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Parent Functions & Transformations · Core principles

Recognize the parent function

Identify a function by its basic equation and graph.

Updated 10/13/2026Permanent link

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The idea
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Common mistake
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The idea

A parent function is the simplest member of a family. Shifts, reflections, and stretches change its position or scale while keeping the family's basic structure.

Family	Parent equation	Graph clue
Linear	$y=x$	Straight line
Quadratic	$y=x^2$	U-shaped parabola
Cubic	$y=x^3$	S-shaped curve through the origin
Absolute value	$y=\lvert x\rvert$	V shape
Square root	$y=\sqrt{x}$	Starts at an endpoint
Cube root	$y=\sqrt[3]{x}$	Sideways S shape
Reciprocal	$y=\frac1x$	Two branches with asymptotes
Exponential	$y=2^x$	Growth curve with horizontal asymptote

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Example 1 of 3

Example 1

In $y=-2(x-3)^2+4$ , the squared expression identifies the quadratic parent, $y=x^2$ . The other numbers describe transformations.

Notice and plan
Pause and identify the mathematical feature or graph clue before following the calculation.
Work it through
Read this as: y equals negative two times the quantity x minus three, squared, plus four. The squared quantity tells us the parent is y equals x squared. The graph is still a parabola. Its vertex has moved to the point three, four, and it opens downward.
Check and understand
At x equal to three, the squared part is zero, leaving four. The vertex changes, but the squared variable still identifies the quadratic family.

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Transformed

Follow the reasoning, then check the result.

Read the narration · spoken math transcript

Example 1. Pause and identify the mathematical feature or graph clue before following the calculation. Read this as: y equals negative two times the quantity x minus three, squared, plus four. The squared quantity tells us the parent is y equals x squared. The graph is still a parabola. Its vertex has moved to the point three, four, and it opens downward. At x equal to three, the squared part is zero, leaving four. The vertex changes, but the squared variable still identifies the quadratic family.

Common mistake

Applying a rule without checking its conditions can give a wrong answer. Check signs, restrictions and units before calculating.

Check your understanding

Check yourself: Name the family before trying to describe its moves. A negative coefficient does not turn a quadratic into a different parent family.

Explore the idea

Try it yourself

Move a parabola

Predict how each slider will move or reshape the curve.

Vertical scale1Horizontal shift0Vertical shift0

y=1(x-(0))^2+(0),\quad (h,k)=(0,0)

The vertex is (0, 0). The parabola opens upward; its vertical scale is 1.

Challenge: Put the vertex at (2, 3) and make the parabola open downward.

Keep learning

Read shifts, reflections, and scale.

Used in practice

Identify the Parent