5 Simple Steps to Create a Circle in Desmos Graph

Creating a circle in Desmos Graphing Calculator is a fundamental skill for visualizing and analyzing mathematical equations. Whether you are a student exploring geometry concepts or a researcher working with complex data, understanding this technique will empower you to effectively represent and explore circular functions.

In this article, we will provide a comprehensive guide on how to draw a circle in Desmos. We will cover the step-by-step process, from defining the center and radius to graphing the equation. We will also explore advanced techniques for customizing the appearance of your circle, such as changing its color, thickness, and transparency.

Creating the Coordinate Plane

To create a coordinate plane in Desmos, you need to first create a new graph. Once you have a new graph, you can click on the “Axes” tab in the top toolbar. This will open a menu with a variety of options for customizing your coordinate plane.

The first option, “Show Axes,” allows you to toggle the visibility of the x- and y-axes. The second option, “Origin,” allows you to change the location of the origin (0,0). The third option, “Scale,” allows you to change the scale of the coordinate plane. The fourth option, “Ticks,” allows you to change the appearance of the tick marks on the x- and y-axes.

In addition to these options, you can also customize the appearance of the coordinate plane by changing the line color, line width, and fill color. To do this, click on the “Style” tab in the top toolbar. This will open a menu with a variety of options for customizing the appearance of your coordinate plane.

Positioning the Coordinate Plane

Once you have created a coordinate plane, you can position it anywhere on the graph by dragging and dropping it with your mouse. You can also resize the coordinate plane by clicking on one of the corners and dragging it. To reset the coordinate plane to its default size and position, click on the “Reset Axes” button in the top toolbar.

Adding Points to the Coordinate Plane

To add points to the coordinate plane, click on the “Points” tab in the top toolbar. This will open a menu with a variety of options for adding points to your coordinate plane.

The first option, “Add Point,” allows you to add a single point to the coordinate plane. The second option, “Add Multiple Points,” allows you to add multiple points to the coordinate plane at once. The third option, “Import Points,” allows you to import points from a CSV file. The fourth option, “Export Points,” allows you to export points to a CSV file.

In addition to these options, you can also customize the appearance of the points on the coordinate plane by changing the point color, point size, and point shape. To do this, click on the “Style” tab in the top toolbar. This will open a menu with a variety of options for customizing the appearance of the points on your coordinate plane.

Plotting Points Using Equations

In Desmos, you can plot points by inputting their coordinates or by using equations. To plot a point using an equation, simply type the equation into the input bar and press enter. For example, to plot the point (2, 3), you would type “x=2” and “y=3” into the input bar.

You can also plot multiple points by using a comma to separate the coordinates. For example, to plot the points (2, 3), (4, 5), and (6, 7), you would type “x={2, 4, 6}” and “y={3, 5, 7}” into the input bar.

Plotting a Circle Using an Equation

To plot a circle using an equation, you can use the following equation:

“`
(x – h)^2 + (y – k)^2 = r^2
“`

where (h, k) is the center of the circle and r is the radius of the circle.

For example, to plot a circle with a radius of 2 and a center at (0, 0), you would type the following equation into the input bar:

“`
(x – 0)^2 + (y – 0)^2 = 2^2
“`

Equation	Graph
y = x^2
y = sin(x)
y = e^x

Tracing the Curve

To trace the curve, it is helpful to break it down into smaller steps:

Determine the Domain and Range: Find the possible input and output values for the curve. This can be determined from the equation or by looking at the graph (if available).
Plot Key Points: Identify important points on the curve, such as intercepts, maxima, and minima. Plot these points on the graph.
Connect the Points: Once you have plotted the key points, connect them using a smooth curve. This can be done by hand or using a graphing calculator or software like Desmos.

Detailed Steps for Connecting the Points:

Examine the Curve’s Behavior: Observe the shape and trends of the curve to determine how the points should be connected.
Use Graphing Tools: Desmos provides tools like the "tangent line" feature to help you draw tangent lines to the curve at specific points. This can help you visualize the direction of the curve.
Consider Continuity: The curve should be drawn so that it is continuous, meaning there are no sudden breaks or discontinuities in the line.
Check for Asymptotes: If the curve has any asymptotes, make sure to draw them as part of the tracing. Asymptotes are lines that the curve approaches but never quite reaches.
Fine-tune the Curve: Adjust the shape and position of the curve as needed to ensure that it aligns with the key points and the original equation or function.

