Parent Functions | Moosa Academy

Parent Functions — Moosa Academy

A small number of functions serve as the ancestors of all the rest. Every graph you will meet is one of these parent functions shifted, stretched or flipped. Learn the six shapes and their properties, and an unfamiliar equation becomes a familiar curve that has simply been moved.

Concept The six shapes

All six pass through the origin except

1/x

, which never reaches it.

Each has a distinctive shape that survives every transformation applied to it.

Note Domain, range and symmetry

Function	Domain	Range	Type
$f(x) = x$	ℝ	ℝ	Odd, increasing
$f(x) = x^2$	ℝ	y ≥ 0	Even, parabola
$f(x) = x^3$	ℝ	ℝ	Odd, increasing
$f(x) = \sqrt{x}$	x ≥ 0	y ≥ 0	Increasing
$f(x) = \|x\|$	ℝ	y ≥ 0	Even, V shape
$f(x) = 1/x$	x ≠ 0	y ≠ 0	Odd, hyperbola

Two of the domains are restricted, and for a clear reason. A square root of a negative number is not real, and division by zero is undefined.

Note Reading each shape

Linear — a straight line through the origin; the coefficient is its slope. Used for proportional relationships.

Quadratic — a parabola opening upward, with its minimum at the origin and the vertical axis as its axis of symmetry.

Cubic — negative for negative inputs, positive for positive ones, with an inflection point at the origin.

Square root — defined only for non-negative inputs; it is the inverse of the quadratic restricted to

x \geq 0

Absolute value — a V; it measures distance from zero, so it never returns a negative.

Reciprocal — a hyperbola in the first and third quadrants, with both axes as asymptotes.

Example Recognising a parent

Which parent function underlies $g(x) = 3\sqrt{x - 2} + 1$ , and what is its domain?

The core operation is a square root, so the parent is

f(x) = \sqrt{x}

The parent needs a non-negative input, so here

x - 2 \geq 0

⟹ parent √x, domain x ≥ 2

The 3 and the +1 change the size and position of the curve but not its shape or its family.

Summary

Six parent functions generate almost every graph met at this level.
x², |x| are even; x, x³ and 1/x are odd.
√x needs x ≥ 0 and 1/x excludes x = 0; the rest are defined for all reals.
Transformations move and resize a parent but never change which family it belongs to.