Parent Functions And Their Graphs
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More Graphs And PreCalculus Lessons
Graphs Of Functions
The following figures show the graphs of parent functions: linear, quadratic, cubic, absolute, reciprocal, exponential, logarithmic, square root, sine, cosine, tangent. Scroll down the page for more examples and solutions.
The following table shows the transformation rules for functions. Scroll down the page for examples and solutions on how to use the transformation rules.
Parent Functions And Their Graphs - How To Graph Elementary Functions?
In math, we often encounter certain elementary functions. These elementary functions include rational functions, exponential functions, basic polynomials, absolute values and the square root function. It is important to recognize the graphs of elementary functions, and to be able to graph them ourselves. This will be especially useful when doing transformations.
Basic Graphs That Every Algebra Student Should Know
Basic graphs that are useful to know for any math student taking algebra or higher.
y = mx + b (linear function)
y = x^{2} (quadratic)
y = x^{3} (cubic)
y = x^{5}
y = |x| (absolute)
y = √x (square root)
y = 1/x (reciprocal)
y = 1/x^{2}
y = log_{b}(x) for b > 1
y = a^{x} for a > 1 (exponential)
y = a^{x} for 0 < a < 1
The Graphs Of Six Basic Functions That You Should Know
f(x) = x
f(x) = x^{2}
f(x) = x^{3}
f(x) = √x
f(x) = cube root(x)
f(x) = |x|
7 Parent Functions With Equations, Graphs, Domain, Range And Asymptotes
y = x
y = x^{2}
y = √x
y = x^{3}
y = 1/x
y = 1/x^{2}
y = |x|
Exploring Properties Of Parent Functions
In math, every function can be classified as a member of a family. Each member of a family of functions is related to its simpler, or most basic, function sharing the same characteristics. This function is called the parent function.
This lesson discusses some of the basic characteristics of linear, quadratic, square root, absolute value and reciprocal functions.
Transformations Of Parent Functions
Learn how to shift graphs up, down, left, and right by looking at their equations.
Vertical Shifts:
f(x) + c moves up,
f(x) - c moves down.
Horizontal Shifts:
f(x + c) moves left,
f(x - c) moves right.
Transforming Graphs And Equations Of Parent Functions
Looking at some parent functions and using the idea of translating functions to draw graphs and write equations.
y = x,
y = x^{2},
y = x^{3},
y = √x,
y = 1/x,
y = |x|,
x^{2} + y^{2} = 9 (circle),
y = b^{x}
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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Parent Functions – Types, Properties & Examples
When working with functions and their graphs, you’ll notice how most functions’ graphs look alike and follow similar patterns. That’s because functions sharing the same degree will follow a similar curve and share the same parent functions.
A parent function represents a family of functions’ simplest form.
This definition perfectly summarizes what parent functions are. We use parent functions to guide us in graphing functions that are found in the same family. In this article, we will:
- Review all the unique parent functions (you might have already encountered some before).
- Learn how to identify the parent function that a function belongs to.
Being able to identify and graph functions using their parent functions can help us understand functions more, so what are we waiting for?
What is a parent function?
Now that we understand how important it is for us to master the different types of parent functions let’s first start to understand what parent functions are and how their families of functions are affected by their properties.
Parent function definition
Parent functions are the simplest form of a given family of functions. A family of functions is a group of functions that share the same highest degree and, consequently, the same shape for their graphs.
The graph above shows four graphs that exhibit the U-shaped graph we call the parabola. Since they all share the same highest degree of two and the same shape, we can group them as one family of function. Can you guess which family do they belong to?
These four are all quadratic functions, and their simplest form would be y = x^{2}. Hence, the parent function for this family is y = x^{2}.
Since parent functions are the simplest form of a given group of functions, they can immediately give you an idea of how a given function from the same family would look like.
What are the different types of parent functions?
It’s now time to refresh our knowledge about functions and also learn about new functions. As we have mentioned, familiarizing ourselves with the known parent functions will help us understand and graph functions better and faster.
Why don’t we start with the ones that we might already have learned in the past?
The first four parent functions involve polynomials with increasing degrees. Let’s observe how their graphs behave and take note of the respective parent functions’ domain and range.
Constant Functions
Constant functions are functions that are defined by their respective constant, c. All constant functions will have a horizontal line as its graph and contain only a constant as its term.
All constant functions will have all real numbers as its domain and y = c as its range. They also each have a y-intercept at (0, c).
An object’s motion when it is at rest is a good example of a constant function.
Linear Functions
Linear functions have x as the term with the highest degree and a general form of y = a + bx. All linear functions have a straight line as a graph.
The parent function of linear functions is y = x, and it passes through the origin. The domain and range of all linear functions are all real numbers.
These functions represent relationships between two objects that are linearly proportional to each other.
Quadratic Functions
Quadratic functions are functions with 2 as its highest degree. All quadratic functions return a parabola as their graph. As discussed in the previous section, quadratic functions have y = x^{2} as their parent function.
The vertex of the parent function y = x^{2} lies on the origin. It also has a domain of all real numbers and a range of [0, ∞). Observe that this function increases when x is positive and decreases while x is negative.
A good application of quadratic functions is projectile motion. We can observe an object’s projectile motion by graphing the quadratic function that represents it.
Cubic Functions
Let’s move on to the parent function of polynomials with 3 as its highest degree. Cubic functions share a parent function of y = x^{3}. This function is increasing throughout its domain.
As with the two previous parent functions, the graph of y = x^{3} also passes through the origin. Its domain and range are both (-∞, ∞) or all real numbers as well.
Absolute Value Functions
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.
The vertex of y = |x| is found at the origin as well. Since it extends on both ends of the x-axis, y= |x| has a domain at (-∞, ∞). Absolute values can never be negative, so the parent function has a range of [0, ∞).
We use absolute value functions to highlight that a function’s value must always be positive.
Radical Functions
The two most commonly used radical functions are the square root and cube root functions.
The parent function of a square root function is y = √x. Its graph shows that both its x and y values can never be negative.
This means that the domain and range of y = √x are both [0, ∞). The starting point or vertex of the parent function is also found at the origin. The parent function y = √x is also increasing throughout its domain.
Let’s now study the parent function of cube root functions. Similar to the square root function, its parent function is expressed as y = ∛x.
The graph shows that the parent function has a domain and range of (-∞, ∞). We can also see that y = ∛x is increasing throughout its domain.
Exponential Functions
Exponential functions are functions that have algebraic expressions in their exponent. Their parent function can be expressed as y = b^{x}, where b can be any nonzero constant. The parent function graph, y = e^{x}, is shown below, and from it, we can see that it will never be equal to 0.
And when x = 0, y passing through the y-axis at y = 1. We can also see that the parent function is never found below the y-axis, so its range is (0, ∞). Its domain, however,can be all real numbers. We can also see that this function is increasing throughout its domain.
One of the most common applications of exponential functions is modeling population growth and compound interest.
Logarithmic Functions
Logarithmic functions are the inverse functions of exponential functions. Its parent function can be expressed as y = log_{b} x, where b is a nonzero positive constant. Let’s observe the graph when b = 2.
Like the exponential function, we can see that x can never be less than or equal to zero for y = log_{2}x. Hence, its domain is (0,∞). Its range, however, contains all real numbers. We can also see that this function is increasing throughout its domain.
We use logarithmic functions to model natural phenomena such as an earthquake’s magnitude. We also apply it when calculating the half-life decay rate in physics and chemistry.
Reciprocal Functions
Reciprocal functions are functions that contain a constant numerator and x as its denominator. Its parent function is y = 1/x.
As can be seen from its graph, both x and y can never be equal to zero. This means that its domain and range are (-∞, 0) U (0, ∞). We can also see that the function is decreasing throughout its domain.
There are many other parent functions throughout our journey with functions and graphs, but these eight parent functions are that of the most commonly used and discussed functions.
You can even summarize what you’ve learned so far by creating a table showing all the parent functions’ properties.
How to find the parent function?
What if we’re given a function or its graph, and we need to identify its parent function? We can do this by remembering each function’s important properties and identifying which of the parent graphs we’ve discussed match the one that’s given.
Here are some guide questions that can help us:
- What is the function’s highest degree?
- Does it contain a square root or cube root?
- Is the function found at the exponent or denominator?
- Is the function’s graph decreasing or increasing?
- What is the function’s domain or range?
If we can answer some of these questions by inspection, we will be able to deduce our options and eventually identify the parent function.
Let’s try f(x) = 5(x – 1)^{2}. We can see that the highest degree of f(x) is 2, so we know that this function is a quadratic function. Hence, its parent function is y = x^{2}.
Why don’t we graph f(x) and confirm our answer as well?
From the graph, we can see that it forms a parabola, confirming that its parent function is y = x^{2}.
