Understanding Reflections and Translations of Functions

1. What They Mean:

   • Reflection: Imagine a reflection like flipping a picture over. When we reflect a function, we turn it over a specific line, like the x-axis or y-axis. For example, if we have a function called $f(x)$, its reflection over the x-axis will be $-f(x)$.
   • Translation: A translation is different. It moves the entire function left, right, up, or down, but it keeps the shape the same. If we change $f(x)$ to $f(x) + k$, this shifts the function up by $k$ units. And if we go for $f(x - h)$, it moves to the right by $h$ units.

2. How Points Change:

   • Reflection: If you have a point on the function, like $(a, b)$, reflecting it over the x-axis will turn it into $(a, -b)$.
   • Translation: If we translate the point $(a, b)$ up by $k$, it becomes $(a, b + k)$.

3. What Happens to the Graph:

   • Reflection: This changes how the function looks, almost like turning it upside down.
   • Translation: This keeps the shape of the function the same but moves it to a different spot on the graph.

4. Different Kinds:

   • Reflections can happen across the x-axis, y-axis, or even other lines.
   • Translations can go up or down, left or right, making the function move in those directions.