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ACT Prep: Tutoring Solution43 chapters | 384 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will familiarize us with parent functions. We will look at many examples to learn the definitions of both families of functions and parent functions. We will also look at identifying a parent function in a family of functions.

When you hear the term parent function, you may be inclined to think of two functions who love each other very much creating a new function.

Well, that's not exactly right; however, there are some similarities that we can observe between our own parents and parent functions. In mathematics, we have certain groups of functions that are called families of functions. Just like our own families have parents, **families of functions** also have a parent function.

The similarities don't end there! In the same way that we share similar characteristics, genes, and behaviors with our own family, families of functions share similar algebraic properties, have similar graphs, and tend to behave alike. An example of a family of functions are the quadratic functions. All quadratic functions have a highest exponent of 2, their graphs are all parabolas so they have the same shape, and they all share certain characteristics.

As mentioned above, each family of functions has a parent function. A **parent function** is the simplest function that still satisfies the definition of a certain type of function. For example, when we think of the linear functions which make up a family of functions, the parent function would be *y* = *x*. This is the simplest linear function.

Furthermore, all of the functions within a family of functions can be derived from the parent function by taking the parent function's graph through various transformations. These transformations include horizontal shifts, stretching or compressing vertically or horizontally, reflecting over the *x* or *y* axes, and vertical shifts. For example, in the above graph, we see that the graph of *y* = 2*x*^2 + 4*x* is the graph of the parent function *y* = *x*^2 shifted one unit to the left, stretched vertically, and shifted down two units. These transformations don't change the general shape of the graph, so all of the functions in a family have the same shape and look similar to the parent function.

Algebraically, these transformations correspond to adding or subtracting terms to the parent function and to multiplying by a constant. For example, the function *y* = 2*x*^2 + 4*x* can be derived by taking the parent function *y* = *x*^2, multiplying it by the constant 2, and then adding the term 4*x* to it.

To illustrate this, let's consider exponential functions. Exponential functions are functions of the form *y* = *a**b*^*x*, where *a* and *b* are both positive (greater than zero), and *b* is not equal to one. Basically, exponential functions are functions with the variable in the exponent. The number *b* is the base of the exponential function. Let's consider the family of functions that are exponential functions with base 2. Some functions that are in this family of functions are shown below.

Notice that the simplest exponential function in the above family is *y* = *2*^*x*. This is the parent function of the family of functions. In general, the simplest exponential function is *y* = *b*^*x* where *b* > 1. This is the parent function of exponential functions with base *b*. The graph below displays the graphs of all of the functions listed above. Observe that they all have the same shape as the parent function and that they can all be derived by performing the transformations previously mentioned to the parent function.

Let's look at some more examples to further our understanding of parent functions.

1.) Cubic functions are functions that are polynomials with highest exponent equal to 3. For example, *y*= 4*x*^3 + 2*x* - 1 is a cubic function. Cubic functions form a family of functions. What would the parent function be for cubic functions?

Solution: The parent function would be the simplest cubic function. That is the simplest polynomial with highest exponent equal to 3. This would be *y*= *x*^3.

2.) In the following graph, which of the following functions is not in the same family of functions as the others?

Solution: In a family of functions, all of the function's graphs have the same shape. It is easy to see that function C does not have the same shape as functions A, B, and D. Therefore, function C is not in the same family as functions A, B, and D.

3.) The following functions are in the family of functions of absolute value functions. Which of them is the parent function?

a.) *y* = |*x* + 4| - 7

b.) *y* = |*x*|

c.) *y* = 2|*x*| + 3

d.) *y* = 4|*x*| - 19

Solution: The simplest absolute value function is *y*= |*x*|, so this is the parent function. We see that we can derive all the other functions shown from *y* = |*x*|. For example, to get to *y* = 2|*x*| + 3, we would take *y* = |*x*|, multiply the absolute value by 2, then add 3 to the result. This would give us *y* = 2|*x*| + 3.

There are many **families of functions**. Some examples of those include linear functions, quadratic functions, cubic functions, exponential functions, logarithmic functions, radical functions, and rational functions, among many more. Each of these families of functions have **parent functions** that are the simplest of the group, and we can derive all of the functions in a family by performing simple transformations to the graph of the parent function. These transformations correspond algebraically to adding or subtracting terms to the function, or multiplying by a constant. Just as all of us share certain characteristics with our parents, graphs in a family of functions all share certain characteristics with their parent function. With the information in this lesson, we should now be familiar with what parent functions are and how to identify them.

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ACT Prep: Tutoring Solution43 chapters | 384 lessons

- What is a Function: Basics and Key Terms 7:57
- Inverse Functions 6:05
- Applying Function Operations Practice Problems 5:17
- How to Compose Functions 6:52
- How to Add, Subtract, Multiply and Divide Functions 6:43
- What Is Domain and Range in a Function? 8:32
- Functions: Identification, Notation & Practice Problems 9:24
- Compounding Functions and Graphing Functions of Functions 7:47
- Understanding and Graphing the Inverse Function 7:31
- Polynomial Functions: Properties and Factoring 7:45
- Polynomial Functions: Exponentials and Simplifying 7:45
- Explicit Functions: Definition & Examples 7:36
- Function Operation: Definition & Overview 6:17
- Function Table in Math: Definition, Rules & Examples 5:53
- Hyperbolic Functions: Properties & Applications 6:39
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- Parent Function in Math: Definition & Examples 7:25
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