Understanding transformations is important for learning how different functions behave. There are three main types of transformations:

1. Shifts:

   • Vertical Shift: This means moving the graph up or down. You can think of it like lifting or lowering a picture on a wall. For example, if you have a function written as $f(x)$ and you add a number $k$, like $f(x) + 3$, it moves the graph up by 3. If you subtract, like $f(x) - 2$, it moves down by 2.

   • Horizontal Shift: This means moving the graph left or right. If we take $f(x)$ and change it to $f(x - h)$, we move the graph to the right by $h$ units. If we use $f(x + 2)$, the graph moves left by 2.

   • Example: If we start with $f(x) = x^2$, adding 3 gives us $f(x) + 3$, which shifts it up. If we use $f(x - 2)$, it shifts to the right by 2.

2. Reflections:

   • Across the x-axis: This flips the graph upside down. For instance, if you have $-f(x)$, it inverts the graph.

   • Across the y-axis: This mirrors the graph. If we rewrite it as $f(-x)$, the graph will look like a reflection in a mirror.

   • Example: If $f(x) = x^2$, the reflection $f(x) = -x^2$ will flip it upside down.

3. Stretching:

   • Vertical Stretch: When we multiply the function by a number greater than 1, like $a \cdot f(x)$, it makes the graph steeper.

   • Horizontal Stretch: With numbers between 0 and 1, like $f(bx)$, where $0 < b < 1$, the graph gets wider or flatter.

   • Example: For $f(x) = x^2$, if we use $f(x) = 2x^2$, it stretches the graph vertically, making it rise faster.

Knowing these transformations helps students understand how changes in a function can change its graph. This skill is really important for doing well in exams like the GCSE.