# Parent Function: Definition, Examples & Graphs

## 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):

## 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:

## 3. Cube Root Parent Function

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

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.

Characteristics:

## 6. Linear function

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

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).

Characteristics: