Learn what a parent function is by understanding the definition and types of parent functions. Explore parent function examples, graphs, and transformations. Updated: 05/22/2022 In mathematics, functions are defined as a relationship between input (independent variable such as x) and an output (dependent variable such as y). This relationship is in terms of rules describing the changes done to the input to create the output. Every function in math can be identified as a member of a family. The parent function of a family of functions is the simplest one in that family. Mathematically, the parent function definition is a function in its most basic form that shows the relationship between the independent and dependent variables in their pre-transformed state. Transformations may be added from this basic form and change the equation into a more complicated form. These changes will create new equations that still follow the basic characteristics of the parent function. Essentially, the properties that are shared among the family of functions that share a parent function are:
Parent FunctionsWhen 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. Parent 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. Quadratic Family of Functions
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 = 2x^2 + 4x 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 = 2x^2 + 4x can be derived by taking the parent function y = x^2, multiplying it by the constant 2, and then adding the term 4x to it.
What are the Types of Parent Functions?Examples of parent functions and their graphs are shown in Fig. 1. Fig. 1 Graphs and equations of parent functions.
The most basic functions are these parent functions, each with its set of properties and characteristics: Fig. 2 Graph of a linear parent function.
Fig. 3 Graph of a quadratic parent function
Fig. 4 Graph of cubic parent function
Fig. 5 Graph of absolute value parent function
Fig. 6 Graph of a reciprocal parent function
Fig. 7 Graph of an exponential parent function
Fig. 8 Graph of logarithmic parent function
Fig. 9 Graph of a square root parent function.
Fig. 10 Graph of a tangent parent function
Fig. 11Graph of a sine parent function.
Fig. 12 Graph of a cosine parent function.
How are Parent Functions Identified and Transformed?The parent functions may transform and change their equations to reflect all the new changes while at the same time still exhibiting the same graph and properties. These new functions can easily be classified under a parent function by understanding the changes. Transformations add constants to the original parent function. Here are the places where changes could take place in a parent function equation:
Example: Exponential FunctionsTo illustrate this, let's consider exponential functions. Exponential functions are functions of the form y = ab^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. Exponential Family of Functions With Base 2
Parent FunctionsWhen 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. Parent 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. Quadratic Family of Functions
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 = 2x^2 + 4x 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 = 2x^2 + 4x can be derived by taking the parent function y = x^2, multiplying it by the constant 2, and then adding the term 4x to it. Example: Exponential FunctionsTo illustrate this, let's consider exponential functions. Exponential functions are functions of the form y = ab^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. Exponential Family of Functions With Base 2
How do you write a parent function?The parent function is normally written in a very basic way, with just the independent and dependent variables and the correct degree or type of function (such as absolute value. For example, a linear function is just y=x. The quadratic parent function is just y=x^2. What is a parent function in graphing?The parent function in graphing is the basic equation where the graph is free from any transformation. For example, y=x is a parent function of a straight line. This graph may be translated, reflected, rotated, or dilated and the equations from such transformations will still be part of the family of lines. What is a parent function and family example?Functions may be classified and grouped into families. These groups each have one basic equation called a parent function. An example is the family of quadratic functions. This family shares the U-shaped graph called the parabola and they all have a second degree with the highest exponent. Register to view this lessonAre you a student or a teacher? Unlock Your EducationSee for yourself why 30 million people use Study.comBecome a Study.com member and start learning now.Become a Member Already a member? Log In Back Resources created by teachers for teachersOver 30,000 video lessons & teaching resources‐all in one place. Video lessons Quizzes & Worksheets Classroom Integration Lesson Plans I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline. Back Create an account to start this course today Used by over 30 million students worldwide Create an account How do you find the parent function of a graph?2. Explore the graphs of linear functions by adding or subtracting values to x (such as y(x) = x + 2) or by multiplying x by a constant (such as y(x) = 3x). Remember the linear parent function is y(x) = x. This is the most basic, simple form of the function.
Which parent function is represented by the graph absolute value parent function?The parent function of absolute value functions is y = |x|. As shown from the parent function's graph, absolute value functions are expected to return V-shaped graphs.
What are the 5 parent functions?These elementary functions include rational functions, exponential functions, basic polynomials, absolute values and the square root function.
Which of these is the quadratic parent function?The parent function of a quadratic equation is y=x2 y = x 2 , whose graph has a vertex at the origin and experiences no vertical stretch or shrink.
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Which parent function is represented by the graph apex
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