Understanding graph transformations helps us see how the shape and position of a graph can change based on its original form.

Types of Transformations

1. Translations:

   • Vertical Shift: When we add or subtract a number (let's call it $k$) to the function, it moves the graph up if we add, and down if we subtract. So, the new function looks like $f(x) + k$.
   • Horizontal Shift: When we change the input of the function by subtracting or adding a number (let's call it $h$), the graph moves left if we subtract and right if we add. This looks like $f(x - h)$.

2. Reflections:

   • Across the x-axis: If we multiply the function by -1, like this: $-f(x)$, it flips the graph upside down.
   • Across the y-axis: If we switch the x's to negative, like this: $f(-x)$, it flips the graph side to side.

3. Stretches:

   • Vertical Stretch/Compression: If we multiply the function by a number (let's call it $a$), it stretches the graph taller if $|a| > 1$ and makes it shorter if $0 < |a| < 1$. So, it looks like $a \cdot f(x)$.
   • Horizontal Stretch/Compression: If we change the 'x' in the function by multiplying it by a number (let's call it $b$), it makes the graph wider if $|b| < 1$ and skinnier if $|b| > 1$. This can be written as $f(bx)$.

By using these transformations step by step, we can draw the graph of any function by starting with its original or "parent" graph.