Transformations of functions can really change how they look on a graph. Here are the main types of transformations:

1. Translations:

   • Horizontal shifts: When we write $f(x) \to f(x - h)$, it means we're moving the graph to the right by $h$ units. If $h$ is negative, we move it to the left.
   • Vertical shifts: When we say $f(x) \to f(x) + k$, we're moving the graph up by $k$ units. If $k$ is negative, we move it down.

2. Scaling (Stretching and Compression):

   • Vertical scaling: When we see $f(x) \to a \cdot f(x)$, it stretches the graph up or down. If $a$ is greater than 1, it stretches. If $a$ is between 0 and 1, it gets squished.
   • Horizontal scaling: In this case, $f(x) \to f(bx)$ changes the width of the graph. If $b$ is greater than 1, it squishes the graph. If $b$ is between 0 and 1, it stretches it.

3. Reflections:

   • Reflection across the x-axis: This is when we write $f(x) \to -f(x)$. It flips the graph over the x-axis.
   • Reflection across the y-axis: Here, we have $f(x) \to f(-x)$. This flips the graph over the y-axis.

These transformations help us see functions in different ways, changing how they look and where they are on the graph.