Key Differences Between Translations and Reflections in Function Transformations

Function transformations are important for understanding how graphs change.

Today, we’ll look at two types of transformations: translations and reflections.

Translations

• What is it?: A translation moves a graph from one place to another. It can go sideways, up, or down, but it doesn’t change the shape or direction of the graph.

• Types of Translations:

  • Horizontal Translation:

    • This moves the graph left or right.
    • If you see $f(x) \rightarrow f(x - h)$:
      • If $h$ is positive (greater than 0), the graph moves to the right.
      • If $h$ is negative (less than 0), the graph moves to the left.

  • Vertical Translation:

    • This moves the graph up or down.
    • If you see $f(x) \rightarrow f(x) + k$:
      • If $k$ is positive, the graph moves up.
      • If $k$ is negative, the graph moves down.

• Effect: The graph’s position changes, but it keeps the same shape.

Reflections

• What is it?: A reflection flips a graph over a specific line, like the x-axis or y-axis.

• Types of Reflections:

  • Reflection over the x-axis:

    • Shown as $f(x) \rightarrow -f(x)$. This changes the y-values, flipping the graph upside down.

  • Reflection over the y-axis:

    • Shown as $f(x) \rightarrow f(-x)$. This changes the x-values, flipping the graph sideways.

• Effect: The graph’s shape changes direction, making it look quite different.

Summary

To sum it up, translations move graphs without changing their shape. In contrast, reflections flip the graphs, changing how they look. Understanding these transformations is important for solving problems in math involving functions.