Skip to main content
Mathovia

Algebra / Common Graphs

Parent Functions

A parent function is the simplest member of a family of graphs: y = x² for every parabola, y = |x| for every V, y = √x for every square root curve. Learn a handful of them and you can sketch thousands of other graphs by shifting, stretching and flipping the ones you already know.

Practice Problems
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Chapter 4 · Lesson 1 of 17Common Graphs
  1. 1.Parent Functions
  2. 2.Parabolas
  3. 3.Vertex and Axis of Symmetry of a Parabola
  4. 4.Ellipses
  5. 5.Hyperbolas
  6. 6.Constant Functions
  7. 7.Square Root Functions
  8. 8.Absolute Value Functions
  9. 9.Cubic Functions
  10. 10.Rational Functions
  11. 11.Vertical Asymptotes
  12. 12.Horizontal Asymptotes
  13. 13.Function Transformations
  14. 14.Horizontal Shifts
  15. 15.Vertical Shifts
  16. 16.Reflections
  17. 17.Symmetry
See all algebra chapters →

There are infinitely many functions, but only a few basic shapes. The parabola , the steep, upside-down and the tiny all have the same U shape as , just moved, stretched or flipped. That simplest version is the parent function, and the others are its family. Know the parents by sight, along with a few points on each, and graphing most functions in algebra stops being a plotting exercise.

The common parent functions: x, x squared, x cubed, absolute value of x, square root of x, cube root of x, 1 over x, b to the x, and log base b of x
The common parent functions

What Is a Parent Function?

A parent function is the most basic function of its type: no coefficient in front, nothing added inside or outside, nothing shifted. Every other function of the same type comes from it by a transformation:

Here is the parent. The number stretches or flips the graph, shifts it left or right, and shifts it up or down. None of them changes the basic shape, which is why the parent is worth memorising.

The Parent Functions and Their Graphs

Ten small graphs of the parent functions: constant (a horizontal line), linear (a diagonal line through the origin), quadratic (a U-shaped parabola), cubic (an S-shaped curve), absolute value (a V), square root (a curve starting at the origin and rising slowly), cube root (a sideways S), reciprocal (two branches with dashed asymptotes on the axes), exponential (a curve rising from the dashed line y = 0) and logarithmic (a curve rising from the dashed line x = 0)
The ten parent functions you'll meet most often
Parent functionEquationDomainRangeKey points
constantall reals
linearall realsall reals, ,
quadraticall reals, ,
cubicall realsall reals, ,
absolute valueall reals, ,
square root, ,
cube rootall realsall reals, ,
reciprocal,
exponentialall reals,
logarithmicall reals,

The key points are the ones to plot first. They’re easy to compute, and after a transformation they tell you exactly where the new graph sits. The domain and range columns matter too: a transformation shifts them, so knowing the parent’s tells you the new ones almost for free.

Getting to Know Each Family

Lines and powers. The linear parent is a straight line through the origin at 45°. The quadratic is the U-shaped parabola, and the cubic is an S-shaped curve that flattens out at the origin and then climbs steeply. The constant function is the odd one out: a flat line with no in the formula at all.

The V. The absolute value function looks like on the right of the origin, and like its mirror image on the left, making a sharp V. It’s the only common parent with a corner.

Roots. The square root function starts at the origin and rises ever more slowly. It only exists for , because you can’t take the square root of a negative number in the real numbers. The cube root has no such limit, so it continues to the left as a sideways S.

The reciprocal. splits into two branches, one in the first quadrant and one in the third, hugging both axes without touching them. Those axes are its vertical and horizontal asymptotes, and it’s the parent of the rational functions.

Exponential and log. The exponential function creeps along just above the -axis on the left, passes through , then shoots upward. The logarithm is its mirror image across the line , because the two are inverse functions.

How to Identify the Parent Function

Look past the numbers and ask what’s being done to . The multiplier, the shift inside and the shift outside are all transformations; the operation that’s left tells you the parent.

Four equations matched to their parents: g(x) = −2√(x + 3) + 1 to the square root √x, h(x) = 4|x − 1| − 5 to absolute value |x|, k(x) = 3/(x + 2) − 1 to the reciprocal 1/x, and m(x) = 5 · 2 to the x − 4 to the exponential 2 to the x
Strip away a, h and k, and the parent is what's left

Example 1: A Square Root

What is the parent of ?