Adjusting Curve Parameters

Desmos Graph provides various parameters which allows users to modify the appearance and behaviour of a curve. These parameters can be accessed by selecting the curve and inspecting the fields in the sidebar. Here are the commonly adjustable parameters:

a: Vertical translation. Shifts the curve up (positive values) or down (negative values) from the x-axis.

h: Horizontal translation. Shifts the curve right (positive values) or left (negative values) from the y-axis.

k: Amplitude. Scales the vertical distance between the maximum and minimum points of the curve. Positive values create an upright curve, while negative values create an inverted curve.

b: Phase shift. Rotates the curve around the origin. A positive value shifts the curve to the left, and a negative value shifts the curve to the right.

d: Damping factor. Controls the decay rate of the curve. A positive value creates a more rapid decay, while a negative value slows down the decay.

c: Frequency. Determines the number of waves in the curve within a given interval. A higher value corresponds to a higher frequency and more frequent oscillations.

Period and Wavelength

The period of a curve refers to the distance between two consecutive peaks or troughs. It is inversely proportional to the frequency, meaning a higher frequency results in a shorter period. The wavelength, on the other hand, is the distance between two consecutive points on the curve that have the same amplitude and oscillation direction.

Amplitude and Asymptote

The amplitude is half the distance between the maximum and minimum points of the curve. It determines the vertical range of the curve’s oscillations. The asymptote, or horizontal asymptote, is the line that the curve approaches as x approaches infinity.

Shifting the Curve

The parameters a and h are used to translate the curve vertically and horizontally, respectively. A positive value of a shifts the curve up, while a negative value shifts it down. Similarly, a positive value of h shifts the curve right, while a negative value shifts it left.

Parameter	Effect
a	Vertical translation
h	Horizontal translation

Defining Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) for which the function is defined.

Finding the Domain

To find the domain of a function, look for any input values that would make the function undefined. For example, if the function involves dividing by x, then x cannot be 0 because division by 0 is undefined.

Finding the Range

To find the range of a function, look for any output values that are not possible for the function to produce. For example, if the function involves taking the square root of x, then the range will be limited to non-negative values because the square root of a negative number is undefined.

Example

Consider the function f(x) = (x-2)/(x+1).

The domain of this function is all real numbers except -1 because division by 0 is undefined.

To find the range, we can use the following approach:

Solve the equation f(x) = y for x in terms of y:

“`
(x-2)/(x+1) = y
(x-2) = y(x+1)
x = yx + y – 2
x = (y – 2)/(1 – y)
“`

Determine the restrictions on y:

Since x must be real, the denominator (1 – y) cannot be zero, so y /= 1.

Substitute the restrictions on y into the equation from step 1:

“`
x = (y – 2)/(1 – y)
x = (-2)/(1 – y)
“`

Therefore, the range of this function is all real numbers except 1.

Function	Domain	Range
f(x) = x^2	All real numbers	Non-negative real numbers
f(x) = 1/(x+1)	All real numbers except -1	All real numbers
f(x) = sin(x)	All real numbers	[-1, 1]

Labeling and Annotating the Graph

To add labels and annotations to your Desmos graph, follow these steps:

1. Title the Graph

Click the “Edit Title” field and enter your desired title.

2. Label Axes

Right-click on the x-axis or y-axis and select “Edit Axis”. In the “Axis Options” window, enter your desired label.

3. Add Text Annotations

Click the “Add Text” button (a capital “A”) in the toolbar. Click on the graph where you want to place the text and type your annotation.

4. Insert Math Expressions

To insert math expressions into annotations, use LaTeX syntax. For example, to add the Greek letter “pi”, type “\pi”.

5. Add Images

To add images, click the “Insert Image” button (a picture) in the toolbar. Select the desired image from your computer or paste an image URL.

6. Floating Text Boxes

To add floating text boxes that are not anchored to the axes, use the “Add Text Box” button (a square with a “T”) in the toolbar. Click on the graph where you want to place the box and type your text.

Floating Text Box Options

Option	Description
Font Size	Adjust the text size.
Font Color	Select the desired text color.
Background Color	Add color to the background of the text box.
Border	Add a border around the text box.
Round Corners	Create rounded corners for the text box.

You can also set the position and size of the text box by dragging its handles.

Adding Equations and Inequalities

7. Entering Inequalities

Inequalities are mathematical statements that show the relative difference between two expressions. In Desmos Graph, inequalities can be entered using a variety of symbols:

Symbol

Meaning

Less than

≤

Less than or equal to

Greater than

≥

Greater than or equal to

To enter an inequality in Desmos Graph, simply type the equation followed by the appropriate inequality symbol. For example, to enter the inequality x < 5, you would type:

x < 5

Desmos Graph will automatically generate a graphical representation of the inequality. The shaded region on the graph represents the solutions to the inequality. In this case, the shaded region will be all values of x less than 5.

