign Change

The x-coordinate sign change is an essential part of reflections across the y-axis. When a point lies on the Cartesian plane, reflection across the y-axis involves altering the sign of the x-coordinate while keeping the y-coordinate constant.

For instance, a point \((x, y)\) becomes \((-x, y)\) when it is reflected over the y-axis. To grasp this, imagine a point on the right side of the y-axis moving to a symmetrical position on the left.

In our example, the given coordinates needed transformation back to their original form. For point \((-2, 2)\), the reflected x-coordinate sign changes to positive, resulting in the original point \((2, 2)\). This concept is consistently applied to all vertices to get each of their original coordinates back.

Graphical Transformations

Graphical transformations, such as reflections, help us visualize and grasp how geometric shapes alter as they undergo transitions. Each transformation follows specific rules which are predictable and easy to apply once understood.

• Reflection across lines: Reflection across the y-axis is one common transformation used for symmetry problems.
• Rotations and translations: These involve moving a shape around a point or along a path, differing from reflections as they don’t necessarily involve flipping.
• Scaling: Changes the size of shapes while preserving proportions.

Reflection is particularly significant as it maintains the lengths and angles of geometric shapes, thus preserving their intrinsic properties. For quadrilaterals, it means each side length and angle remains constant before and after transformation, only the orientation and positions on the graph change. Understanding how to effectively apply graphical transformations can make solving complex geometry problems much simpler.