Featured Image:
In the realm of mathematics, fractions hold a pivotal role, representing parts of a whole or ratios between quantities. Desmos, a powerful online graphing calculator, provides an intuitive platform to explore and manipulate fractions with ease. Delve into this guide to uncover the intricacies of creating and working with fractions on Desmos, a journey that will empower you to seamlessly tackle a wide range of mathematical equations and scenarios.
To craft a fraction on Desmos, commence by entering the numerator and denominator within a pair of curly brackets, separated by a forward slash. For instance, to represent the fraction three-fifths, simply type {3}/{5} into the calculator’s input field. Desmos will automatically display the fraction in a clear and visually pleasing manner, making it easy to read and manipulate throughout your calculations. Moreover, you can employ the fraction’s numerator and denominator independently by accessing them within the curly brackets. By clicking on the numerator, you can modify its value, while clicking on the denominator allows you to alter that component. This flexibility empowers you to swiftly adjust fractions to suit your needs.
Desmos goes beyond mere fraction representation, offering a comprehensive suite of tools to perform operations on fractions. Seamlessly add, subtract, multiply, and divide fractions using the calculator’s built-in functions. To add two fractions, simply type the fractions within parentheses and separate them with a plus sign. For example, to add one-half and three-quarters, enter ({1}/{2}) + ({3}/{4}) into the input field. Desmos will automatically calculate and display the result, providing you with the sum of the fractions. Similarly, you can perform subtraction, multiplication, and division operations by employing the minus sign (-), asterisk (*), and forward slash (/), respectively. Desmos will guide you through these operations with precision, ensuring accurate and efficient fraction manipulation.
Creating the Fraction in Desmos
1. Understand the Fraction Syntax
Desmos uses a specific syntax to represent fractions: numerator/denominator. For example, the fraction 1/2 would be written as 1/2. The numerator represents the top number of the fraction, while the denominator represents the bottom number.
Additional Details on Fraction Syntax:
- The numerator and denominator must be separated by a forward slash
/. - The numerator and denominator can be any valid mathematical expression.
- If the denominator is negative, the fraction will be flipped (i.e., thenumerator becomes the denominator, and vice versa).
- Fractions can be nested within other fractions or expressions.
2. Enter the Fraction
Once you understand the syntax, you can enter the fraction into Desmos.
- To create a simple fraction: Type the numerator and denominator separated by a
/. For example, to create the fraction 3/4, type3/4. - To create a complex fraction: Use parentheses to enclose the numerator and denominator as separate expressions. For example, to create the fraction (x+1)/(y-2), type
(x+1)/(y-2).
3. Display the Fraction
By default, Desmos will display fractions in the simplified form. To display a fraction in its original unsimplified form, use the fraction() function. For example, to display the fraction 3/4 in its unsimplified form, type fraction(3,4).
Expressing the Fraction in Simplified Form
To express a fraction in simplified form, the numerator and denominator must share no common factors other than 1. To simplify a fraction, follow these steps:
- Find the greatest common factor (GCF) of the numerator and denominator.
- Divide both the numerator and denominator by the GCF.
- The resulting fraction is in simplified form.
Example: Simplifying 12/18
The GCF of 12 and 18 is 6. Dividing both the numerator and denominator by 6 gives 12/18 = 12 ÷ 6 / 18 ÷ 6 = 2/3.
Therefore, the simplified form of 12/18 is 2/3.
Common Divisible Factors
The following table lists common divisible factors for numbers 1-12:
| Number | Common Divisible Factors |
|---|---|
| 1 | 1 |
| 2 | 1, 2 |
| 3 | 1, 3 |
| 4 | 1, 2, 4 |
| 5 | 1, 5 |
| 6 | 1, 2, 3, 6 |
| 7 | 1, 7 |
| 8 | 1, 2, 4, 8 |
| 9 | 1, 3, 9 |
| 10 | 1, 2, 5, 10 |
| 11 | 1, 11 |
| 12 | 1, 2, 3, 4, 6, 12 |
Using the Decimal Equivalents
To make a fraction on Desmos using decimal equivalents, follow these steps:
- Convert the decimal to a fraction.
- Open Desmos and click on the “Graphs” tab.
- Enter the fraction in the input field.
