Graphs of Parent Functions

A parent function is the simplest form that a function can be. Its basic shape is not in any way altered.

For instance, when you see a u-shaped graph that is inverted and vertically stretched, you should still recognize that it is a parabola which has undergone different transformations.

the graph of a parabola opening upward and to its right a parabola opening downward

That is, if [latex]y = a{x^2} + bx + c[/latex] is the general form of a quadratic function, then its parent function is simply [latex]y=x^2[/latex] since it’s the simplest of its kind in the family.


List of Parent Functions

The graphs of the most frequently used parent functions are shown below. It’s a useful mathematical skill to be able to recognize them just by looking at their fundamental shapes.

  1. Constant Function

[latex]\large{f\left( x \right) = c}[/latex]

where [latex]\large{c}[/latex] is a number

Graph of Constant Function

2. Linear Function

[latex]\large{f\left( x \right) = x}[/latex]

Graph of Parent Function of Linear Function

3. Absolute Value Function

[latex]\large{f\left( x \right) = \left| x \right|}[/latex]

Graph of Parent Function of Absolute Value Function

4. Quadratic Function

[latex]\large{f\left( x \right) = {x^2}}[/latex]

Graph of Parent Function of Quadratic Function

5. Square Root Function

[latex]\large{f\left( x \right) = \sqrt x }[/latex]

Graph of Parent Function of Square Root Function

6. Cubic Function

[latex]\large{f\left( x \right) = {x^3}}[/latex]

Graph of Parent Function of Cubic Function

7. Cube Root Function

[latex]\large{f\left( x \right) = \sqrt [3] {x}}[/latex]

Graph of Parent Function of Cube Root Function

8. Rational Function

[latex]\large{f\left( x \right) = \Large{{1 \over x}}}[/latex]

Graph of Parent Function- Rational Function

9. Exponential Function

[latex]\large{f\left( x \right) = {e^x}}[/latex]

Graph of Parent Function - Exponential Function

10. Logarithmic Function

[latex]\large{f\left( x \right) = \ln \left( x \right)}[/latex]

Graph of Parent Function - Logarithmic Function

Take a quiz:

Identifying Function by its Graph Quiz