How to Find Mode in Math—Explained

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How to Find Mode in Math—Explained

How to Find the Mode in Math Step-by-Step

Math Skills: How to find mode of a data set

 

Are you ready to learn how to find mode?

 

In math, it is important for students to understand different ways of measuring and describing central tendency when it comes to sets of data. One of the simplest ways to do this is a method called finding the mode, or most common number, of a given data set.

This free step-by-step guide on How to Find Mode will teach you everything you need to know about the mode of a set of numbers, how to find it, and what it represents.

You can use the links below to jump to any section of this guide, or you can work through each section in order.

Ready to get started?

 

Lesson Preview: How to find mode of a given data set.

 

What is the Mode of a Data Set?

In math, the term data set refers to a collection of values that represent some type of data.

For example, the amount of money in savings for five different students is $18, $21, $9, $13, and $21.

These dollar amounts can be expressed as a data set simply by writing the collection of numbers inside of curly brackets (separated by commas) as follows:

  • {18, 21, 9, 13, 21}

This scenario where we are using a data set to represent the savings of five students in illustrated in Figure 01 below.

 

Figure 01: How to Find Mode of a Data Set.

 

In math, the mode of a data set is the value that has the highest frequency. In other words, the mode is the value in the data set that is present more than any other number (i.e. the mode is the most common number).

Note that: a data set can have one mode, no mode, or multiple modes.

While finding the mode of a data set is a simple process, it is an important skill because it will help you to identify trends within a set of data and how it is distributed.

How to Find Mode

To find the mode of a data set, simply look at all of the values in the set and determine which value appears the most. Remember that a data set can have one mode, multiple modes, or no mode (we will see instances of all three scenarios in the examples below).

For example, how can we find the mode of the data set that represented student’s savings?

Remember that the data set included the following values:

  • {18, 21, 9, 13, 21}

In this case, notice that one value is present twice (21)), while all of the other values only occur once.

Therefore, we know that 21 is the most common number in the data set and we can conclude that:

Final Answer: The mode is 21.

The step-by-step process for how to find the mode in math are shown in Figure 02 below.

 

Figure 02: How to Find Mode: Carefully look for the most common value.

 

Now that you know what the mode of a data set represents and how to identify the mode, you are ready to work through several problems where you will gain more experience with identifying the mode of a given set of data.

How to Find the Mode Examples

How to Find the Mode Example #1

Example: What is the mode of the data set?

Data Set: 5, 2, 8, 2, 7, 3, 2

Remember that the mode of a data set is the value with the highest frequency (i.e. the value that occurs the most times).

Given the data set above, only one value occurs more than once, and that value is 2 (which occurs 3 times). So, we can conclude that the data set has a mode of 2.

Final Answer: The mode is 2.

Notice that the data set in this example has one, and only one, mode. Figure 03 below illustrates how to find the mode for this first example.

 

Figure 03: How to Find Mode of a Data Set.

 

Now that we have solved this first example, we can see that determining the mode of a given data set is relatively easy to do.

However, there are a few nuances to finding the mode as you will see in the next few examples.

How to Find Mode Example #2

Example: What is the mode of the data set?

Data Set: 14, 14, 17, 9, 11, 17, 24, 22, 17, 12

For this next example, it may be tempting to say that the mode of the data set is 14 since 14 occurs more than once. However, this is actually not the case. While it is true that 14 occurs twice in this data set, there is a number that occurs more than twice: 17 occurs 3 times.

Since 17 occurs more than any other number (it occurs 3 times), we can conclude that:

Final Answer: The mode is 17.

Just like the previous example, this data set has only one mode.

 

Figure 04: How to Find Mode in Math

 

How to Find Mode Example #3

Example: What is the mode of the data set?

Data Set: 5, 11, 9, 8, 9, 9, 11, 11, 7, 5, 11, 10, 9, 3, 11, 12

Notice that the given data set in this example is much larger than the data sets from the first two examples. In this case, we have several numbers that occur multiple times, so it would be helpful to make a quick tally chart for each value so we can accurately determine which value in this data set occurs the most:

  • 3: once

  • 5: 2 times

  • 7: once

  • 8: once

  • 9: 4 times

  • 10: once

  • 11: 5 times

  • 12: once

Now that we have logged the frequency of each value in the data set, we can see that the number with the highest frequency is 11 (it occurs 5 times).

Final Answer: The mode is 11.

Again, the data set in this example has only one mode.

 

Figure 05: How to Find Mode Explained.

 

How to Find Mode Example #4

Example: What is the mode of the data set?

Data Set: 5, 5, 5, 7, 7, 9, 9, 9, 7, 7, 9

Just like the last example, it is difficult to determine the mode of this data set simply by looking at it, so we will again create a tally chart to keep track of the frequency of each value:

  • 5: 3 times

  • 7: 4 times

  • 9: 4 times

Notice that this data set does not have one single value that occurs more than any other value. In fact, there is a “tie” in this case, where both 7 and 9 occur 4 times each in the data set.

In cases such as this, the data set can be considered multimodal, meaning that it has more than one mode (in this case, the data set has 2 modes).

Final Answer: The mode is 7 and 9.

 

Figure 06: Example of a multimodal data set (i.e. a data set that has more than one mode).

 

How to Find Mode Example #5

Example: What is the mode of the data set?