For example, to make the fraction 0.5 on Desmos, you would:
- Convert 0.5 to a fraction: 0.5 = 5/10 = 1/2
- Open Desmos and select the “Graphs” tab.
- Enter the fraction 1/2 in the input field.
### Converting Decimals to Fractions
Converting decimals to fractions can be tricky. There are several methods you can use to convert a decimal to a fraction. Here are three common methods:
1. Divide the decimal by 1
Place a 1 underneath the decimal and add a decimal point above the 1. Bring down all the numbers behind the decimal point. Divide the decimal by 1. The numbers in the quotient will be the numerator and the numbers in the divisor will be the denominator. For example, 0.5 can be converted by dividing 0.5 by 1:
| 0.5 |
|---|
| 1 |
Multiply the decimal by 10, so 0.5 becomes 5, and the 1 becomes 10.
| 5 |
|---|
| 10 |
Divide 5 by 10, you get 0.5. Therefore, the fraction of 0.5 is 5/10, which can be simplified to 1/2.
2. Multiply the decimal by a power of 10
Multiply the decimal by a power of 10 that will make the decimal a whole number. For example, to convert 0.5 to a fraction, multiply 0.5 by 10 to get 5.
| 5 |
|---|
| 10 |
Therefore, the fraction of 0.5 is 5/10, which can be simplified to 1/2.
3. Use a fraction calculator
If you’re not sure how to convert a decimal to a fraction, you can use a fraction calculator. There are many free online fraction calculators available.
Converting to a Percentage
To convert a fraction to a percentage, we multiply the numerator by 100 and then divide by the denominator.
For example, to convert the fraction 1/2 to a percentage, we would do the following:
1. Multiply the numerator (1) by 100: 1 x 100 = 100
2. Divide the result by the denominator (2): 100 / 2 = 50
Therefore, 1/2 as a percentage is 50%.
Decimal Fraction to Percentage
If the fraction has a decimal point in the numerator, we can convert the fraction to a percentage by moving the decimal point two places to the right.
For example, to convert the fraction 0.75 to a percentage, we would move the decimal point two places to the right:
0.75 = 75%
Percentage to Decimal Fraction
To convert a percentage to a decimal fraction, we divide the percentage by 100.
For example, to convert the percentage 50% to a decimal fraction, we would do the following:
50% / 100 = 0.5
Therefore, 50% as a decimal fraction is 0.5.
Fraction to Percentage Table
| Fraction | Percentage |
|---|---|
| 1/2 | 50% |
| 1/4 | 25% |
| 3/4 | 75% |
| 1/5 | 20% |
| 2/5 | 40% |
Representing the Fraction as a Graph
To represent a fraction as a graph on Desmos, you need to create two functions: one for the numerator and one for the denominator. Then, you can use the “f(x)/g(x)” command to divide the numerator function by the denominator function and create the graph of the fraction.
Step 1: Create the Numerator Function
In the input bar, enter the numerator of the fraction. For example, if the numerator is 2x, enter “2x”.
Step 2: Create the Denominator Function
Below the numerator function, enter the denominator of the fraction. For example, if the denominator is x+1, enter “x+1”.
Step 3: Divide the Functions
Use the “f(x)/g(x)” command to divide the numerator function by the denominator function. For example, if the numerator is 2x and the denominator is x+1, enter “(2x)/(x+1)”.
Step 4: Graph the Fraction
Click on the “Graph” button to graph the fraction. The graph will show the behavior of the fraction as the variable x changes.
Step 5: Use Table to See Specific Values
To view specific values of the fraction, you can create a table by using the “Table” function. In the input bar, enter the “Table” command followed by the fraction expression. For example, to create a table for the fraction (2x)/(x+1), enter “Table((2x)/(x+1))”. The table will display the values of the fraction for different values of x.
| x | (2x)/(x+1) |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 1.33 |
| 3 | 1.5 |
Finding the Derivative of a Fraction on Desmos
To find the derivative of the fraction f(x) = g(x) / h(x) on Desmos, we can use the quotient rule, which states that the derivative of a fraction is given by the formula:
“`
f'(x) = (h(x)g'(x) – g(x)h'(x)) / h(x)^2
“`
We can input this formula into Desmos using the following steps:
1. Create a new graph by typing “y=” into the input box.
2. Enter the numerator g(x) in the top line of the input box.
3. Press the “/” key to create a division symbol.
4. Enter the denominator h(x) in the bottom line of the input box.
5. Press the “Enter” key to create the graph of the fraction.
6. To find the derivative of the fraction, click on the “f(x)” button in the top right corner of the screen and select “Derivative” from the menu.
