Graph transformations are a pivotal concept in offering a window into the dynamic nature of functions. This section provides an in-depth exploration of various graph transformations, including vertical and horizontal translations, stretches, compressions, reflections, and their combinations. These transformations are key to understanding how alterations in a function's equation reflect on its graph.

**Graph Transformations**

**1. Translation**

**Vertical Translation**:**Upward**: $y = f(x) + c$ (Graph moves up)**Downward**: $y = f(x) - c$ (Graph moves down)

Image courtesy of cuemath

**Horizontal Translation**:**Rightward**: $y = f(x - c)$ (Graph moves right)**Leftward**: $y = f(x + c)$ (Graph moves left)

Image courtesy of cuemath

**2. Stretch**

**Vertical Stretch**:- $y = af(x)$ (Stretches vertically)

Image courtesy of openlibrary

**Horizontal Stretch**:- $y = f(\frac{x}{a})$ (Stretches horizontally)

Image courtesy of openlibrary

**3. Reflection**

**Over X-axis**:- $y = -f(x)$ (Flips over x-axis)

**Over Y-axis**:- $y = f(-x)$ (Flips over y-axis)

Image courtesy of lumen

**Examples**

**Example 1**:

Transform $y = x^2$ by vertically stretching by 2, and translating 3 units up and 2 units right.

**Solution**:

- New function: $y = 2(x - 2)^2 + 3$
- The graph is stretched, moved right and up.

The blue curve represents the original quadratic function $y = x^2$. It is stretched vertically by a factor of 2, and translating 3 units up and 2 units to the right, then the red curve would represent the function $y = 2(x - 2)^2 + 3$.

**Example 2**:

Transform $y = \sqrt{x}$ by reflecting over the x-axis and stretching horizontally by 2.

**Solution**:

- New function: $y = -\sqrt{\frac{x}{2}}$
- The graph is reflected and stretched horizontally.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.