Functions visual reference

Parent Functions Chart

A parent function is the simplest member of a function family. Recognizing its shape makes transformations, domain, range, intercepts, symmetry, and end behavior much easier to predict.

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Parent functions chart with small graphs and equations for linear quadratic cubic absolute value square root cube root reciprocal exponential and logarithmic functions
Open the chart for a full-size, printable version.
How to read it

Use the chart as a decision tool.

A printable parent functions chart with graphs, equations, domains, ranges, and identifying features for nine essential function families. Read the label first, identify the matching structure in your problem, then confirm that any stated conditions apply before substituting values.

01

Polynomial and piecewise shapes

These are defined for every real input.

Linear

f(x)=x,D=R, R=R\displaystyle f(x)=x,\quad D=\mathbb R,\ R=\mathbb R

Quadratic

f(x)=x2,D=R, R=[0,∞)\displaystyle f(x)=x^2,\quad D=\mathbb R,\ R=[0,\infty)

Cubic

f(x)=x3,D=R, R=R\displaystyle f(x)=x^3,\quad D=\mathbb R,\ R=\mathbb R

Absolute value

f(x)=∣x∣,D=R, R=[0,∞)\displaystyle f(x)=|x|,\quad D=\mathbb R,\ R=[0,\infty)
02

Radical and rational shapes

Watch endpoints and excluded inputs.

Square root

f(x)=x,D=[0,∞), R=[0,∞)\displaystyle f(x)=\sqrt{x},\quad D=[0,\infty),\ R=[0,\infty)

Cube root

f(x)=x3,D=R, R=R\displaystyle f(x)=\sqrt[3]{x},\quad D=\mathbb R,\ R=\mathbb R

Reciprocal

f(x)=1x,x≠0, y≠0\displaystyle f(x)=\frac1x,\quad x\ne0,\ y\ne0
03

Inverse growth families

Exponential and logarithmic functions are inverses.

Exponential

f(x)=bx,D=R, R=(0,∞)\displaystyle f(x)=b^x,\quad D=\mathbb R,\ R=(0,\infty)

Logarithmic

f(x)=log⁡bx,D=(0,∞), R=R\displaystyle f(x)=\log_b x,\quad D=(0,\infty),\ R=\mathbb R
How to remember it

Identify the shape before reading the shifts

  1. 1Lines have constant rate of change; parabolas have a turning point.
  2. 2Absolute value makes a V; square root begins at an endpoint.
  3. 3Reciprocal, exponential, and logarithmic parents have asymptotes.
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