7. Desmos will return the derivative of the fraction in the form f'(x) = (h(x)g'(x) – g(x)h'(x)) / h(x)^2.
Integrating the Numerator and Denominator
To integrate the numerator and denominator of a fraction on Desmos, we can use the following steps:
1. Create a new graph by typing “y=” into the input box.
2. Enter the numerator g(x) in the top line of the input box.
3. Press the “/” key to create a division symbol.
4. Enter the denominator h(x) in the bottom line of the input box.
5. Press the “Enter” key to create the graph of the fraction.
6. To integrate the numerator, click on the “g(x)” button in the top right corner of the screen and select “Integral” from the menu.
7. Desmos will return the integral of the numerator in the form ∫g(x)dx.
8. To integrate the denominator, click on the “h(x)” button in the top right corner of the screen and select “Integral” from the menu.
9. Desmos will return the integral of the denominator in the form ∫h(x)dx.
We can then use these integrals to find the indefinite integral of the fraction by dividing the integral of the numerator by the integral of the denominator. That is:
“`
∫f(x)dx = ∫g(x)dx / ∫h(x)dx
“`
We can input this formula into Desmos using the following steps:
1. Create a new graph by typing “y=” into the input box.
2. Enter the integral of the numerator in the top line of the input box.
3. Press the “/” key to create a division symbol.
4. Enter the integral of the denominator in the bottom line of the input box.
5. Press the “Enter” key to create the graph of the indefinite integral of the fraction.
Calculating the Area under the Fraction’s Curve
Method 1: Utilizing the Integral Calculator
Desmos provides an integral calculator that can directly compute the area under the curve of a fraction. To access this feature, click on the “Integral” button and select the fraction function you want to integrate. Desmos will display the integral solution, including the antiderivative and the indefinite integral.
Once you have the indefinite integral, you can evaluate it at the desired endpoints of the interval to determine the area under the curve within that range. For instance, if you want to calculate the area under the curve of a fraction from x = 0 to x = 2, you would evaluate the integral at these two endpoints and subtract the results.
Method 2: Approximating the Area with a Sum of Rectangles
This method involves dividing the area under the curve into a series of rectangles, where the width of each rectangle represents a small interval on the x-axis and the height represents the value of the fraction at the midpoint of that interval.
By summing the areas of all the rectangles, you can obtain an approximation of the total area under the curve. As you increase the number of rectangles, the approximation becomes more precise.
Method 3: Using a Numerical Integration Tool
If the fraction function is particularly complex, you can use a numerical integration tool to approximate the area under the curve. Desmos offers a numerical integration feature that allows you to specify the fraction function, the interval of integration, and the number of trapezoids to use for the approximation.
This method provides a convenient and efficient way to calculate the area under the curve of a fraction, especially when analytical integration is impractical.
| Method | Advantages |
|---|---|
| Integral Calculator | Exact solution, applicable to various fraction functions |
| Sum of Rectangles | Simple and intuitive, allows for visual representation |
| Numerical Integration | Convenient and efficient for complex functions |
Deriving the Inverse of the Fraction
To derive the inverse of a fraction, you can use the following steps:
1. Swap the numerator and denominator.
2. Simplify the fraction by dividing both the numerator and denominator by their greatest common factor (GCF).
For example, to find the inverse of the fraction 3/4:
1. Swap the numerator and denominator: 4/3
2. Simplify the fraction by dividing both the numerator and denominator by 1: 4/3
Therefore, the inverse of 3/4 is 4/3.
Here is a more detailed explanation of step 2:
To simplify a fraction, you need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest number that divides both the numerator and denominator without leaving a remainder. Once you have found the GCF, you can divide both the numerator and denominator by the GCF to simplify the fraction.
For example, to simplify the fraction 12/18:
1. Find the GCF of 12 and 18. The GCF is 6.
2. Divide both the numerator and denominator by the GCF: 12/18 ÷ 6/6 = 2/3
Therefore, the simplified fraction is 2/3.
| Fraction | Inverse |
|---|---|
| 1/2 | 2/1 |
| 3/4 | 4/3 |
| 5/6 | 6/5 |
Exploring the Asymptotes of the Fraction
Asymptotes are lines that the graph of a function approaches but never quite reaches. In the case of a fraction, there can be two types of asymptotes: vertical and horizontal.
