A parent function is the simplest function of a family, before any shifts, stretches or reflections. For example, y = x2 is the parent of every quadratic, and y = |x| is the parent of every absolute value function. Every other member of the family is a transformation of its parent’s graph.
Learn these six shapes. Every function in a family is one of them moved, stretched or flipped.
Formal definition and notation
A family of functions is a set of functions that can all be written as g(x) = af(b(x − h)) + k for one fixed function f and real constants a ≠ 0, b ≠ 0, h and k. The fixed function f, with a = b = 1 and h = k = 0, is the parent function of the family. Common parents are x, x2, x3, |x|, √x, 1x and bx.
Symbol
Read it as
Meaning
f(x) + k
f of x plus k
shift the parent up k units (down if k is negative)
f(x − h)
f of x minus h
shift right h units (left if h is negative)
af(x)
a times f of x
vertical stretch if |a| > 1, compression if |a| < 1, reflection over the x-axis if a < 0
A transformation is the move (shift, stretch, reflection). The parent function is the starting graph the move is applied to.
Parent Functionvs.Inverse function
An inverse undoes a function: the inverse of y = x3 is y = 3√x. A parent is the simplest member of a family, not an undoing.
Quick check: 3 questions
Question 1 of 3
What is the parent function of f(x) = 12(x + 1)3 − 4?
Show answer and explanation
Answer: C. The cube on (x + 1) puts it in the cubic family, parent y = x3. The 12 is a compression, not a reciprocal, and the +1 and −4 are shifts.
Question 2 of 3
How is g(x) = |x − 2| + 6 related to y = |x|?
Show answer and explanation
Answer: B.x − 2 inside moves the graph right 2, and + 6 outside moves it up 6, so the corner is at (2, 6). Left 2 reads the inside sign the wrong way, down 6 flips the outside sign, and the last choice swaps the two shifts.
Question 3 of 3
Which parent function has domain x ≥ 0?
Show answer and explanation
Answer: A. You cannot take the real square root of a negative number, so √x starts at 0. The absolute value, cubic and exponential parents accept every real input; |x| and 2x only have restricted ranges.
Parent Function: frequently asked questions
What are the main parent functions?
The ones used most in algebra are linear y = x, quadratic y = x2, cubic y = x3, absolute valuey = |x|, square rooty = √x, reciprocaly = 1x, exponential y = bx and logarithmic y = logbx. Constant functions y = c and cube roots are sometimes included. Each has a shape you can sketch from memory.
How do you identify the parent function of an equation?
Look at the operation applied to x after you ignore all the added and multiplied constants. A square means quadratic, a cube means cubic, absolute value bars mean the absolute value family, a radical means square root, and x in an exponent means exponential. If x appears only to the first power, the parent is y = x.
Why are parent functions useful?
They let you graph a complicated function quickly. Sketch the parent, then apply the shifts, stretches and reflections one at a time. They also tell you the domain, range, end behavior and key point before you compute anything, which helps you check graphing calculator results and spot errors on tests.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.B.3
Cite this entry
Effortless Math Team. “Parent Function.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/parent-function/
Flashcard
TermParent Functiony = x2, y = |x|, y = √x
DefinitionThe simplest function of a family. y = (x − 4)2 + 1 is the parent y = x2 moved right 4 and up 1.
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The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusLinear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Absolute ValueThe absolute value of a number is its distance from zero on the number line, so it is never negative. It is written with two vertical bars: |−6| = 6 and |6| = 6. A positive number and 0 keep their value; a negative number loses its minus sign.Number System · Grade 6, Grade 7, Grade 8Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2GraphA graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.Functions · Grade 5, Grade 6, Grade 7Transformation of FunctionsTransformation of functions is changing a function’s rule so that its graph moves, flips or resizes. Adding k outside, f(x) + k, shifts the graph up; replacing x with x − h shifts it right; −f(x) reflects it over the x-axis; and a · f(x) stretches it vertically. The basic graph being changed is the parent function.Functions · Algebra 1, Algebra 2, PrecalculusTranslationA translation is a slide: a transformation that moves every point of a figure the same distance in the same direction, without turning or flipping it. The image is congruent to the original. On the coordinate plane, a translation adds the same amounts to every point: (x, y) → (x + a, y + b).Geometry · Grade 8, GeometryQuadratic FunctionA quadratic function is a function that can be written f(x) = ax2 + bx + c with a ≠ 0. Its graph is a U-shaped curve called a parabola, which opens up when a > 0 and down when a < 0. Its highest or lowest point is the vertex.Functions · Algebra 1, Algebra 2, Precalculus