Data Set: 13, 0, 24, 83, 81, 55, 16, 42, 22

For this final example, we have a data set with nine values. We can track the frequency of each value as follows:

  • 0: once

  • 13: once

  • 16: once

  • 22: once

  • 24: once

  • 42: once

  • 55: once

  • 81: once

  • 83: once

Notice that each value in the data set only occurs once. In cases like this, we have a data set with no mode.

Final Answer: There is no mode.

 

Figure 07: Example of a data set that has no mode.

 

Conclusion: How to Find Mode

Understanding how to find the mode in math is a simple, yet important skill that will help you analyze and draw conclusions about sets of data.

In math, the mode of a data set is the value with the highest frequency (i.e. the value that occurs the most in the data set).

There are three ways to classify the mode of a data set:

  • One Mode: There is one value that occurs more than any other value in the data set.

  • Multimodal: There is more than one value that occurs the most (i.e. there is a “tie” for the highest frequency).

  • No Mode: Each value in the data set occurs only once.

With these nuances in mind, determining the mode of a data set is a pretty simple skill that all students can master with some study and practice.

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How to Find Mean in 3 Easy Steps

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How to Find Mean in 3 Easy Steps

How to Find the Mean in Math Step-by-Step

Math Skills: How to find the mean of a data set in 3 easy steps

 
 

Understanding how to find mean is an important and fundamental math skill that every student must learn at one point or another.

In math, the term mean is synonymous with average, meaning that the mean of a data set represents the average value of that data set.

This free guide will teach you the mathematical definition of mean and how to find the mean of a data set using a simple 3-step process. The following topics related to how to find mean will be covered:

While you can use the hyperlinks above to jump to any section of this guide, we highly recommend that you follow each section in order, starting with the first section below, where we will review the definition of mean and what it means in relation to a given data set.

Let’s get started!

 

Lesson Preview: How to find the mean in math in 3 easy steps.

 

What is the Mean of a Data Set?

In math, a data set is a collection or group of data in the form of numbers.

For example, consider the following data set which represents the saving amounts, in dollars, of five high school students:

  • $20, $25, $5, $25, and $15

These savings amounts can be expressed as a data set by removing the “$” signs and writing the numbers inside of curly brackets as follows:

  • {20, 25, 5, 25, 15}

This example of expressing the savings amounts of five high school students is illustrated in Figure 01 below.

 

Figure 01: What is a data set in math?

 

In math, the mean of a data set is one single value that represents the average of the entire data set.

In other words, the mean is a measure of central tendency (i.e. the average of a given data set). The mean can be found by taking the sum of all of the numbers in a data set and dividing that value by the total amount of data points.

So, you can find the mean of any data set by adding up all of the numbers and dividing that sum by the total amount of numbers in the data set.

How to Find Mean in 3 Easy Steps:

  • Step One: Add together all the numbers in the data set to find the sum.

  • Step Two: Determine how many numbers are in the data set.

  • Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

To find the mean of the data set that represented student savings (as shown in Figure 01 above), we can simply follow these three steps to find the mean as follows:

Find the Mean of: {20, 25, 5, 25, 15}

Step One: Add together all the numbers in the data set to find the sum.

We can start by finding the sum of all of the numbers in the data set as follows:

  • 20 + 25 + 5 + 25 + 15 = 90

Step Two: Determine how many numbers are in the data set.

  • This particular data set has 5 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Finally, we just have to take our result from step one (the sum of all of the numbers in the data set) and divide it by the result of step two (the total number of numbers in the data set) as follows:

  • 90 ÷ 5 = 18

So, we can conclude that the average savings amount per student is $18.

Final Answer: The mean is 18.

The step-by-step process for how to find the mean in math are shown in Figure 02 below.

 

Figure 02: You can find the mean of any data set by adding up all of the numbers and dividing the sum by the total amount of numbers in the set.

 

Now that you understand what the mean of a data set represents and how to find it, let’s use our three-step process to solve three examples of how to find the mean of a data set.

How to Find the Mean of a Data Set

How to Find Mean Example #1

Example: Find the Mean of a Group of Siblings’ Ages

In this first example, we have a data set that represents the ages of six siblings with the following ages: 20, 22, 24, 26, 28, and 30.

So, our data set can be expressed as: {20, 22, 24, 26, 28, 30)

Step One: Add together all the numbers in the data set to find the sum.

Start by adding all of the numbers in the data set together as follows:

  • 20 + 22 + 24 + 26 + 28 + 30 = 150

Step Two: Determine how many numbers are in the data set.

  • The data set in this example has 6 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Now we are ready to find the mean by dividing the sum of all of the numbers by the total amount of numbers as follows:

  • 150 ÷ 6 = 25

Now we can conclude that the mean age of the siblings is 25 years old.

Final Answer: The mean is 25.

The entire step-by-step process for how to find the mean for this first example is illustrated in Figure 03 below.

 

Figure 03: How to Find Mean Using 3 Easy Steps.

 

As you can see from this first example, the process of finding the mean of a data set is an easy process that can be completed using just three steps.

The more that you practice using these steps on how to find mean, the easier these types of problems will become. Now, let’s move onto another example of how to find the mean of a data set.

