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 \(1 / 2\) followed by a vertical stretch by a factor of \(3,\) compared to (b) the same operations but in opposite order

Short answer

Yes, the transformations result in the same shape regardless of the order.

Step-by-step solution

Understanding the Unit Square

The unit square on a coordinate plane typically has vertices at \((0, 0), (1, 0), (1, 1), (0, 1)\). We will apply the transformations to each vertex to see how the square is changed.

Set (a): Horizontal Stretch by Factor of \(1/2\)

Apply the horizontal stretch to each vertex of the unit square. The new vertices will be \((0, 0), (1/2, 0), (1/2, 1), (0, 1)\). This transformation compresses the width of the square by half.

Set (a): Vertical Stretch by Factor of \(3\)

Now apply the vertical stretch. Multiply the y-coordinates of each vertex by 3. The transformed vertices will be \((0, 0), (1/2, 0), (1/2, 3), (0, 3)\). This stretches the height of the rectangle obtained in the previous step, not altering its width.

Set (b): Vertical Stretch by Factor of \(3\)

Start with a vertical stretch on the original unit square. Multiply the y-coordinates by 3 to get \((0, 0), (1, 0), (1, 3), (0, 3)\). The square is stretched vertically into a taller rectangle.

Set (b): Horizontal Stretch by Factor of \(1/2\)

Now apply the horizontal stretch on the vertices obtained from the vertical stretch. The new vertices will be \((0, 0), (1/2, 0), (1/2, 3), (0, 3)\). The width is halved without changing the height.

Comparison of Transformations

Both sets (a) and (b) result in the same final shape: a rectangle with vertices \((0, 0), (1/2, 0), (1/2, 3), (0, 3)\). Thus, the order of operations does not affect the final result in this case because stretches along orthogonal axes are independent of each other.

Key concepts

Unit Square

The unit square is a simple yet powerful concept in geometry, especially when dealing with linear transformations. As the name suggests, the unit square is a square on the coordinate plane with sides of unit length. Its four corners are situated at the points

• (0,0)
• (1,0)
• (1,1)
• (0,1)

This neatly boxed shape is often used as a reference in transformations because its regular structure makes it ideal for observing changes. By applying various transformations like stretching or rotating to this square, we can easily visualize, calculate, and understand the effect of transformations on other shapes.
In this exercise, we started with the unit square as our foundational shape. It helps us illustrate the changes resulting from horizontal and vertical stretches separately. These transformations essentially modify the lengths without altering the original positioning of the vertices, making the unit square a fundamental example in transformation exercises.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is the playing field of geometry. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical), intersecting at a common point called the origin, marked as (0,0). This setup allows us to plot any point with a unique pair of coordinates (x, y), which correspond to their respective distances from the two axes.
In our exercise, the coordinate plane serves as the reference backdrop against which transformations occur. It provides a systematic way to understand how each point of a shape, like the unit square, moves during a transformation.
Shapes on the coordinate plane often encounter transformations that involve changes to these coordinates, whether they are stretches, rotations, or translations. By having such a structure, it becomes much simpler to track how shapes change and analyze the outcomes of those transformations effectively.

Geometric Transformations

Geometric transformations encompass a variety of operations that you can apply to shapes in geometry. They include among others, translations, rotations, reflections, and stretches. When a transformation is applied, it changes the position, size, or orientation of a shape, while preserving the basic properties like parallelism and proportional distances.
In our exercise, specifically, we explored linear transformations using stretches. These transformations altered the size of our unit square. Applying a horizontal stretch modifies the x-coordinates, while a vertical stretch alters the y-coordinates. Here's a brief overview:

• Horizontal Stretch: Reduces or expands the shape along the x-axis. In our exercise, a factor of 1/2 compressed the square horizontally.
• Vertical Stretch: Alters the shape along the y-axis. A factor of 3 expanded it vertically, creating a taller shape.

These operations highlight a key property of linear transformations on orthogonal axes: they are commutative. This means that performing a horizontal stretch first and then a vertical stretch yields the same result as performing them in the opposite order. Understanding this flexibility is crucial when working with transformations, especially in more complex shapes or transformations that might involve multiple steps.