The sits under a square root. The , the inside and the outside are all transformations, so the parent is . Reading them off: the graph is shifted 3 left, stretched by 2, flipped upside down and shifted 1 up.

Example 2: A Quadratic in Standard Form

What is the parent of ?

The highest power of is 2, so it’s a quadratic and the parent is . Completing the square rewrites it as , which shows the same parabola stretched by 2 and moved to the vertex .

Example 3: A Fraction

What is the parent of ?

appears in the denominator of a fraction, so the parent is the reciprocal function . The asymptotes move with the graph: and .

Parent Functions and Transformations

Once you know the parent, a whole family of graphs follows from it. Here are three members of the quadratic family drawn against their parent:

The parent parabola y = x² in blue with three relatives: y = (x − 2)² + 1 shifted right 2 and up 1 to a vertex at (2, 1), y = 2x² narrower because it is stretched, and y = −x² + 3 opening downward with its vertex at (0, 3)
Every parabola is the parent y = x², moved, stretched or flipped

To graph a family member, move the parent’s key points. For , the parent’s points , , shift 2 right and 1 up to , , . Plot those and draw the familiar U through them. The function transformations lesson covers the order to apply several moves in.

Symmetry of the Parent Functions

Some parents are mirror images of themselves, which halves the plotting. The quadratic, absolute value and constant parents are even: symmetric about the -axis, so . The linear, cubic, cube root and reciprocal parents are odd: symmetric about the origin, so . The square root, exponential and log parents are neither. The symmetry lesson shows how to test any function.

Common Mistakes to Avoid

  • Letting the numbers decide. is still an absolute value function; the 5 and the only transform it.
  • Mixing up and . is a parabola, with as the base. is exponential, with in the exponent, and grows much faster.
  • Forgetting restricted domains. and don’t exist to the left of the origin, and is undefined at 0.
  • Calling every curve a parabola. The cubic and the square root curve aren’t parabolas; check the power or the operation.
  • Drawing the asymptote as part of the graph. The dashed lines for , and are guides the curve approaches, never touches.

Practice Problems

Work each one before opening the answer.

Problem 1. What is the parent function of ?

Show answer

is squared, so the parent is , shifted 4 left and 7 up and flipped upside down.

Answer:

Problem 2. What is the parent function of ?

Show answer

is under a cube root, so the parent is , shifted 1 right and stretched by 3.

Answer:

Problem 3. Which parent function has domain and range all real numbers?

Show answer

Only the logarithm is defined just for positive inputs yet takes every real value as output.

Answer:

Problem 4. Which parent functions have the range ?

Show answer

The quadratic , the absolute value and the square root never give negative outputs.

Answer: , and

Problem 5. The parent is shifted 2 right and 3 down. Write the new equation and give its starting point.

Show answer

Shifting right 2 replaces with ; shifting down 3 subtracts 3. The starting point moves to .

Answer: , starting at

Problem 6. What are the asymptotes of ?

Show answer

The parent’s asymptotes and move 5 right and 2 up.

Answer: and

Parent functions are the starting point for the rest of this chapter. Each of the graphs that follow, from parabolas to rational functions, is one of these parents with transformations applied.

Frequently Asked Questions

What is a parent function?+

It's the simplest function of a given type, with no shifts, stretches or reflections. For example, is the parent of every quadratic function, including . Every member of the family is a transformation of the parent.

What are the main parent functions?+

The ones you meet most often are constant , linear , quadratic , cubic , absolute value , square root , cube root , reciprocal , exponential and logarithmic .

How do you identify the parent function of an equation?+

Ignore the numbers that multiply, add or shift, and look at what is done to . In , sits under a square root, so the parent is . In it's inside absolute value bars, so the parent is .

What is the parent function of a quadratic?+

. Its graph is a U-shaped parabola with its vertex at the origin, domain all real numbers and range . Every quadratic can be written as a transformation of it.

Which parent functions have asymptotes?+

The reciprocal function has two: and . The exponential function has the horizontal asymptote , and the logarithmic function has the vertical asymptote . The polynomial, absolute value and root parents have none.

Related lessons

Function Transformations: Shifts, Stretches, and ReflectionsParabolas: How to Graph a Quadratic FunctionAbsolute Value Functions: The V-Shaped GraphSquare Root Functions: Graph, Domain, and Transformations