Transformations of functions are a fun way to see how one function can turn into another! Let’s look at the different types of transformations and how they change the original characteristics of functions.

1. Translations

• Horizontal Translations: This means moving a function left or right! When you add $c$ to $f(x)$, it shifts the function to the left, making it $f(x+c)$. If you subtract $c$, the function moves to the right.

• Vertical Translations: This means moving a function up or down! When you add $d$ to $f(x)$, it lifts the entire graph up by $d$. This changes the function to $f(x) + d$.

2. Reflections

• Across the x-axis: If you put a negative in front of the function, like $-f(x)$, it flips the graph over the x-axis.

• Across the y-axis: If you replace $x$ with $-x$, it flips the graph over the y-axis. This changes the function to $f(-x)$.

3. Stretches and Compressions

• Vertical Stretch: If you multiply the function by a number bigger than 1, like $k \cdot f(x)$ (where $k > 1$), it makes the graph taller.

• Vertical Compression: If you multiply the function by a number less than 1, like $k \cdot f(x)$ (where $0 < k < 1$), it squashes the graph down.

• Horizontal Stretch/Compression: You can change $x$ to $cx$ (where $c > 1$ squashes the graph, and $0 < c < 1$ stretches it).

Isn’t it amazing how these transformations can change how we see functions while keeping their main relationships and behaviors? Jump into the world of transformations and discover the wonders of mathematics!