How to Find the Mean in Math

How to Find Mean Example #2

Example: Find the Mean of a Data Set Representing Monthly Income

In this example, John has been tracking his monthly income and recorded the data in the following data set:

  • $1,200, $1,350, $1,250, $1,400, $1,300, $1,000, $900

For this example, we can make the values easier to work with by removing the dollar signs and expressing the values in a data set as follows:

  • {1200, 1350, 1250, 1400, 1300, 1000, 900}

Now we can use our 3-step process for how to find mean as follows:

Step One: Add together all the numbers in the data set to find the sum.

Just like the previous example, we can start by finding the sum of all of the numbers in the data set:

  • 1200 + 1350 + 1250 + 1400 + 1300 + 1000 + 900 = 8400

Step Two: Determine how many numbers are in the data set.

  • This data set has 7 numbers.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

The last step is to divide the sum of all of the values in the data set by the total number of values in the data set:

  • 8400 ÷ 7 = 1,200

Finally, we can conclude that the mean monthly income is $1,200.

Final Answer: The mean is 1,200.

The process for finding the solution to this second example is shown in Figure 04 below.

 

Figure 04: How to Find the Mean of a Data Set.

 

Notice how our final answers in our first two examples ended up being whole numbers. This, however, will not always be the case. Now, let’s work through one more example where our final result will be a decimal.

How to Find the Mean of a Data Set

How to Find Mean Example #3

Example: Find the Mean of a Data Set Representing Student Test Scores

In this third example, we are looking to find the mean (or average) test score of a student with the following test scores: 78, 85, 92, 88, 80, 63, 90, 89, 71, and 95

Let’s go ahead and use our 3-step method for how to find mean of a data set to solve this problem and determine the mean test score.

Step One: Add together all the numbers in the data set to find the sum.

Our first step is to find the sum of all of the numbers in the data set:

  • 78 + 85 + 92 + 88 + 80 + 63 + 90 + 89 + 71 + 95 = 831

Step Two: Determine how many numbers are in the data set.

  • There are 10 values (or test scores) in this data set.

Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Finally, we just have to take our result from step one (831) and divide it by the total number of values in the data set (10) as follows:

  • 831 ÷ 10 = 83.1

Now we can make the conclusion that the student had an average test score of 83.1

Final Answer: The mean is 83.1

Notice how our result for this example was a decimal, which is totally fine!

Figure 05 below illustrates how we solved this third example.

 

Figure 05: How to Find Mean with Decimal Answers.

 

How to Find Mean Challenge

Challenge Problem

Scenario: On their last five science exams, students had the following scores:

  • Bonnie: 83, 100, 95, 61, 68

  • Celeste: 88, 83, 79, 86, 90

  • Michelle: 97, 64, 60, 89, 73

Which student had the highest mean test score?

For this challenge problem, you are tasked with determining which of the three students had the highest mean (or average) test score.

To solve this problem, you will have to calculate the mean test score of each student and then compare the means to see which on was the highest value.

 

Figure 06: Which student had the highest mean test score?

 

We can find the mean test score for each student by adding all five of their scores together and then dividing that sum by 5 as follows:

  • Bonnie: 83 + 100 + 95 + 61 + 68 = 407 → 407 ÷ 5 = 81.4

  • Celeste: 88 + 83 + 79 + 86 + 90 = 426 → 426 ÷ 5 = 85.2

  • Michelle: 97 + 64 + 60 + 89 + 73 = 393 → 393 ÷ 5 = 78.6

Now we can conclude that:

  • Bonnie’s mean test score was 81.4

  • Celeste’s mean test score was 85.2

  • Michelle’s mean test score was 78.6

And, we can conclude that Celeste’s 85.2 mean test score was the highest of the three students.

Final Answer: Celeste had the highest mean test score, which was 85.2.

 

Figure 07: How to Find Mean in Math: Test Scores

 

Conclusion: How to Find Mean

Knowing how to find the mean of a data set is an important math skill that every student should be familiar with.

In math, a data set is a collection of data values and the mean of a data set represents the average value of that set. You can find the mean of any data set by following the following three steps:

  • Step One: Add together all the numbers in the data set to find the sum.

  • Step Two: Determine how many numbers are in the data set.

  • Step Three: Divide the sum by the total amount of numbers in the data set to determine the mean.

Once you gain enough practice with finding the mean, you can simplify this process for how to find the mean by simply adding together all of the numbers in the data set and then dividing that sum by the total number of values.

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How to Complete the Square in 3 Easy Steps

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How to Complete the Square in 3 Easy Steps

How to Complete the Square in 3 Easy Steps

Step-by-Step Guide: How to do Completing the Square

 

Step-by-Step Guide: How to Complete the Square Explained.

 

As you continue onto more advanced problems where you have to factor quadratics, you will have to learn how to complete the square in order to find correct solutions. Completing the square is a special technique that you can use to factor quadratic functions.

This free step-by-step guide on How to Complete the Square will teach you an easy 3-step method for factoring any quadratic using a technique called “completing the square.”

This guide will focus on the following topics and sections. You can click on any of the text links below to jump to one particular section, or you can follow each section in sequential order.

Let’s begin by exploring the meaning of completing the square and when you can use it to help you to factor a quadratic function.

 

Figure 01: Are you ready to learn how to find solutions to quadratic equations by completing the square?

 

What is Completing the Square?

Completing the square is a method that you can use to solve quadratic equations of the form ax² + bx + c = 0 (where a, b, and c are all not equal to zero).

