How To Find Parent Functions

Parent functions in mathematics represent the basic function types and resulting graphs that a function can have. Parent functions do not have any of the transformations that a full function can have such as translation or dilation. You can use parent functions to determine the basic behavior of a function: the possibilities for axis intercepts, the number of solutions, etc. However, you cannot use parent functions to solve any problems for the original equation.

We can think of parent functions as the most distilled form for a given family of functions. This might apply to many types of functions, with varying usefulness. The biggest advantage of parent functions lies in their visual representation; we can use parent function graphs to learn about the general behavior of any function within a family in the simplest form.

Suppose we have a function ‌y‌ that is dependent on a variable ‌x‌; that is, we have a relationship to a function ‌f(x)‌ through ‌y‌:

\(y = f(x)\)

we can look at varying types of parent functions using this basis. We will assume to be only using the real numbers for simplicity.

All linear functions on a direct relationship between ‌y‌ and ‌x‌ described by the linear parent function:

\(y = x\)

We can further expand on this relationship with the standard form:

\(y = mx + b\)

Where ‌m‌ acts to rotate the parent function, and ‌b‌ acts to translate the parent function. All linear functions are straight lines that have an x-intercept and a y-intercept (assuming no restrictions on the domain or range).

Polynomial Functions

Polynomial functions cover a much wider range of possible relationships. We can have a practically infinite number of different types of functions, but they do still share similar attributes. The general parent function for a polynomial function is

\(y = x^n\)

where ‌n‌ is the degree of the polynomial, or the highest exponential power on any term. We can further reduce this with a quadratic parent function for even degrees and a cubic parent function for odd degrees.

Even Degree Polynomial Functions

When the degree (or highest power of any one term) of a polynomial is even, we can use the parent function

\(y = x^2.\)

All functions of this type will have a the general shape of a parabola (u-shaped), with different inflection points and variations. The geometry of these functions does change as the degree increases; quadratic functions have a single vertex and focus, while higher degree polynomials will often have multiple minima and maxima.

When the degree of a polynomial is odd, we can use the parent function

\(y = x^3.\)

All functions of this type have the general sideways-s shape of a cubic function, with different inflection points and variations.

Different from a parent function, standard form for a polynomial gives the expanded form for any possible polynomial function:

\(f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1x^1 + a_0x^2\)

Where the ‌a‌ coefficient terms can be any number (including zero), and they control which degrees of terms influence the function.

Exponential functions, where the ‌x‌ variable (independent variable) is in the exponent, are harder to represent simply with a parent function because the simplest function (1^x) is arbitrary – always being equal to one.

Instead we look to Euler's number, a familiar mathematical constant represented using the letter ‌e‌. Using this we have a parent function,

\(y = e^x\)

that gives a general shape and behavior for exponential functions.

These functions are easy to represent with a parent function, we simply only use ‌x‌ and ‌y‌, to give

\(y = |x|.\)

Similarly to absolute value functions, we use the simplest form that still gives a general indication of shape and behavior for radical functions. This gives us the following parent function:

\(y = \sqrt{x}.\)

This applies to cube root functions and beyond, we just scale the degree of the radical.

Logarithmic functions can be represented with two different, but valid, parent functions. We can refer back to Euler's number, and use a natural log (log base ‌e‌):

\(y = \ln{x}.\)

We can also use the default logarithmic expression (log base 10):

\(y = \log{x}.\)

We group these functions differently because their behavior varies quite a bit. If we are dealing with sine and cosine functions, we use sine as our parent function:

\(y = \sin{x}.\)

If we are dealing with tangent, we use tangent as our parent function:

\(y = \tan{x}.\)

The reciprocal functions (cosecant, secant, cotangent) and inverse functions (arcsin, arccos, arctan) all follow the same groupings.

We start by expanding and simplifying the function, so we can recognize the type of parent function to use. For example, expand the function

\(y=(x+1)^2 \rightarrow y=x^2+2x+1\)

Then we can recognize this as an even degree polynomial, and we reduce to a parent function to get:

\(\text{Parent function: } y = x^2\)

Graph the result on a graphing calculator, and this is the parent function. The other parent functions include the simple forms of the trigonometric, cubic, linear, absolute value, square root, logarithmic, and reciprocal functions that we have reference above.