When I learned about how functions change in Year 12, I found out that stretching and compressing these functions is super helpful for understanding graphs. It’s like using a magic lens that shows us different views of functions!

Visualizing Changes

The first big idea I got was from simply visualizing these changes. When you stretch or compress a function, it’s not only about making it bigger or smaller. It actually changes the graph’s shape and where it sits. For example, think about the basic function $f(x) = x^2$.

• Vertical Stretch: If we make the function bigger by multiplying it by a number greater than 1, like $f(x) = 2x^2$, the graph becomes taller. It looks steeper and more focused, which makes it easier to find the highest and lowest points.

• Horizontal Compression: On the other hand, if we make it wider by using a number between 0 and 1, like $f(x) = (0.5)x^2$, the graph spreads out. This helps us see how fast or slow a function gets to its highest or lowest point.

Functional Relationships

Understanding these changes also helped me learn about how different functions relate to each other. When we stretch or compress functions, we can see connections between them. For instance, by comparing $f(x) = x^2$ and $g(x) = \frac{1}{2} x^2$, we can notice how a vertical compression (making it smaller by a factor of 2) affects the whole function, including where it crosses the axes.

Real-world Applications

In real life, stretching and compressing functions help us understand different situations. For example, in physics problems about how objects move through the air, or using quadratic functions to look at area compared to perimeter, knowing about these transformations can help us find the best answers or the highest points.

Curiosity and Creativity

This part of math also gets creative! I remember playing around with the function $f(x) = \sin(x)$. When I stretched it vertically to $f(x) = 2\sin(x)$, I saw how it changed the height of the waves. I started to think about how these changes in wave functions relate to real-life things like sound waves or ocean waves. This opened up new ideas for how math connects with physics and even music!

Summary

In conclusion, stretching and compressing functions is not just a math skill; it’s a way to understand mathematics better. It changes how we look at functions and graphs, helps us find connections, solves problems, and even inspires creativity. Learning about these transformations not only builds our math skills but also helps us think more broadly about the math concepts we encounter in Year 12 and beyond.