What are the different types of transformations you can have with a function?

Functions can undergo various transformations that alter the position, orientation, or size of their graphs while preserving their fundamental shape. Understanding these changes is crucial for analyzing and predicting function behavior.

Understanding Function Transformations

A transformation applied to a function changes its graph in a predictable way based on how the function's equation is modified. These transformations are categorized into four main types: translation, reflection, dilation, and rotation. Each type affects the graph differently, shifting it, flipping it, stretching or shrinking it, or turning it around a point.

Here's a summary of the common types of transformations:

Key Types of Transformations and Their Effects

Let's explore each transformation in more detail, considering its impact on a function y = f(x).

1. Translation

Translation involves moving the graph of a function without changing its orientation or size. It's a simple shift.

• Vertical Translation:
  • Rule: y = f(x) + k
  • Effect: The graph shifts up by k units if k > 0, or down by |k| units if k < 0.
  • Example: If f(x) = x^2, then g(x) = x^2 + 3 shifts the parabola up by 3 units, and h(x) = x^2 - 2 shifts it down by 2 units.
• Horizontal Translation:
  • Rule: y = f(x - h)
  • Effect: The graph shifts right by h units if h > 0, or left by |h| units if h < 0. Note the inverse sign relationship.
  • Example: If f(x) = |x|, then g(x) = |x - 4| shifts the V-shape 4 units to the right, and h(x) = |x + 1| (which is |x - (-1)|) shifts it 1 unit to the left.

For more on translations, you can refer to resources on horizontal and vertical shifts.

2. Reflection

Reflection flips the graph of a function across a line, creating a mirror image. This line is typically the x-axis or the y-axis.

• Reflection Across the X-axis:
  • Rule: y = -f(x)
  • Effect: The graph is flipped vertically. Every positive y-value becomes negative, and every negative y-value becomes positive.
  • Example: If f(x) = x^2, then g(x) = -x^2 opens downwards instead of upwards.
• Reflection Across the Y-axis:
  • Rule: y = f(-x)
  • Effect: The graph is flipped horizontally. Points on the right side of the y-axis move to the left, and vice-versa.
  • Example: If f(x) = 2^x, then g(x) = 2^{-x} reflects the exponential growth across the y-axis, resulting in exponential decay.

Further details on reflections can be found in guides about reflections of graphs.

3. Dilation (Stretching and Shrinking)

Dilation changes the size of the graph by stretching or shrinking it, either vertically or horizontally. This is often referred to as scaling.

• Vertical Stretch or Compression:
  • Rule: y = a * f(x)
  • Effect:
    • If |a| > 1, the graph is stretched vertically by a factor of a.
    • If 0 < |a| < 1, the graph is compressed (shrunk) vertically by a factor of a.
    • If a < 0, a reflection across the x-axis also occurs.
  • Example: If f(x) = sin(x), then g(x) = 2sin(x) stretches the sine wave vertically, doubling its amplitude, while h(x) = 0.5sin(x) compresses it.
• Horizontal Stretch or Compression:
  • Rule: y = f(bx)
  • Effect:
    • If |b| > 1, the graph is compressed (shrunk) horizontally by a factor of 1/b.
    • If 0 < |b| < 1, the graph is stretched horizontally by a factor of 1/b.
    • If b < 0, a reflection across the y-axis also occurs.
  • Example: If f(x) = cos(x), then g(x) = cos(2x) compresses the cosine wave horizontally, halving its period, while h(x) = cos(0.5x) stretches it.

For more information on stretches and compressions, see resources on stretching and compressing graphs.

4. Rotation

Rotation turns the pre-image (the original graph) around a fixed point, typically the origin (0,0). While rotations are fundamental geometric transformations, their direct application to general functions y = f(x) can be complex because a rotated function's graph may not always represent a function (it might fail the vertical line test).

Common rotation rules for a point (x, y) about the origin are:

• 90° Counter-Clockwise Rotation: (-y, x)
• 180° Rotation: (-x, -y)
• 270° Counter-Clockwise (or 90° Clockwise) Rotation: (y, -x)

When a function's graph is rotated, the new relation might require implicit equations or parametric forms rather than remaining a simple y = f(x) form. For instance, rotating the parabola y = x^2 by 90 degrees counter-clockwise about the origin results in the equation x = -y^2, which is a parabola opening to the left and is not a function of x.

Combining Transformations

It's common for functions to undergo multiple transformations simultaneously. When applying a sequence of transformations, the order often matters, especially for combinations of reflections, dilations, and translations. A general approach involves applying dilations and reflections first, followed by translations.