What is the Formula for Transformed Functions?

The formula for transformed functions isn't a single universal equation but rather a set of specific rules that describe how a parent function's graph changes. These rules depend on the type of transformation applied, such as translation, dilation (stretching or compressing), or reflection. Each transformation alters the original function, denoted as y = f(x), in a distinct way.

Understanding Function Transformations

Function transformations involve modifying the shape, size, or position of the graph of a basic function without changing its fundamental nature. These changes are represented by specific mathematical formulas applied to the original function f(x).

Here are the key formulas for common function transformations:

Details of Each Transformation Formula

Let's explore each transformation type and its corresponding formula in more detail, where f(x) represents the original (parent) function:

1. Translation

Translation involves shifting the entire graph of a function without changing its orientation or size.

• Vertical Translation: y = f(x) + k
  • When a constant k is added to the function, the graph shifts vertically.
  • If k > 0, the graph shifts upward by k units.
  • If k < 0, the graph shifts downward by |k| units.
  • Example: If f(x) = x^2, then y = x^2 + 3 shifts the parabola 3 units up, and y = x^2 - 2 shifts it 2 units down.

2. Dilation (Stretching and Compressing)

Dilation changes the size of the graph, either stretching it out or compressing it in.

• Horizontal Dilation: y = f(kx)

  • When the input x is multiplied by a constant k inside the function, the graph stretches or compresses horizontally.
  • If |k| > 1, the graph is compressed horizontally by a factor of 1/k.
  • If 0 < |k| < 1, the graph is stretched horizontally by a factor of 1/k.
  • Example: If f(x) = sin(x), then y = sin(2x) compresses the sine wave horizontally, completing a cycle twice as fast. y = sin(0.5x) stretches the wave horizontally, making it complete a cycle half as fast.

• Vertical Dilation: y = k f(x)

  • When the entire function f(x) is multiplied by a constant k, the graph stretches or compresses vertically.
  • If |k| > 1, the graph is stretched vertically by a factor of k.
  • If 0 < |k| < 1, the graph is compressed vertically by a factor of k.
  • Example: If f(x) = x^2, then y = 3x^2 stretches the parabola vertically, making it narrower. y = 0.5x^2 compresses the parabola vertically, making it wider.

3. Reflection

Reflection flips the graph across an axis, creating a mirror image.

• Reflection About the x-axis: y = -f(x)
  • When the entire function f(x) is multiplied by -1, the graph is reflected across the x-axis. Every positive y-value becomes negative, and every negative y-value becomes positive, effectively flipping the graph upside down.
  • Example: If f(x) = sqrt(x), which graphs the top half of a parabola opening right, then y = -sqrt(x) graphs the bottom half, reflected across the x-axis.

Understanding these formulas allows for the precise manipulation and prediction of how a function's graph will appear after various transformations are applied.