❗Note that the equations of the form ax² + bx + c = 0 are called quadratic equations and they can be rewritten as follows:

  • ax² + bx + c = 0

  • ax² + bx = -c

Both of these equations are equivalent to each other, and understanding the relationship between these two equations will help you to understand how to complete the square later on in this guide.

In the next section, we will work through three examples of how to complete the square using the following 3-step method:

  • Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

  • Step #2: Add (b/2)² to both sides of the equal sign.

  • Step #3: Factor and solve.

By solving a quadratic equation by completing the square, you are identifying values where the parabola that represents the equation crosses the x-axis.

As long as you understand how to follow and apply these three steps, you will be able to solve quadratics by completing the square (provided that they are solvable). Now, let’s gain some experience with using the three step method on how to complete the square by working through some step-by-step practice problems.

 

Figure 02: The solutions to a quadratic equation are the values where the graph crosses the x-axis.

 

How to Complete the Square: Example #1

Solve: x² - 6x -16 = 0

For this first example (and all of the practice problems in this guide), we can solve the problem by completing the square using our three step method as follows:

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

Let’s start off by noticing that our given quadratic function is indeed in ax² +bx + c = 0 form, where a=1, b=-6, and c=-16.

To complete the first step, we have to move all of the constants (all of the values not attached to variables, to the right side of the equals sign as follows:

  • x² - 6x -16 = 0

  • x² - 6x -16 (+16) = 0 (+16)

  • x² - 6x = 16

Now we have completed the first step and we are left with a new equivalent equation:

  • x² - 6x = 16

Step #2: Add (b/2)² to both sides of the equal sign.

For the next step, we have to find the value of (b/2)² and add it to both sides of the equals sign.

Since we know that b=-6, we can find the value of (b/2)² by substituting -6 for b as follows:

  • (b/2)²

  • (-6/2)²

  • (-3)²

  • =9

In this case, (b/2)² = 9, so, to complete Step #2, we simply have to add 9 to both sides of the equal sign as follows:

  • x² - 6x = 16

  • x² - 6x + 9 = 16 + 9

  • x² - 6x + 9 = 25

 

Figure 03: How to Complete the Square in 3 Easy Steps.

 

Step #3: Factor and solve.

Finally, we are ready for the third and final step where we just need to factor and solve.

Notice that the left side of the equation of x² - 6x + 9 = 25 is a trinomial that is factorable as follows:

  • x² - 6x + 9 = (x-3)(x-3)

  • x² - 6x + 9 = (x-3)²

In this example, the factors of x² - 6x + 9 are (x-3)(x-3), which we will express as (x-3)² since it will allow us to solve the problem as follows:

  • x² - 6x + 9 = 25

  • (x-3)² = 25

  • √[(x-3)²] = √[25]

  • x -3 = ± 5

In the third step above, we took the square root of both sides of the equation to remove the exponent and we are left with x -3 = ± 5, which means that

  • x - 3 = 5

  • x - 3 = -5

We can now find the solutions to this first example by solving both equations as follows:

  • x - 3 = 5 → x = 8

  • x - 3 = -5 → x = -2

All three steps for how to do completing the square are shown in Figure 03 above.

Now, we can conclude that the original quadratic equation x² - 6x -16 = 0 has two solutions:

Final Answer: x = 8 and x = -2

This means that the graph of the equation x² - 6x -16 = 0 will be a parabola that crosses the x-axis at both (-2,0) and (8,0) as shown in Figure 04 below.

 

Figure 04: How to solve by completing the square: graph explanation

 

How to Complete the Square: Example #2

Solve: x² +12x +32 = 0

We can solve this next example using our 3-step method, just as we did in the previous example, as follows:

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

Notice for our given quadratic equation, x² +12x +32 = 0, that a=1, b=12, and c=32.

Since our constant c is on the left side of the equation, we simply have to move it to the right side using inverse operations to complete Step #1.

  • x² +12x + 32 = 0

  • x² +12x + 32 (-32) = 0 (-32)

  • x² + 12x = -32

After completing the first step, we now have:

  • x² + 12x = -32

Step #2: Add (b/2)² to both sides of the equal sign.

Next, we have to add (b/2)² to both sides of our new equation.

In this example, b=12, so we can find the value of (b/2)² as follows:

  • (b/2)²

  • (12/2)²

  • (6)²

  • =36

Since (b/2)² = 36, we can complete Step #2 by adding 36 to both sides of the equation as follows:

  • x² + 12x = -32

  • x² + 12x +36 = -32 +36

  • x² + 12x +36 = 4

 

Figure 05: How to complete the square to find the solutions to a quadratic equation.

 

Step #3: Factor and solve.

For the final step, we just have to factor and solve for any potential values of x.

Just like example #1, we can finish completing the square by factoring the trinomial on the left side of the equation and then solving.

In this case, the trinomial on the left side of the equation can be factored as follows:

  • x² + 12x +36 = (x+6)(x+6)

  • x² + 12x +36 = (x+6)²

❗Note that whenever you solve a problem using the complete the square method, you will always end up with two identical factors when you complete Step #3.