Review the first few sections of this article and your own notes, then let’s try out some questions to check our knowledge on parent functions.
Example 1
Graphs of the five functions are shown below. Which of the following functions do not belong to the given family of functions?
Solution
The functions represented by graphs A, B, C, and E share a similar shape but are either translated upward or downward. In fact, these functions represent a family of exponential functions. This means that they also all share a common parent function: y=b^{x}.
On the other hand, the graph of D represents a logarithmic function, so D does not belong to the group of exponential functions.
Example 2
Which of the following functions do not belong to the given family of functions?
- y = 5x^{2}
- y = -2x^{2} + 3x – 1
- y = x(3x^{2})
- y = (x – 1)(x + 1)
Solution
The function y = 5x^{2} has the highest degree of two, so it is a quadratic function. This means that its parent function is y = x^{2}. The same goes for y = -2x^{2} + 3x – 1. From this, we can confirm that we’re looking at a family of quadratic functions.
Applying the difference of perfect squares on the fourth option, we have y = x^{2} – 1. This is also a quadratic function. That leaves us with the third option.
When expanded, y = x(3x^{2}) becomes y = 3x^{3,} and this shows that it has 3 as its highest degree. Hence, it can’t be part of the given family of functions.
Example 3
Identify the parent function of the following functions based on their graphs. Define each function’s domain and range as well.
Solution
Let’s start with f(x). We can see that it has a parabola for its graph, so we can say that f(x) is a quadratic function.
- This means that f(x) has a parent function of y = x^{2}.
- The graph extends on both sides of x, so it has a domain of(-∞, ∞).
- The parabola never goes below the x-axis, so it has a range of [0,∞).
Based on the graph, we can see that the x and y values of g(x) will never be negative. They also show an increasing curve that resembles the graph of a square root function.
- Hence, the parent function of g(x) isy = √x.
- The graph extends to the right side of x and is never less than 2, so it has a domain of[2, ∞).
- The parabola never goes below the x-axis, so it has a range of [0,∞).
The h(x) graph shows that their x and y values will never be equal to 0. The symmetric curves also look like the graph of reciprocal functions.
- This means that h(x) has a parent function of y = 1/x.
- As long as the x and y are never equal to zero, h(x) is still valid, so it has both a domain and range of(-∞, ∞).
The straight lines representing i(x) tells that it is a linear function.
- It has a parent function of y = x.
- The graph extends on both sides of x and y, so it has a domain and range of(-∞, ∞).
Example 4
Identify the parent function of the following functions.
- f(x) = x^{3} – 2x + 1
- g(x) = 3√x + 1
- h(x) = 4/ x
- i(x) = e ^{x + 1 }
Solution
- The highest degree of f(x) is 3, so it’s a cubic function. This means that it has a parent function of y = x^{3}.
- The function g(x) has a radical expression, 3√x. Since it has a term with a square root, the function is a square root function and has a parent function of y = √x.
- We can see that x is found at the denominator for h(x), so it is reciprocal. Hence, its parent function is y = 1/x.
- The function’s exponents contain x, so this alone tells us that i(x) is an exponential function. Hence, its parent function can be expressed as y = b^{x}, where b is a constant. For the case of i(x), we have y = e^{x} as its parent function.
Practice Questions
1. The graphs of the five functions are shown below. Which of the following functions do not belong to the given family of functions?
2. Which of the following functions do not belong to the given family of functions?
- y = 4x^{3}
- y = -3x^{3} + 4x^{2} + 5x – 1
- y = x(5x^{2})
- y = (x – 1)(x + 1)(x + 2)
3. Identify the parent function of the following functions.
- f(x) = x^{3} – 2x + 1
- g(x) = 3√x + 1
- h(x) = 1/ (x + 1)
- i(x) = e ^{x + 1 }
4. Identify the parent function of the following functions based on their graphs. Define each function’s domain and range as well.
5. Describe the difference between f(x) = -5(x – 1)^{2} and its parent function. What is the domain and range of f(x)?
6. Let a and b be two nonzero constants. Describe the difference between g(x) = ax + b and its parent function. What is the domain and range of f(x)?
Images/mathematical drawings are created with GeoGebra.
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Parent function
This article needs attention from an expert in Mathematics. The specific problem is: The definition is not precise, and lacks authoritative references. See the talk page for details. WikiProject Mathematics may be able to help recruit an expert.(March 2013) |
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, for the family of quadratic functions having the general form
the simplest function is
- .
