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.