Vertical Asymptotes
Vertical asymptotes occur when the denominator of the fraction is equal to zero. This is because when the denominator is zero, the fraction is undefined. The graph of the fraction will have a vertical asymptote at any point where the denominator is equal to zero.
Horizontal Asymptotes
Horizontal asymptotes occur when the numerator and denominator of the fraction are both very large. In this case, the fraction will approach a constant value. The graph of the fraction will have a horizontal asymptote at this constant value.
Example: Finding the Asymptotes of f(x) = (x-1)/(x+2)
To find the asymptotes of this fraction, we first need to find the zeros of the denominator. The denominator is x+2, which is equal to zero when x=-2. Therefore, the graph of the fraction will have a vertical asymptote at x=-2.
Next, we need to find the constant value that the fraction will approach as x gets very large. To do this, we can divide both the numerator and denominator by x. This gives us:
“`
f(x) = (x-1)/(x+2) = (x/x – 1/x)/(x/x + 2/x) = 1 – 1/x)/(1 + 2/x)
“`
As x gets very large, the terms 1/x and 2/x will both get very small. Therefore, the fraction will approach the value of 1. The graph of the fraction will have a horizontal asymptote at y=1.
| Asymptote Type | Equation |
|---|---|
| Vertical | x=-2 |
| Horizontal | y=1 |
Creating a Fraction on Desmos
To create a fraction on Desmos, enter the numerator followed by a forward slash, then the denominator. For example, to enter the fraction 1/2, type 1/2.
Simplifying Fractions
Desmos can simplify fractions automatically. To simplify a fraction, select it and press the “Simplify” button that appears.
Operating with Fractions
Desmos supports basic arithmetic operations with fractions. To add, subtract, multiply, or divide fractions, use the corresponding operators (+, -, *, /). For example, to add the fraction 1/2 to 1/4, type 1/2 + 1/4.
Applying the Fraction in Real-World Scenarios
Example 1: Dividing a Pizza
If you have a pizza cut into 8 slices and want to divide it evenly among 4 people, how many slices should each person get? Use a fraction to represent each person’s share.
| Person | Fraction of Pizza |
|---|---|
| Person 1 | 1/4 |
| Person 2 | 1/4 |
| Person 3 | 1/4 |
| Person 4 | 1/4 |
Each person would receive 1/4 of the pizza, or 2 slices.
Example 2: Measuring Ingredients
A recipe calls for 3/4 of a cup of flour. If you double the recipe, how much flour will you need? Multiply the original amount of flour by 2.
(3/4) cup * 2 = 6/4 cup = 1 1/2 cups
You will need 1 1/2 cups of flour for the doubled recipe.
Example 3: Calculating Speed
A car travels 180 miles in 3 hours. What is the average speed of the car in miles per hour? Use a fraction to represent the speed.
Speed = Distance / Time
Speed = 180 miles / 3 hours
Speed = 60 miles per hour
The average speed of the car is 60 miles per hour.
How to Make a Fraction on Desmos Testing
Creating fractions on Desmos Testing is a straightforward and versatile process. To input a fraction, use the following syntax: “numerator/denominator”. For instance, to enter the fraction 1/2, simply type “1/2” into the input field.
Desmos Testing supports both proper and improper fractions. To create an improper fraction, the numerator should be greater than or equal to the denominator. For example, to enter the improper fraction 7/3, type “7/3”.
People Also Ask
How do I create a mixed number on Desmos Testing?
To create a mixed number on Desmos Testing, follow these steps:
- Type the whole number part.
- Press the “/” key.
- Type the numerator of the fractional part.
- Press the “/” key again.
- Type the denominator of the fractional part.
For example, to create the mixed number 2 1/2, type “2/1/2”.
How do I use fractions in calculations on Desmos Testing?
To perform fractional calculations on Desmos Testing, use the following operators:
- +
- –
- ×
- ÷
- % (for finding a percentage)
For example, to calculate the sum of 1/2 and 1/4, type “(1/2) + (1/4)”.
