Question

Two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not. (a) a horizontal stretch by a factor of 2 followed by a reflection across the line \(y=x,\) compared to (b) a vertical stretch by a factor of 2

Short answer

The transformations are not the same; they produce different shapes.

Step-by-step solution

Understand the Initial Unit Square

The unit square is a square with vertices at (0,0), (1,0), (1,1), and (0,1). We will apply each set of transformations to this square and obtain new images.

Perform Transformation (a)

The first transformation is a horizontal stretch by a factor of 2. This affects the x-coordinates of our vertices:- (0,0) stays (0,0)- (1,0) becomes (2,0)- (1,1) becomes (2,1)- (0,1) stays (0,1)The second transformation reflects these new points across the line \(y = x\):- (0,0) stays (0,0) (Invariance)- (2,0) becomes (0,2)- (2,1) becomes (1,2)- (0,1) becomes (1,0)The transformed vertices form a skewed quadrilateral with vertices at (0,0), (0,2), (1,2), and (1,0).

Perform Transformation (b)

This is a vertical stretch by a factor of 2, which affects the y-coordinates of the unit square: - (0,0) stays (0,0) - (1,0) stays (1,0) - (1,1) becomes (1,2) - (0,1) becomes (0,2) The transformed unit square is now a rectangle with vertices at (0,0), (1,0), (1,2), and (0,2).

Compare the Transformed Shapes

From Step 2, the new shape is a quadrilateral with vertices (0,0), (0,2), (1,2), and (1,0), resembling a parallelogram. From Step 3, the new shape is a rectangle with vertices (0,0), (1,0), (1,2), and (0,2). Compare these two shapes to determine their equivalence.

Conclusion: Determine if the Transformations are the Same

The first transformation gives a parallelogram, while the second gives a rectangle. These shapes are not congruent or equivalent as their angles and side orientations differ.

Key concepts

Unit Square

The unit square is a simple geometrical shape that serves as a foundation for applying transformations. It is a square with vertices at the points (0,0), (1,0), (1,1), and (0,1). This makes it precisely one unit long along both the x-axis and y-axis. Using the unit square, we can easily visualize how different transformations affect it, allowing us to see the impact on a known shape.
Understanding how the initial unit square changes under various transformations is key to grasping the effect of matrix transformations. The simplicity of its dimensions helps learners focus on the transformation's effects instead of complex numeric calculations.
Visualizing a unit square before and after a transformation helps in seeing differences such as stretching, skewing, or changing orientation.

Horizontal Stretch

A horizontal stretch transformation changes the x-coordinates of a shape while keeping the y-coordinates unchanged. In simpler terms, it "stretches" the shape along the x-axis. For example, applying a horizontal stretch by a factor of 2 to the unit square means multiplying each x-coordinate by 2.

• The point (0,0) remains as (0,0), as any number multiplied by zero stays zero.
• The point (1,0) transforms to (2,0), as the x-coordinate is doubled while the y-coordinate stays the same.
• The point (1,1) transforms to (2,1), affecting only the x-value.
• (0,1) remains unchanged at (0,1).

By applying this transformation to the unit square, you see a change in width, making it wider. This makes it easier to understand how horizontal distances in shapes are scaled by transformation matrices.

Vertical Stretch

Vertical stretch involves modifying the y-coordinates while keeping x-coordinates static. For a vertical stretch by a factor of 2, each y-coordinate of the unit square is doubled. This is the opposite principle of a horizontal stretch, focusing on vertical change.

• The point (0,0) stays the same at (0,0).
• (1,0) also remains unchanged as (1,0).
• The point (1,1) moves to (1,2), showing a doubling in the y-value.
• (0,1) becomes (0,2).

Such transformation leads to the creation of a rectangle from the initial square, elongated vertically. By observing this, one can see how a vertical stretch transforms the visual and mathematical properties of shapes in geometry.

Reflection Across Line y=x

Reflecting a shape across the line \(y = x\) swaps the x and y coordinates of every point on the shape. This reflection creates a mirror image of the original figure where each point flips over the line \(y = x\). For a unit square, let's consider the reflection:

• (0,0) remains unchanged, as it lies on the line \(y = x\).
• The point (2,0) becomes (0,2), reflecting across the line.
• (2,1) reflects to (1,2).
• (0,1) reflects to (1,0).

This forms a different configuration where the unit square is changed to a shape more like a skewed rectangle or parallelogram, highlighting how reflection shifts orientation. Understanding reflection is fundamental, as it is frequently used along with other transformations like scaling and rotation in geometric manipulation and computer graphics.