Now that we know that the factors of x² + 12x +36 are equal to (x+6)², we can solve for x as follows:

  • x² + 12x +36 = 4

  • (x+6)² = 4

  • √[(x+6)²] = √[4]

  • x + 6 = ± 2

In this case, the original quadratic function x² +12x +32 = 0 will have two solutions:

  • x + 6 = 2

  • x + 6 = -2

We can determine these two solutions by solving each equation as follows:

  • x + 6 = 2 → x = -4

  • x + 6 = -2 → x = -8

The entire 3-step method for completing the square for Example #2 is shown in Figure 05 above.

Final Answer: x = -4 and x = -8

Figure 06 below shows the graph of the parabola represented by x² +12x +32, with x-intercepts at -4 and -8.

 

Figure 06: Solving a quadratic by completing the square helps you to find the x-intercepts of the parabola that represents the equation.

 

How to Complete the Square: Example #3

Solve: x² +2x -7 = 0

Are you starting to get the hang of how to complete the square? Let’s gain some more experience with this next example.

Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

For this problem, we know that a=1, b=2, and c=-7.

For our first step, let’s rearrange the equation so that all of the constants are on the right side:

  • x² +2x -7 = 0

  • x² +2x -7 (+7) = 0 (+7)

  • x² + 2x = 7

Now we have a new equivalent function:

  • x² + 2x = 7

Step #2: Add (b/2)² to both sides of the equal sign.

Now we need to add (b/2)² to both sides of the equation. Since b=2 in this example, (b/2)² is equal to:

  • (b/2)²

  • (2/2)²

  • (1)²

  • =1

Since (b/2)² = 1, we can complete the second step by adding 1 to each side of the equation as follows:

  • x² + 2x = 7

  • x² + 2x +1 = 7 +1

  • x² + 2x +1 = 8

 

Figure 07: How to Complete the Square: Solutions will not always be rational numbers.

 

Step #3: Factor and solve.

Now we are ready to factor and solve the equation.

We have a trinomial on the left side of the equation that can be factored as follows:

  • x² + 2x +1 = (x+1)(x+1)

  • x² + 2x +1 = (x+1)²

With these factors in mind, we can solve for x as follows:

  • x² + 2x +1 = 8

  • (x+1)² = 8

  • √[(x+1)²] = √[8]

  • x +1 = ± √[8]

For this third example, the quadratic function x² +2x -7 = 0 will have two solutions:

  • x + 1 = √[8]

  • x + 1 = - √[8]

If we continue onto solving these two equations, we will see that, unlike Examples #1 and #2, we do not end up with a rational answer:

  • x + 1 = √[8] → x = -1 + √8

  • x + 1 = - √[8] → x = -1 - √8

In cases like this, you can often conclude that:

Final Answer: x = -1 + √8 and x = -1 - √8

The solutions above are considered exact answers. However, if you are trying to estimate where the parabola will cross the x-axis on the coordinate plane, you could take the problem a step further by approximating for √8 as follows:

  • √8 ≈ 2.83

  • x = -1 + √8 → x = -1 + 2.83 → x =1.83

  • x = -1 + √8 → x = -1 - 2.83 → x =-3.83

Now we have two approximate solutions for x:

  • x =1.83 and x=-3.83

What does this mean? Just like we saw in Examples #1 and #2, the solutions tell you where the graph of the parabola crosses the x-axis. In this example, the graph crosses the x-axis at approximately 1.83 and -3.83, as shown in Figure 08 below.

 

Figure 08:  How to complete the square to determine the solutions to a quadratic equation.

 

Conclusion: How to Complete the Square

When learning how to solve quadratic equations of the form ax² + bx +c=0, understanding how to complete the square to find the values where x=0 is an important and useful algebra skill that you can use to solve a variety of problems.

Whenever you have an equation in ax² + bx +c=0 form, you can solve it by following these 3-simple steps to completing the square:

  • Step #1: Rearrange the quadratic equation so that all of the constants are on one side of the equals sign.

  • Step #2: Add (b/2)² to both sides of the equal sign.

  • Step #3: Factor and solve.

Note that the above 3-step method for completing the square can be used to find the solutions of any quadratic equation of the form ax² + bx +c=0. These solutions represent the x-values where the parabola that represents the equation crosses the x-axis.

Keep Learning:

How to Find the Vertex of a Parabola in 3 Easy Steps

Learn how to find the coordinates of the vertex point of any parabola with this free step-by-step guide.

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How to Find Horizontal Asymptotes in 3 Easy Steps

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How to Find Horizontal Asymptotes in 3 Easy Steps

How to Find Horizontal Asymptotes in 3 Easy Steps

Step-by-Step Guide: How to Find the Horizontal Asymptote of a Function

 

Step-by-Step Guide: How to Find the Horizontal Asymptotes of Functions.

 

During your study of algebra, you will eventually learn how to analyze and understand the behavior functions when they are represented graphically on the coordinate plane. The graph of a function gives you a visual portrayal of how the function behaves—including the location of any potential horizontal asymptotes.

This free step-by-step guide on How to Find Horizontal Asymptotes of a Function will demonstrate and explain everything you need to know about horizontal asymptotes and how to find the horizontal asymptote of any function (assuming that the function does indeed have a horizontal asymptote).

The following sections will be covered in this guide. While we recommend that you work through each section in order, you can use the quick-links below to jump to a particular section":

Let’s start off by learning more about horizontal asymptotes and what they look like.

 

Figure 01: The horizontal asymptote of a function tells you the behavior of a function as it approaches the edges of a graph.

