 ### Parent Functions and Transformations: Explanations

The parent functions are the simplest group of functions in mathematics. Find out what this means by exploring the definition and examples of the parent function in math. Study exponential functions, linear functions, cubic functions, rational functions, and others.

## Basic Parent Functions

You will probably study a few “popular” parent functions and work with them to learn how to transform them – how to resize and move them. We call these basic functions “parent” functions because they are the simplest form of that type of function, meaning they are as close as possible to the origin (0,0).

You should be familiar with the following basic parent functions. As well as the significant points, I have included the critical points with which to graph the parent function. Also known as “reference points” or “anchor points”, these serve as points of reference.

Know the shapes of these parent functions well! In order to understand how to transform t-charts correctly, you need to know the general shape of the parent functions.

The Greatest Integer Function, also known as the Step Function, returns the greatest integer that is less or equal to a given number (imagine rounding down to an integer). In addition, there is a Least Integer Function, indicated by y=⌈x⌉, which returns the least integer that is greater than or equal to a number (think rounding up to an integer).

## Generic Transformations of Functions

As always, the “parent functions” assume that we have the simplest form of the function; in other words, the function either passes through the origin (0,0) or, if it does not pass through the origin, it is not shifted. In any case, when a function is shifted, stretched (or compressed), or flipped in any way from its “parent function”, it is said to be transformed and is called a transformation of a function.

An extremely useful tool when dealing with functions is a T-chart. For example, if you know that the quadratic parent function y=x2 is being transformed 2 units to the right, and 1 unit down (only a shift, not a stretch or a flip), we can create the original t-chart, followed by the transformation points on the outside of the original points.

However, we need to be cautious when looking at the equation of the transformed function. Whenever functions are transformed on the outside of the f(x) part, you move the function up and down and do the “normal” math, as we will see in the examples. Translations or vertical transformations affect the y component of the function. If transformations are made on the inside of the f(x) part, you move the function back and forth (but do the “opposite” math – since if you were to isolate the x, you would move everything to the other side). The transformations or translations affect the x part of the function.

There are several ways to perform transformations of parent functions; I like to use T-charts since they work consistently with every function. You can see that in most t-charts, I’ve included more than just the critical points above, just to highlight the graphs better.

## Vertical Transformations

The following are the rules and examples of when functions are transformed on the “outside” (note how the y values are affected). T-charts contain the points (ordered pairs) of the original parent functions as well as the transformed or shifted points. The first two transformations are translations, the third is dilation, and the last are forms of reflection. Absolute value transformations will be discussed more in depth in the Absolute Value Transformations Section!

## Horizontal Transformations

These are the rules and examples of when functions are transformed on the “inside” (notice how the x-values are affected). You do the math in the opposite direction when you’re affecting the x-values: if you’re adding on the inside, you subtract from the x; if you’re subtracting on the inside, you add to the x; if you’re multiplying on the inside, you divide from the x; if you’re multiplying on the inside, you multiply to the x.

If you have a negative value on the inside, you flip across the y axis (notice that you still multiply the x by −1 just like you do with the y for vertical flips). The first two transformations are translations, the third is dilation, and the last are forms of reflection.