## What is a Parent Function?

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

- Absolute Value
- Constant
- Cube Root
- Cubic polynomial
- Exponential
- Linear (Identity)
- Logarithmic
- Quadratic polynomial

## 1. Absolute value parent function

The absolute value function is an even function with the parent **p(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):

**Characteristics**:

- Domain: (-∞, ∞).
- Range: [c, c], where c is a real number.
- Undefined (asymptotic) inverse Function.
- General form: Ay + B = 0

## 3. Cube Root Parent Function

The cube root function is an odd function that has the parent

**f(x) = ∛x**.

**Characteristics**:

- Domain: (-∞, ∞).
- Range: (-∞, ∞).
- Inverse Function: g(x) = x
^{3} - General form: f(x) = ∛√(b(x – h)) + k.

## 4. Cubic Polynomial Parent Function

Cubic functions are odd functions. The parent is:** f(x) = x ^{3}.**

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

**Characteristics**:

- Domain: (-∞, ∞).
- Range: (-∞, ∞).
- Inverse Function: g(x) = ∛x
- General form: f(x) = a((b(x – h))
^{3}+ k.

## 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) = e ^{x}** or

**f(x) = 10**.

^{x}**Characteristics**:

- Domain: (-∞, ∞).
- Range: (0, ∞).
- Inverse Functions: g(x) = ln(x) or g(x) = log(x).
- General form: f(x) = a 10
^{(b(x – h))}+ k.

## 6. Linear function

The linear function is an odd function with the parent:

**f(x) = x.**

More info: Linear parent functions.

**Characteristics**:

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

## 7. Logarithmic function

**Parent: f(x) = log _{b}x; Where b is the base. **

For example, the parent f(x) = log_{2}x is different from the parent f(x) = log_{10}x.

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

**Characteristics**:

- Domain: (0, ∞).
- Range: (-∞, ∞).
- Inverse Function: g(x) = 10
^{x}or g(x) = e^{x} - General form: f(x) = a log(b(x – h)) + k.

## Quadratic or Square Function

**Parent: F(x) = x ^{2}**

**Characteristics**:

- Domain: (-∞, ∞).
- Range: (0, ∞).
- Inverse Function: g(x) = √x
- General form: Ax
^{2}+ By + Cx + D = 0.

## 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