Exploring Transformations of Curves

Desmos Graph offers a powerful toolset for exploring transformations of curves to understand how they modify the shape and position of graphs.

8. Transformations Using Sinusoidal Functions

Sinusoidal functions are of the form y = a*sin(bx + c) + d, where a, b, c, and d are constants. Transformations applied to sinusoidal functions include:

Vertical Shift: Adding a constant to d shifts the graph vertically. For example, y = sin(x) + 3 shifts the graph up by 3 units.
Horizontal Shift: Subtracting a constant from c shifts the graph horizontally. For example, y = sin(x – π/2) shifts the graph to the right by π/2 units.
Amplitude Change: Multiplying the function by a constant a greater than 0 changes the amplitude of the graph. For example, y = 2*sin(x) doubles the amplitude of the graph.
Period Change: Dividing the argument of the sine function by a constant b greater than 0 decreases the period of the graph. For example, y = sin(2x) halves the period of the graph.
Phase Shift: Adding a constant to the argument of the sine function shifts the graph horizontally. For example, y = sin(x + π/4) shifts the graph to the left by π/4 units.

To better understand these transformations, explore the following table:

Transformation	Equation	Effect
Vertical Shift	y = sin(x) + d	Shifts the graph vertically by d units
Horizontal Shift	y = sin(x – c)	Shifts the graph horizontally by c units
Amplitude Change	y = a*sin(x)	Changes the amplitude of the graph by a factor of a
Period Change	y = sin(bx)	Changes the period of the graph by a factor of 1/b
Phase Shift	y = sin(x + c)	Shifts the graph horizontally by c units

Exporting a Curve

When you’re done creating your curve, you can export it to share it with others or to use it in other software. To do so, click the "Share" button in the top right corner of the screen. This will generate a URL that you can share with others, or you can click the "Export as PNG" or "Export as SVG" buttons to download the curve as an image or SVG file, respectively.

Sharing the Curve

Once you’ve exported your curve, you can share it with others by sending them the URL that you generated. They can then click on the link to view the curve in their own browser. If they don’t have Desmos installed, they will be prompted to download it.

Exporting and Sharing the Curve

To export your curve, click the "Share" button in the top right corner of the screen. This will generate a URL that you can share with others, or you can click the "Export as PNG" or "Export as SVG" buttons to download the curve as an image or SVG file, respectively.

To share your curve with others, send them the URL that you generated. They can then click on the link to view the curve in their own browser. If they don’t have Desmos installed, they will be prompted to download it.

You can also export your curve as a PNG or SVG file by clicking the appropriate button in the "Share" menu. This will download the curve as an image or SVG file that you can save to your computer or upload to a website.

Here is a table summarizing the different export and sharing options:

Export Format	Description
PNG	A raster image format that is suitable for sharing on the web.
SVG	A vector image format that is suitable for printing or using in design software.
URL	A link that you can share with others to view the curve in their own browser.

Using Advanced Tools in Desmos Graph

10. Exploring the Graph Gallery

Desmos Graph features an extensive Graph Gallery, a treasure trove of user-created and curated graphs that cover a wide range of mathematical concepts, real-world applications, and stunning visual displays. Use the search bar to explore specific topics or browse the various categories to discover intriguing and instructive graphs. The Graph Gallery is a great source of inspiration, learning, and sharing your own graphical creations.

Tips for Navigating the Graph Gallery:

Feature	Description
Featured Gallery	Showcases a curated selection of graphs based on popularity, quality, and relevance.
Trending Graphs	Displays graphs that are gaining popularity and receiving attention from the community.
Newest Uploads	Lists the latest graphs uploaded by users, offering a glimpse into the newest creations.
Categories	Organizes graphs into specific categories, such as Algebra, Calculus, Geometry, and Science.
Search Bar	Allows you to search for specific graph titles, keywords, or creators.
Unofficial Graphs	Includes graphs not officially curated by Desmos but still worth exploring.

How to Make a Circle in Desmos Graph

Desmos is a free online graphing calculator that allows you to create and share graphs of mathematical functions. It is a powerful tool that can be used for a variety of purposes, including teaching, learning, and research. One of the most basic shapes that you can create in Desmos is a circle.

To make a circle in Desmos, you can use the following steps:

Open Desmos in your web browser.
Click on the “New Graph” button.
In the function entry field, type the following equation: (x - h)^2 + (y - k)^2 = r^2
Replace h, k, and r with the coordinates of the center of the circle and the radius of the circle, respectively.
Click on the “Graph” button.

Desmos will now display the circle on the graph. You can use the zoom and pan tools to adjust the view of the circle.