What is a parent function?

A parent function is the simplest form of a function family. It's a basic function that is transformed to create other, more complex functions within the same family. Understanding parent functions is crucial for grasping how transformations affect the graph and equation of a function.

Here's some key information about parent functions:

• Definition: A parent function is the simplest function in a family of functions. Other functions in the family can be derived from the parent function through transformations such as translations, reflections, stretches, and compressions.

• Examples of Common Parent Functions:

  • Linear Function: f(x) = x (A straight line passing through the origin with a slope of 1).
  • Quadratic Function: f(x) = x^2 (A parabola with its vertex at the origin).
  • Cubic Function: f(x) = x^3
  • Square Root Function: f(x) = √x
  • Absolute Value Function: f(x) = |x| (A V-shaped graph with its vertex at the origin).
  • Exponential Function: f(x) = a^x (where a is a constant, usually a > 0 and a ≠ 1)
  • Logarithmic Function: f(x) = log_a(x) (where a is a constant, usually a > 0 and a ≠ 1)
  • Reciprocal Function: f(x) = 1/x

• Transformations: Understanding how transformations affect a parent function is key to graphing and analyzing functions. Transformations include:

  • Translations: Shifting the graph horizontally or vertically.
  • Reflections: Flipping the graph over the x-axis or y-axis.
  • Stretches/Compressions: Vertically or horizontally stretching or compressing the graph.

• Importance: Parent functions provide a foundation for understanding the behavior of more complex functions. By recognizing the parent function within a given function, you can more easily predict its graph and properties. They are helpful in solving equations and understanding the relationship between equations and their graphical representations.