Calculus How To

Parent Function: Definition, Examples & Graphs

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Types of Functions >

What is a Parent Function?

Every function in the Cartesian plane stems from a particular parent function.

a family of functions parents

Parent functions: square function

For example, every linear function can be generated from the parent function f(x) = x; Every other possible linear function of the form y = mx + b is a child function of this parent. Together, parent functions and child functions make up families of functions.

To put this another way, every function in a family is a transformation of a parent function. For example, the function f(x) = 2x is the linear parent function vertically stretched by a factor of 2; Instead of the function passing through (1, 1) the graph passes through (2, 1):

Common Parent Function List

  1. Absolute Value
  2. Constant
  3. Cube Root
  4. Cubic polynomial
  5. Exponential
  6. Linear (Identity)
  7. Logarithmic
  8. Quadratic polynomial

1. Absolute value parent function

The absolute value function is an even function with the parent p(x) = |x|.

absolute value parent function

Graph of the absolute value parent function f(x) = x.


  • Domain: (-∞, ∞).
  • Range: [0, ∞]; If x ≥ 0, then f(x) = x and if x <0, then f(x) = -x.
  • Inverse Function: f(x) = x, for x≥ 0.
  • General form: f(x) = a|b(x – h) + k.

2. Constant Parent Function

The constant function is an even function that has the parent f(x) = c.

The graph depends on the value of c. For example, the following graph shows two constant functions where c = 3 (red) and c = 2.5 (blue):

constant parent function

Two constant functions y = 3 and y = 2.5.


3. Cube Root Parent Function

The cube root function is an odd function that has the parent
f(x) = ∛x.

Graph of the cube root parent function f(x) = ∛x.


4. Cubic Polynomial Parent Function

Cubic functions are odd functions. The parent is: f(x) = x3.

The cubic parent function is strictly increasing, which basically means it’s always headed upwards.

Graph of the cubic parent function f(x) = x3.


5. Exponential Parent Function

The exponential function has no restrictions: inputs can be real numbers or imaginary numbers. The parent function is either f(x) = ex or f(x) = 10x.

exponential parent function

Graph of the exponential parent functions f(x) = ex(red) and f(x) = 10x (black).


6. Linear function

The linear function is an odd function with the parent:
f(x) = x.

linear parent function

The linear parent function f(x) = x is a straight line that goes through the origin (0, 0). Graphed with

More info: Linear parent functions.


  • Domain: (-∞, ∞).
  • Range: (-∞, ∞).
  • Inverse Function: g(x) = x.
  • General form: y = mx + b (m ≠ 0). Alternative: Ax + By + C = 0 or y – y0 = m(x – x0). For an overview of the different forms, see: Standard Form.

7. Logarithmic function

Parent: f(x) = logbx; Where b is the base.

For example, the parent f(x) = log2x is different from the parent f(x) = log10x.

The parent function for the natural logarithm function is ln(x).

logarithmic parent function

Two logarithmic functions with different bases: f(x) = log10 (blue) and f(x) = log2. (red)


Quadratic or Square Function

Parent: F(x) = x2

quadratic parent function

The quadratic parent function takes on a classic parabola shape.



Desmos Graphing Calculator.
UF Library of Functions. Retrieved March 11, 2021 from:
Harold’s Parent Functions “Cheat Sheet” (2016). Retrieved March 11, 2021 from:

Stephanie Glen. "Parent Function: Definition, Examples & Graphs" From Calculus for the rest of us!

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