Stretches and Compressions on a Graph

Welcome back to Professor Baker's Math Class! Today, we're diving into the fascinating world of stretches and compressions on the coordinate plane. Unlike translations and reflections, which preserve the size and shape of a figure (resulting in congruent figures), stretches and compressions change the shape, creating figures that are no longer congruent to the original.

Understanding Transformations

A transformation is a change in the position, size, or shape of a figure. We've previously explored translations (slides) and reflections (flips). Now, let's focus on how we can stretch or compress a graph. These transformations can be either horizontal or vertical.

Horizontal Stretches and Compressions

A horizontal stretch or compression affects the $x$-coordinate of a point. The transformation takes the form $(x, y) ightarrow (ax, y)$.

  • If $|a| > 1$, the transformation is a horizontal stretch, making the graph wider.
  • If $0 < |a| < 1$, the transformation is a horizontal compression, squeezing the graph closer to the $y$-axis.

For example, consider the function $f(x) = x^2$. If we apply a horizontal stretch by a factor of 2, the transformation becomes $(x, y) ightarrow (2x, y)$. This means every $x$-coordinate is doubled, while the $y$-coordinate remains the same.

Vertical Stretches and Compressions

A vertical stretch or compression affects the $y$-coordinate of a point. The transformation takes the form $(x, y) ightarrow (x, ay)$.

  • If $|a| > 1$, the transformation is a vertical stretch, making the graph taller.
  • If $0 < |a| < 1$, the transformation is a vertical compression, squeezing the graph closer to the $x$-axis.

For example, consider the function $f(x) = x^2$. If we apply a vertical compression by a factor of $\frac{1}{2}$, the transformation becomes $(x, y) ightarrow (x, \frac{1}{2}y)$. This means every $y$-coordinate is halved, while the $x$-coordinate remains the same.

Class Resources

Don't forget to check out the Class Notes for a detailed overview of these concepts. The worksheets (Examples of each Transformation, Core Math Activity) provide great practice, and the homework (Page 11-14 # 20-31, 37-41, 45-51) will solidify your understanding.

Discussion Question

Think about this: What would be a situation in your trade or future career where you might offer a discount on a list of services? How would you represent this discount graphically (hint: think about stretches or compressions!)?

Keep up the great work, and remember to ask questions if anything is unclear! We're here to help you succeed!