# Parent Function: Definition, Examples & Graphs

Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.

## What is a Parent Function?

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

## 1. Absolute value parent function

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

Characteristics:

• 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): Two constant functions y = 3 and y = 2.5.

Characteristics:

## 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.

Characteristics:

## 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.

Characteristics:

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

Characteristics:

## 6. Linear function

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

Characteristics:

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

Characteristics:

## Quadratic or Square Function

Parent: F(x) = x2 The quadratic parent function takes on a classic parabola shape.

Characteristics:

## References

Desmos Graphing Calculator.
UF Library of Functions. Retrieved March 11, 2021 from: https://xronos.clas.ufl.edu/mac1140nowell/PrecalculusXourse/graphing/parentFunctions.
Harold’s Parent Functions “Cheat Sheet” (2016). Retrieved March 11, 2021 from: https://people.clas.ufl.edu/srnatkins/files/ParentFunctionChart.pdf

CITE THIS AS:
Stephanie Glen. "Parent Function: Definition, Examples & Graphs" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/parent-function-definition-examples-graphs/
---------------------------------------------------------------------------  Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!