Graph transformations help us see how the roots of a function change.

A root, also called a zero, is the point where the graph crosses the x-axis. This means the output of the function is zero at that spot. When we change the graph, it can change its shape, position, or direction. This also affects the roots of the function.

Types of Transformations

1. Vertical Translations: This happens when we add or subtract a number from a function. For example, in the equation $f(x) = g(x) + k$, $k$ is a constant.

   • Example: If $g(x)$ has roots at $x = a$ and $x = b$, and we change it to $f(x) = g(x) + 2$, the roots will disappear. The whole graph moves up, meaning it no longer touches the x-axis.

2. Horizontal Translations: This change happens when we adjust the input, like in $h(x) = g(x - k)$, which moves the graph left or right.

   • Example: If $g(x)$ has a root at $x = a$, changing it to $h(x) = g(x - 3)$ moves the root to $x = a + 3$. This shifts the graph to the right.

3. Reflections: Reflecting a graph over the x-axis with $f(x) = -g(x)$ also changes the roots.

   • Example: If $g(x)$ crosses the x-axis at certain points, reflecting it will keep those roots at the same x-values, but change their signs in the function.

4. Stretching or Compressing: This happens when we change the function like in $f(x) = a \cdot g(x)$. If $a$ is bigger than 1, the graph gets squished down. If $a$ is between 0 and 1, the graph stretches up. Both ways can change how we see the roots.

   • Example: If there’s a root at $x = b$ and it has an even multiplicity, stretching could make this root show up differently based on how we transform it.

Conclusion

In short, graph transformations change how we see the roots of a function. Whether we are moving, flipping, or stretching the graph, each change can create new roots, remove some, or change their places along the x-axis. It’s important to understand these transformations, especially when studying functions in Year 10 Math.