The Linear Parent Function is f(x) = x. It is the most basic form of a linear function f(x) = mx + b.
Linear Parent Function Characteristics
The Linear Parent Function has the following characteristics:
- A domain and range of all real numbers (from negative to positive infinity).
- A constant slope, or rate of change.
Graph of the Linear Parent Function
Graph of the linear parent function (graphed with Desmos).
The above graph shows the basic linear parent function f(x) = x, which creates a diagonal line when graphed. The function is the simplest linear function possible, with a = 1 and b = 0:
- f(x) = ax + b becomes f(x) = 1x + 0 or simply f(x) = x.
Why is it called a “Parent Function?”
It’s called a “parent function” because every linear function in the Cartesian plane stems from the linear parent function f(x) = x. Every other possible linear function of the form y = mx + b is a child function of this parent. Together, the parent linear function and child functions make up a family: the family of linear functions.
To put this another way, every linear function f(x) = mx + b is a transformation of the linear parent function f(x) = x. For example, the function f(x) = 3x is the parent function vertically stretched by a factor of 3; Instead of the function passing through (1, 1) the graph passes through (3, 1):
References
Beezer, Robert A. Linear Transformations. from A First Course in Linear Algebra, version 3.50. Created on December 12, 2015. Retrieved from http://linear.ups.edu/html/section-LT.html on November 9, 2019
Larson, R. & Edwards, B. (2009). Calculus. Cengage Learning.
Oldham, K. et al. (2008). An Atlas of Functions: with Equator, the Atlas Function Calculator 2nd Edition. Springer.
Stephanie Glen. "Linear Parent Function" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/linear-parent-function/
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