Function Family Vocabulary

Function Family Vocabulary

Function

A relation where for each value in the domain, there exists one and only one resulting value in the range. A function can be described with an equation where the resultant value (y) is based on modifications to the input value (x).

y = 3(x-2)²

Parent Function

In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family.

For example: in the family of quadratic functions having the same general form, the simplest function is

y = x²

Domain

The set of all valid values for the input value of a function.

Range

The set of all valid values for the output value of a function.

Transformation

The specific variations from a parent function that create a specific member of the same function family.

Transformations can include:

horizontal shift

vertical shift

stretching and compressing

reflections

Constant

A value that is unchanging. For example, in

f(x) = 3

for any x value, the value of the function at that input is 3.

Similarly, a quantity added to a function that does not vary based on the input is a constant.

f(x)=2x -7

Linear

A function with a constant slope which is represented by a line.

Quadratic

A function with noted for its "u" shape (parabola).

f(x)=x²

This function has a clear vertex on the axis of symmetry.

Cubic

A function with noted for its sideways "s" shape.

f(x)=x³

This function usually has a clear vertex on the axis of symmetry.

Square Root

A function based on the form

 $f\left(x\right)=\sqrt{x}$  

Cube Root

A function based on the form

 $f\left(x\right)=^3\sqrt{x}$  

Absolute Value

A function based on the form

 $f\left(x\right)=\left|x\right|$  

Reciprocal

A function based on the form

 $f\left(x\right)=\frac{1}{x}$  

Step

A function based on rounding either up or down to an integer, such as 

Exponential

A function where a base (b) is raised to a variable exponent.

 $f\left(x\right)=2^x$   

If the base is >1, then the function increases. 
If the base is between 0 and 1, it decreases.

Exponential functions have an asymptote.

Reflection

A function where the variable (x) is replaced with its negative equivalent (-x).  

 $f\left(-x\right)$ 

Or the entire function is "flipped" $f\left(x\right)=2x^2\ \ \ \ vs\ -f\left(x\right)=-2x^2$    

Translation

Any transformation where the entire function is "slid" either horizontally or verbally or a combination of the two.

Stretch / Compress

Any transformation where the entire function is modified by changing the coefficient "slope."

A coefficient larger than 1 stretches the function vertically.

A coefficient between 0 and 1 compresses it vertically.