 

What is a Horizontal Asymptote?

Before you learn how to find horizontal asymptotes, it is important for you to understand some key foundational concepts.

In algebra, a function is an equation of the form y=f(x) that represents the relationship between two sets. A relation can be defined as a function when there is only one output value for each input value (i.e. the x-values do not repeat). Functions can be visually represented on a graph (i.e. a coordinate plane).

A horizontal asymptote of a function is a horizontal line (↔) that tells you the behavior of a function as it approaches the edges of a graph. We can also say that horizontal asymptotes allow us to identify the “end behavior” of a function.

It is also important to note that horizontal asymptotes occur when functions are rational expressions, meaning that the function is a quotient of two polynomials (i.e. the function is a fraction where both the numerator and the denominator are polynomials.

For example, consider the following function:

  • y = (3x²)/x³

 

Figure 02: The graph of the function y = (3x²)/x³

 

Our function, y = (3x²)/x³, is the quotient of two polynomials (i.e. it is a rational expression) and its corresponding graph is shown in Figure 02 above.

Since our function is a fraction, we know that we can not have zero in the denominator (otherwise the function would be undefined at this point). So, the graph of the function, as you can see in Figure 02, does something interesting at zero.

In this example, we have a horizontal asymptote at y=0.

Notice how the graph of the function y = (3x²)/x³ gets closer and closer to the line y=0, without ever touching it, as it approaches the ends of the graph (horizontally), as shown in Figure 03 below.

 

Figure 03: The function has a horizontal asymptote at y=0.

 

If we take a look at a few points on the graph of y = (3x²)/x³ and the corresponding table, as shown in Figure 04 below, we can see that as the graph moves to the left of the coordinate plane, the y-values get closer and closer to zero. And, we can also see that as the graph moves to the right of the coordinate plane, the y-values get closer and closer to zero.

This type of observation is what we call determining the “end behavior” and it helps us to understand why the graph has a horizontal asymptote at y=0.

 

Figure 04: How to Find a Horizontal Asymptote of a Function

 

Now that you understand what a horizontal asymptote is and what it looks like, you are ready to work through a few step-by-step examples of how to find a horizontal asymptote.

Before moving forward, here are a few quick key points you should be familiar with:

  • Not all functions that are rational expressions have a horizontal asymptote.

  • It is possible for a function to have 0, 1, or 2 asymptotes (i.e. a function can have a maximum of 2 asymptotes).

  • When horizontal asymptotes are shown on a graph, they are typically drawn using a dashed line, which is what you will see in this guide.

  • The horizontal asymptote of a function is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

  • A horizontal asymptote can be thought of as an imaginary dashed line on the coordinate plane that helps you to visual a “gap” in a graph.

 

Figure 05: A horizontal asymptote is an imaginary line that is not a part of the graph of a function.

 

How to Find Horizontal Asymptotes Example #1

Now you are ready to learn how to find a horizontal asymptote using the following three steps:

  • Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Now, let’s go ahead and apply these three steps to our first practice problem.

Example #1: Find the horizontal asymptote of the function f(x)=4x/(x+4)

For this first example, we can start by taking a look at the graph in Figure 05 above. By looking at the graph, we can see that the function f(x)=4x/(x+4) has one horizontal asymptote at y=4.

Let’s now apply our three steps to see if confirm the at the function has a horizontal asymptote at y=4.

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ 4x/(x+4)

  • = 4 lim x→∞ [1/(1+(4/x))]

  • = 4 (lim x→∞ (1)) / (lim x→∞ (1+(4/x))

  • = 4 ( (1)/(1))

  • =4

After completing the first step, we can see that, as the function has a horizontal asymptote at y=4 as the function approaches positive ∞ on the graph.

Next, let’s confirm that the function has the same behavior as it approaches negative ∞.

Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→∞ 4x/(x+4)

  • = 4 lim x→-∞ [1/(1+(4/x))]

  • = 4 (lim x→-∞ (1)) / (lim x→∞ (1+(4/x))

  • = 4 ( (1)/(1))

  • =4

Completing the second step gives us the same result as the first step (i.e. there is a horizontal asymptote at y=4).

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Finally, based on our results from steps one and two, we can conclude that the function has one horizontal asymptote at y=4, as shown on the graph in Figure 06 below.

Final Answer: The function has one horizontal asymptote at y=4.

 

Figure 06: How to find a horizontal asymptote using limits.

 

How to Find Horizontal Asymptotes Example #2

Find the horizontal asymptote of the function f(x)=3ˣ+5

For this next example, we want to see if the exponential function f(x)=3ˣ+5 has any horizontal asymptotes.

We can solve this problem the same as we did the first example by using our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ 3ˣ+5

  • =lim x→∞ (3ˣ) + lim x→∞ (5)

  • lim x→∞ (3ˣ) = ∞

  • lim x→∞ (5) = 5

  • = ∞ + 5 = ∞

Notice that our result for the first step is that, as the function approaches positive ∞, the limit is ∞. And, since ∞ is not a real number, we can not yet determine whether or not this function will have any horizontal asymptotes. But we can not be completely sure until we complete the second step.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→-∞ 3ˣ+5

  • =lim x→-∞ (3ˣ) + lim x→-∞ (5)

  • lim x→-∞ (3ˣ) = 0

  • lim x→-∞ (5) = 5

  • = 0 + 5 = 5

Our result from step number two results in a real number, so we can conclude that the function does have a horizontal asymptote at y=5.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Again, even though Step One did not result in a real number limit, Step Two did, so we can conclude that:

Final Answer: The function has one horizontal asymptote at y=5.

