Parent Functions Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Identify the parent function of f(x)=3(x−2)2+5f(x) = 3(x - 2)^2 + 5 and describe the transformations.

Solution

  1. 1
    The parent function is y=x2y = x^2 (the basic quadratic function).
  2. 2
    The transformation (x−2)(x-2): horizontal shift 22 units to the right.
  3. 3
    The coefficient 33: vertical stretch by a factor of 33.
  4. 4
    The +5+5: vertical shift 55 units up. The vertex moves from (0,0)(0, 0) to (2,5)(2, 5).

Answer

Parent: y=x2; right 2, vertical stretch by 3, up 5\text{Parent: } y = x^2; \text{ right 2, vertical stretch by 3, up 5}
A parent function is the simplest form of a function family. Common parent functions include y=xy = x (linear), y=x2y = x^2 (quadratic), y=∣x∣y = |x| (absolute value), y=xy = \sqrt{x} (square root), and y=1/xy = 1/x (reciprocal). All other functions in the family are transformations of the parent.

About Parent Functions

A parent function is the simplest, most basic version of a function family — the unshifted, unstretched, unreflected template. All other functions in the family are transformations of this parent. Memorizing parent function shapes allows rapid graphing of transformed versions.

Learn more about Parent Functions →

More Parent Functions Examples

Example 2 medium

Match each equation to its parent function: (a) [formula], (b) [formula], (c) [formula], (d) [formul

Example 3 medium

Sketch the key features (domain, range, intercepts, asymptotes) of the parent function [formula].

Example 4 hard

Write a single equation using transformations of the parent function [formula] that has domain [form

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