This is therefore the parent function of the family of quadratic equations.
For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes. For example, the graph of y = x^{2} − 4x + 7 can be obtained from the graph of y = x^{2} by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)^{2}.
For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R^{2} = A^{2} + B^{2}. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities).
The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as x^{n}, or, to simplify further, x^{2} when n is even and x^{3} for odd n. Turning points may be established by differentiation to provide more detail of the graph.
See also[edit]
External links[edit]
Parent Functions And Transformations
Parent 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. 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.
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An example of a family of functions is the quadratic functions. All quadratic functions have the highest exponent of 2, their graphs are all parabolas so they have the same shape, and they all share certain characteristics. 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.
Parent Functions Worksheet
*The Greatest IntegerFunction, sometimes called the Step Function, returns the greatest integer less than or equal to a number (think of rounding down to an integer). There’s also a Least IntegerFunction, indicated by \(y=\left\lceil x \right\rceil \), which returns the least integer greater than or equal to a number (think of rounding up to an integer).
Again, the “parent functions” assume that we have the simplest form of the function; in other words, the function either goes through the origin \(\left( {0,0} \right)\) or if it doesn’t go through the origin, it isn’t shifted in any way.
Notes on End Behavior: To get the end behavior of a function, we just look at the smallest and largest values of \(x\), and see which way the \(y\) is going. Not all functions have end behavior defined; for example, those that go back and forth with the \(y\) values and never really go way up or way down (called “periodic functions”) don’t have end behaviors.
Most of the time, our end behavior looks something like this:\(\displaystyle \begin{array}{l}x\to -\infty \text{, }\,y\to \,\,?\\x\to \infty \text{, }\,\,\,y\to \,\,?\end{array}\) and we have to fill in the \(y\) part. So the end behavior for a line with a positive slope is: \(\begin{array}{l}x\to -\infty \text{, }\,y\to -\infty \\x\to \infty \text{, }\,\,\,y\to \infty \end{array}\).
There are a couple of exceptions; for example, sometimes the \(x\) starts at 0 (such as in the radical function), we don’t have the negative portion of the \(x\) end behavior. Also, when \(x\) starts very close to 0 (such as in in the log function), we indicate that \(x\) is starting from the positive (right) side of 0 (and the \(y\) is going down); we indicate this by \(\displaystyle x\to {{0}^{+}}\text{, }\,y\to -\infty \).
What are the 8 parent functions?
8 Parent Functions
- The Exponential Function is f(x)=e.
- x.
- The Linear Function is f(x)=x.
- 8 Parent Functions.
- The Logistic Function is f(x)= 1. 1+e.
- Looking for boundedness.
- The Cubing Function is f(x)=x.
- The Cosine Function is f(x)=cos x.
What is an example of a parent function?
An example of a family of functions is the quadratic functions. … 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.
Parent Functions Graphs
Includes basic parent functions for linear, quadratic, cubic, rational, absolute value, and square root functions.
Match graphs to equations. Match family names to functions. Match graphs to the family names. Read cards carefully so that you match them correctly.
This is designed to be a matching activity. It will not work well as a flashcard activity.
The following table shows the transformation rules for functions. Scroll down the page for examples and solutions on how to use the transformation rules.
Parent Functions Chart
T-charts are extremely useful tools when dealing with transformations of functions. For example, if you know that the quadratic parent function \(y={{x}^{2}}\) is being transformed 2 units to the right, and 1 unit down (only a shift, not a stretch or a flip yet), we can create the original t-chart, following by the transformation points on the outside of the original points. Then we can plot the “outside” (new) points to get the newly transformed function:
Transformation | T-chart | Graph | ||||||||
Quadratic Function
\(y={{x}^{2}}\) Transform function 2 units to the right, and 1 unit down. |
Transformed: Domain: \(\left( {-\infty ,\infty } \right)\) Range: \(\left[ {-1,\,\,\infty } \right)\) |
When looking at the equation of the moving function, however, we have to be careful.
When functions are transformed on the outside of the \(f(x)\) part, you move the function up and down and do the “regular” math, as we’ll see in the examples below. These are verticaltransformations or translations, and affect the \(y\) part of the function.
When 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’d move everything to the other side). These are horizontal transformations or translations, and 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 ever function. And note that in most t-charts, I’ve included more than just the critical points above, just to show the graphs better.
Parent Functions And Transformations Worksheet
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.
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