Transformations are important for understanding how a function’s graph looks and acts. They can move, stretch, shrink, or flip the graph in different ways. Let’s break down these transformations:

1. Shifts

• Vertical Shifts: When we add or subtract a number $k$ to a function $f(x)$, it shifts the graph up or down. For example, if we do $f(x) + k$, the graph moves up by $k$ units. If we do $f(x) - k$, it moves down by $k$ units.

• Horizontal Shifts: Changing the input $x$ by adding or subtracting $h$ will shift the graph left or right. The function $f(x - h)$ moves the graph to the right by $h$ units, while $f(x + h)$ moves it to the left.

2. Stretches and Compressions

• Vertical Stretch or Compression: If we multiply the function by a number $a$, it changes how tall or short the graph is. For example, $a \cdot f(x)$ stretches the graph if $a > 1$, and it squishes it if $0 < a < 1$.

• Horizontal Stretch or Compression: To stretch or compress the graph sideways, we change the input. The function $f(bx)$ stretches the graph if $0 < b < 1$, and compresses it if $b > 1$.

3. Reflections

• Over the x-axis: The function $-f(x)$ flips the graph over the x-axis.
• Over the y-axis: Changing the input to $f(-x)$ flips the graph over the y-axis.

By understanding these transformations, we can easily guess how to draw and analyze graphs!