We can confirm the location of a horizontal asymptote at y=5 for the exponential function f(x)=3ˣ+5 by looking at the completed graph in Figure 07 below.

 

Figure 07: How to find horizontal asymptotes given a function explained.

 

How to Find Horizontal Asymptotes Example #3

Find the horizontal asymptote of the function f(x)=(3x^2+x)/(x+2)

For this third and final example, we have to see if the rational function f(x)=(3x^2+x)/(x+2) has one, two, or zero horizontal asymptotes.

We can determine whether or not the function has any horizontal asymptotes by following our three steps as follows:

Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • lim x→∞ f(x) = lim x→∞ (3x^2+x)/(x+2)

  • =lim x→∞ (3x+1)/(1+(2/x))

  • = (lim x→∞ (3x+1)) / (lim x→∞(1+(2/x)))

  • lim x→∞ (3x+1) = ∞

  • lim x→∞(1+(2/x)) = 1

  • = ∞ / 1 = ∞

Since our result is not a real number, Step One does not help us to determine the possible existence of a horizontal asymptote.

Step Two: Determine lim x→-∞ f(x. In other words, find the limit for the function as x approaches negative ∞.

  • lim x→-∞ f(x) = lim x→-∞ (3x^2+x)/(x+2)

  • =lim x→-∞ (3x+1)/(1+(2/x))

  • = (lim x→-∞ (3x+1)) / (lim x→-∞(1+(2/x)))

  • lim x→-∞ (3x+1) = -∞

  • lim x→-∞(1+(2/x)) = 1

  • = -∞ / 1 = -∞

Again, our result is not a real number so we can not determine the location of a horizontal asymptote.

Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Since neither Step One nor Step Two resulted in a real number value for k, we can conclude that the function does not have any horizontal asymptotes.

Final Answer: The function does not have any horizontal asymptotes.

The graph of the function f(x)=(3x^2+x)/(x+2) is shown in Figure 08 below. Do you notice how there is no horizontal asymptote?

 

Figure 08: How to Find Horizontal Asymptotes: Not all functions will have a horizontal asymptote.

 

Conclusion: How to Find Horizontal Asymptotes

A horizontal asymptote of a function is an imaginary horizontal line (↔) that helps you to identify the “end behavior” of the function as it approaches the edges of a graph.

Not every function has a horizontal asymptote. Functions can have 0, 1, or 2 horizontal asymptotes.

If a function does have any horizontal asymptotes, they will be displayed as a dashed line. A horizontal asymptote is an imaginary line that is not a part of the function, and it is not a requirement to include the horizontal asymptote of a function when you graph it on the coordinate plane.

You can determine whether or not any function has horizontal asymptotes by following these three simple steps:

  • Step One: Determine lim x→∞ f(x). In other words, find the limit for the function as x approaches positive ∞.

  • Step Two: Determine lim x→-∞ f(x). In other words, find the limit for the function as x approaches negative ∞.

  • Step Three: If one of or both of the limits determined above is a real number, k, then the equation of the horizontal asymptote is y=k for each unique value of k (a function can have 0, 1, or 2 horizontal asymptotes).

Keep Learning:

How to Find the Vertex of a Parabola in 3 Easy Steps

Learn how to find the coordinates of the vertex point of any parabola with this free step-by-step guide.

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Exponent Rules Explained! (Free Chart)

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Exponent Rules Explained! (Free Chart)

Everything you need to know about the exponent rules!

Exponent Rules Explained

In algebra, exponent rules (also known as the laws of exponents), are a set of laws that you can use to simplify mathematical expressions that have exponents. In other words, exponent rules can be used to make complicated math expressions containing exponents much easier to work with.

Once you learn the rules of exponents, you will be able to use them to quickly and easily perform operations (adding, subtracting, multiplying, dividing, etc.) on expressions that have exponents.

The following free guide to Exponent Rules will describe and explain the rules of exponents are, what they look like, and examples of how you can use them to simplify expressions and solve math problems involving exponents.

You can use the quick links below to jump to any section of this guide, or you can follow along step-by-step.

By the time that you reach the conclusion of this free guide on the rules of exponents, you will have a deep understanding of how to apply the exponent rules to simplify and solve math problems.

Are you ready to get started?

Before we start learning about the exponent rules, let’s do a quick review of exponents and cover some key vocabulary terms that you will need to be familiar with in order to make the most out of this guide.

 

Figure 01: The number 5 raised to a power of 3.

 

Quick Review: Exponents

In math, an exponent is a number or a variable at the top right of a base number or expression. The value of the exponent tells you what power the base number will be raised to.

For example, the expression 5³ has a base number of 5 and an exponent of 3. This expression means that the base number, 5, is being raised to a power of 3, meaning:

  • 5³ = 5 x 5 x 5 = 125

In short, an exponent tells you how many times to multiply the base number by itself.

Now that you understand the basic definition of an exponent, it’s time to move onto learning about the Exponent Rules.

What are the Exponent Rules?

The rules of exponents are a collection of specific ways to simplify math expressions involving exponents.

