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.

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.

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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

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2. Linear Function

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

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3. Absolute Value Function

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

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4. Quadratic Function

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

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5. Square Root Function

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

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6. Cubic Function

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

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7. Cube Root Function

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

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8. Rational Function

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

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9. Exponential Function

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

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10. Logarithmic Function

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