Parent functions

8 min read · Updated 20 October 2026

A parent function is the simplest member of a family of functions. y = x² is the parent of every parabola, y = |x| of every V-shape. Learn the dozen parent graphs and most other graphs become a matter of shifting, stretching and flipping one of them.

The idea

Every function in a family is its parent with some combination of shifts, stretches and reflections applied. y = 3(x − 1)² + 2 is y = x² stretched by 3, moved right 1 and up 2. Knowing the parent tells you the shape; the numbers tell you where it sits. The table below lists the parents you meet most often, with links to a full page on each.

The parent functions at a glance

Family	Parent	Domain	Range
Linear	y = x	All reals	All reals
Quadratic	y = x²	All reals	y ≥ 0
Cubic	y = x³	All reals	All reals
Square root	y = √x	x ≥ 0	y ≥ 0
Cube root	y = ∛x	All reals	All reals
Absolute value	y = \|x\|	All reals	y ≥ 0
Reciprocal	y = 1/x	x ≠ 0	y ≠ 0
Exponential	y = bˣ	All reals	y > 0
Logarithmic	y = log_b x	x > 0	All reals
Sine	y = sin x	All reals	−1 ≤ y ≤ 1
Cosine	y = cos x	All reals	−1 ≤ y ≤ 1
Greatest integer	y = ⌊x⌋	All reals	Integers

Polynomial parents

The line, the parabola and the cubic are the first three powers of x. Even powers (x², x⁴) are symmetric about the y-axis and have a minimum; odd powers (x, x³) pass through the origin and rise from left to right. All of them pass through (0, 0) and (1, 1).

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The linear, quadratic and cubic parents all pass through the origin and (1, 1).

Roots, absolute value and reciprocals

The square root is half of a sideways parabola and only exists for x ≥ 0; the cube root continues to negative x. The absolute value is a V with a sharp corner. The reciprocal 1/x is the only parent here with asymptotes: the graph never touches either axis.

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√x, |x| and 1/x.

Exponential, logarithmic and trigonometric parents

The exponential bˣ and the logarithm log_b x are inverses of each other, mirror images in the line y = x. Sine and cosine are waves of period 2π. These parents are not polynomials, so their shapes cannot be produced by any finite combination of powers of x.

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2ˣ and log₂ x are reflections of each other in the dashed line y = x.

Identifying the parent of a graph

A U-shape with a rounded bottom: quadratic. A sharp corner: absolute value.
Starts at a point and curves away in one direction: square root.
An S-shape through a flat point: cubic. Through a vertical point: cube root.
Two separate branches with asymptotes: reciprocal.
Flat on one side and shooting up on the other: exponential. Vertical asymptote and slow growth: logarithmic.
Repeating waves: sine or cosine.

Practice

A new problem every time. Answer in any form you like: fractions, decimals, √ and π all work, and equivalent equations count as right.

Problem 1

Which parent function is y = 4√(x − 5) + 5 built from?

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Answer: Square root, y = √x

The 4 in front stretches the graph, x − 5 moves it sideways and 5 at the end moves it up or down.
Take those away and the shape left is y = √x: the square root parent.

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Frequently asked questions

What is a parent function?

The simplest function in a family, such as y = x² for quadratics. Every other member of the family is a transformation of it.

What are the main parent functions?

Linear, quadratic, cubic, square root, cube root, absolute value, reciprocal, exponential, logarithmic, sine, cosine and the greatest integer function.

Which parent functions have asymptotes?

The reciprocal (both axes), the exponential (y = 0) and the logarithm (x = 0). Tangent, from the trig family, has vertical asymptotes too.

How do I find the parent function of y = −2|x + 3| − 1?

Strip away the numbers: the parent is y = |x|. The −2 flips and stretches it, and the +3 and −1 move the vertex to (−3, −1).