The chart in Figure 02 below shows all of the exponent rules. You can click the link below the chart to download a PDF copy which you can print out and use as a reference whenever you need to.

Below the chart, you will find a section that explains each individual exponent rule along with examples.

 

Figure 02: The Exponent Rules

 

Exponent Rules: Zero Exponent Rule

The first exponent rule described in this guide will be the zero exponent rule:

  • a⁰=1

 

Figure 03: The zero exponent rule.

 

The zero exponent rule states that any number, variable, or expression raised to the zero power is equal to 1.

For example:

  • 8⁰=1

  • x⁰=1

  • (a+b)⁰=1

Pretty simple, right?

To understand the zero exponent rule, you only have to remember that anything raised to the zero power is equal to one.

Exponent Rules: Product Rule

The next exponent rule that we will cover is called the product rule:

  • a^b x a^c = a^(b+c)

 

Figure 04: The product exponent rule.

 

The product rule of exponents applies whenever you have to multiply two or more expressions that have the same base.

The rule goes as follows: whenever you have to multiply two expressions with the same base value, you can simplify the expression by adding the exponents together and keeping the base value the same.

For example:

  • 4³ x 4² = 4⁵

  • m² x m² = m⁴

  • (a+b)⁵ x (a+b)³ = (a+b)⁸

Are you wondering where the product exponent rule comes from, let’s take a closer look at the example:

  • 4³ x 4² = 4⁵

We can rewrite each individual expression as follows:

  • 4³ = 4 x 4 x 4

  • 4² = 4 x 4

Therefore,

  • 4³ x 4² = 4 x 4 x 4 x 4 x 4 = 4⁵

The product rule allows you to simplify the expression 4³ x 4² as 4⁵. They both mean the same thing!

Before we move onto the next rule of exponents, note that the product rule only applies when the expressions have the same base value!

Exponent Rules: Quotient Rule

Our next stop on our chart of the rules of exponents is called the quotient rule.

The exponent rule comes into play when you have to divide two expressions with exponents that have the same base value.

The quotient rule of exponents goes as follows:

  • a^b ➗ a^c = a^(b-c); or

  • (a^b)/(a^c) = a^(b-c)

 

Figure 05(a): The quotient exponent rule with a division symbol.

Figure 05(b): The quotient exponent rule with a fraction.

 

Whether two expressions with the same base are being divided using a division sign (as shown in Figure 05(a)) or with a fraction (as shown in Figure 05(b)), you can simplify the expression by subtracting the second exponent from the first exponent and keeping the base value the same.

For example:

  • 8⁸ ➗ 8² = 8⁶

  • 8⁵ / 8² = 8³

  • n¹⁰ / n⁸ = n²

  • (a+b)⁹ ➗ (a+b)² = (a+b)⁷

Remember that the quotient exponent rule only applies if both expressions have the same base value.

Exponent Rules: Power of a Power Rule

What happens when you take an expression with an exponent and raise it to another power?

In case like this, you can use the power of a power exponent rule, which states that, whenever you have a base number, variable, or expression with an exponent raised to another exponent, the expression can be simplified by multiplying the two exponents together and keeping the base value the same.

  • (a^b)^c = a^(bc)

 

Figure 06: The power of a power exponent rule.

 

Whenever you have a single base with two exponents in a row, you can simplify the expression by multiplying the two exponents together, for example:

  • (9³)⁴ = 9¹²

  • (y²)² = y⁴

  • (x⁷)³ = x²¹

The power of a power exponent rule is a useful law of exponents that you can use to simplify complicated expressions involving multiple exponents.

Exponent Rules: Power of a Product Rule

What happens when you take an entire product and raise the entire thing to a power?

In cases like this, whenever you are raising a product of two numbers inside of a set of parentheses by an exponent, you can distribute the exponent and apply it to each term of the product as follows:

  • (ab)^c = a^b x a^c

 

Figure 07: The power of a product exponent rule.

 

In cases when your base is a product raised to an exponent, you can distribute the exponent to each term, for example:

  • (ab)³ = a³b³

  • (xy²)³ = x³y⁶

  • (m²n³)⁴ = m⁸n¹²

The power of a product exponent rule is one of the most important and use exponent rules on our list! Make sure that you feel comfortable with how to use it before moving onto the next law of exponents.

Exponent Rules: Negative Exponent Rule

The final exponent rule that we will cover in this guide is the negative exponent rule, which states that:

  • a^-b = 1/(a^b)

 

Figure 08: The negative exponent rule.

 

The negative exponent rule states that any number, variable, or expression raised to a negative power can be rewritten as a fraction with one in the numerator and the same original expression in the denominator, but with the negative sign removed from the exponent.

For example:

  • 8^-2 = 1/(8^2)

  • x^-7 = 1/(x^7)

  • (a+b)^-4 = 1/((a+b)^4)

That’s all that there is to it! If you ever have to solve a problem that requires you to give an answer that includes only positive exponents, the negative exponent rule will come in handy.

Conclusion: Exponent Rules

In math, you will need to be able to work with expressions that have exponents, and being able to simplify them quickly and correctly is an important skill. By learning and understanding the rules of exponents, you can easily simplify exponents in a variety of situations. Whenever you need to review the exponent rules, we encourage you to revisit this guide for a quick refresher.

Need More Help?

Check out our animated video lessons on the power to a power rule and the product to a power rule.

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