Parent Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Parent Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A parent function is the simplest base graph in a function family before transformations.

It is the original template shape you move, stretch, or reflect.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Knowing a parent function's key features (shape, intercepts, domain, range) lets you immediately describe all transformed versions without recomputing from scratch.

Common stuck point: Students try to graph transformed forms from scratch each time.

Sense of Study hint: Identify the family first, sketch parent, then apply one transformation at a time.

Worked Examples

Example 1

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

Solution

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

Answer

\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 = x (linear), y = x^2 (quadratic), y = |x| (absolute value), y = \sqrt{x} (square root), and y = 1/x (reciprocal). All other functions in the family are transformations of the parent.

Example 2

medium
Match each equation to its parent function: (a) y = -\sqrt{x+3}, (b) y = \frac{2}{x-1}, (c) y = |x| - 4, (d) y = 2^{x+1} - 3.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Sketch the key features (domain, range, intercepts, asymptotes) of the parent function y = \frac{1}{x}.

Example 2

hard
Write a single equation using transformations of the parent function y = \sqrt{x} that has domain (-\infty, 4], range [-2, \infty), and passes through the point (0, 0).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function familiestransformationmultiple representations