What is a parent function?

A parent function is the simplest form of a family of functions. It's the basic building block from which other functions (called "transformed" or "child" functions) are derived through transformations like shifting, stretching, compressing, and reflecting. Understanding parent functions is crucial because their properties and graphs directly inform the properties and graphs of their transformations.

Here are some key characteristics and examples:

• Simplicity: Parent functions contain no parameters (constants other than 1 or 0) that alter their basic shape. Their equations are generally simple and easy to understand.

• Building Blocks: They serve as a foundation. By applying transformations, you can create a vast array of more complex functions. For example, you can create many different quadratic functions from the parent function f(x) = x².

• Key Properties: Knowing the domain, range, intercepts, asymptotes, and increasing/decreasing intervals of a parent function makes analyzing its transformations straightforward.

Here are some examples of common parent functions:

• Linear: f(x) = x (straight line through the origin)

  • Domain: (-∞, ∞)
  • Range: (-∞, ∞)

• Quadratic: f(x) = x² (parabola opening upwards)

  • Domain: (-∞, ∞)
  • Range: [0, ∞)

• Cubic: f(x) = x³ (S-shaped curve)

  • Domain: (-∞, ∞)
  • Range: (-∞, ∞)

• Square Root: f(x) = √x (half of a parabola lying on its side)

  • Domain: [0, ∞)
  • Range: [0, ∞)

• Absolute Value: f(x) = |x| (V-shaped graph)

  • Domain: (-∞, ∞)
  • Range: [0, ∞)

• Reciprocal: f(x) = 1/x (hyperbola)

  • Domain: (-∞, 0) U (0, ∞)
  • Range: (-∞, 0) U (0, ∞)

• Exponential: f(x) = aˣ (where a > 0 and a ≠ 1) (exponentially increasing or decreasing curve)

  • Domain: (-∞, ∞)
  • Range: (0, ∞) (if a>1) or (∞,0) (if 0<a<1)

By studying these parent functions and their transformations, you can gain a deeper understanding of a wide range of functions and their behaviors. You'll be better equipped to sketch graphs, solve equations, and analyze function properties.