Understanding how different changes can mix together to create new graphs is pretty interesting! Here's a simple breakdown of what I've learned:

Types of Changes

1. Translations: This means moving the graph up, down, left, or right. For example, if you take the function $f(x)$ and move it up by 3, it becomes $f(x) + 3$.

2. Reflections: This flips the graph over a specific line. If you flip it over the x-axis, it changes from $f(x)$ to $-f(x)$. If you flip it over the y-axis, it changes to $f(-x)$.

3. Stretching/Compressing: This changes how the graph looks. If you squish it down, you might multiply by a factor like $1/2$, turning it into $1/2 f(x)$. Stretching it out is just the opposite!

Combining Changes

When you mix these changes together, the order makes a difference! For example, if you reflect $f(x)$ over the x-axis first and then move it up by 4 units, it works like this:

1. Reflect: $-f(x)$
2. Move up: $-f(x) + 4$

Seeing this combination on a graph can really help us understand how the functions change and connect, which is pretty cool!