Transformations are super important for understanding how graphs of functions behave. There are different types of transformations that can change these graphs. These include translations, reflections, stretches, and compressions.

Types of Transformations:

1. Translations:

   • Vertical Translation: This means moving the graph up or down. It changes the $y$-coordinate of each point on the graph.

     • If we write it as $f(x) + k$, where $k$ is a number:
       • If $k > 0$, the graph moves up by $k$ units.
       • If $k < 0$, the graph moves down by $|k|$ units.

   • Horizontal Translation: This means moving the graph left or right. It changes the $x$-coordinate.

     • When we write it as $f(x - h)$:
       • If $h > 0$, the graph moves right by $h$ units.
       • If $h < 0$, the graph moves left by $|h|$ units.

2. Reflections:

   • Reflecting a graph means flipping it:
     • Over the x-axis: The transformation $-f(x)$ flips the graph upside down.
     • Over the y-axis: The transformation $f(-x)$ flips the graph sideways.

3. Stretches and Compressions:

   • Vertical Stretch/Compression: The transformation $af(x)$ works like this:
     • If $a > 1$, the graph stretches away from the $x$-axis.
     • If $0 < a < 1$, the graph gets squished towards the axis.
   • Horizontal Stretch/Compression: The transformation $f(bx)$ does this:
     • If $b > 1$, the graph squeezes towards the $y$-axis.
     • If $0 < b < 1$, the graph stretches away from the axis.

Summary

In short, learning about these transformations helps students change graphs easily, predict how they will look, and understand their features better.