10 Steps to Master Piecewise Functions on Desmos Graphing

Piecewise Function on Desmos Graphing

Piecewise functions are a type of function that is defined by different rules for different intervals of its domain. This can be a useful way to model situations where the relationship between the input and output variables changes depending on the value of the input variable. For example, you could use a piecewise function to model the cost of shipping a package, where the cost depends on the weight of the package. In this article, we’ll show you how to graph a piecewise function on Desmos, a free online graphing calculator. We’ll start by creating a simple piecewise function and then we’ll add more complex functions as we go along.

To create a piecewise function on Desmos, you first need to create a new graph. Then, click on the “Functions” tab and select the “Piecewise” function. You will then be prompted to enter the different rules for the different intervals of the function’s domain. For example, if you want to create a piecewise function that models the cost of shipping a package, you could enter the following rules:
“`
if x ≤ 10: 5
if x ≤ 20: 10
if x ≤ 30: 15
“`

This function states that if the weight of the package is less than or equal to 10 pounds, the cost of shipping is $5. If the weight of the package is greater than 10 pounds but less than or equal to 20 pounds, the cost of shipping is $10. And if the weight of the package is greater than 20 pounds but less than or equal to 30 pounds, the cost of shipping is $15. Note that the intervals in the function’s domain are closed on the left and open on the right. This means that the function is defined for all values of x that are less than or equal to the endpoints of the intervals.

Understanding Piecewise Functions

Piecewise functions are a type of function made up of multiple parts, each part defined over a specific interval of the input. These sub-functions combine to form the overall piecewise function.

The most common type of piecewise function is defined by a set of linear functions. For example, the following function is piecewise linear with three parts:

Interval	Function
x < -1	y = -2x + 1
-1 ≤ x ≤ 1	y = x
x > 1	y = 2x – 1

In this piecewise function, the function changes its behavior depending on the value of x. For x values less than -1, the function is a line with a slope of -2 and a y-intercept of 1. For x values between -1 and 1, the function is a line with a slope of 1 and a y-intercept of 0. Finally, for x values greater than 1, the function is a line with a slope of 2 and a y-intercept of -1.

To graph a piecewise function, you can break it down into its individual parts and graph each part separately.

Other Types of Piecewise Functions

Piecewise functions can also be defined using other types of functions, such as quadratic functions, exponential functions, or logarithmic functions. These functions can be used to create more complex piecewise functions that exhibit a wider variety of behaviors.

Creating a Piecewise Function on Desmos

Define the Function

To create a piecewise function on Desmos, start by entering the different segments of your function in the equation editor. Each segment should be written as an expression, followed by its domain using the | character. For example, to define the function:

f(x) = 2x + 3, x < 0

f(x) = x^2, x >= 0

You would enter the following into Desmos:

2x + 3 | x < 0

x^2 | x >= 0

Graph the Function

Once you have defined the function, Desmos will automatically graph it. The graph will show the different segments of the function as separate lines, connected at the boundaries of their domains. In the example above, the graph would show a line with a slope of 2 for x < 0, and a parabola for x >= 0.

Additional Features

Desmos offers several additional features that can be useful when working with piecewise functions. For example, you can:

Adjust the domain of each segment by dragging the boundaries on the graph.
Add or remove segments by clicking on the plus or minus buttons in the equation editor.

Preview the graph of each segment individually by selecting it in the legend.

Defining Multiple Conditions

When defining multiple conditions for a piecewise function, you can use the following syntax in the Desmos graphing calculator:

piecewise([condition 1]: [expression 1], [condition 2]: [expression 2], …, [condition n]: [expression n])

For example, to define a piecewise function that returns 1 when x is less than 0, 2 when x is between 0 and 1, and 3 when x is greater than 1, you would use the following:

piecewise(x < 0: 1, 0 ≤ x < 1: 2, x ≥ 1: 3)

You can also use inequalities in your conditions. For example, to define a piecewise function that returns 1 when x is less than or equal to 0, 2 when x is between 0 and 1, and 3 when x is greater than 1, you would use the following:

piecewise(x ≤ 0: 1, 0 < x < 1: 2, x > 1: 3)

Here is a table summarizing the syntax for defining multiple conditions in a piecewise function:

Syntax	Meaning
piecewise([condition 1]: [expression 1], [condition 2]: [expression 2], …, [condition n]: [expression n])	Defines a piecewise function with multiple conditions.
[condition]	A condition that evaluates to true or false.
[expression]	The expression to be evaluated if the corresponding condition is true.

Graphing Different Function Segments

To graph different function segments on Desmos Graphing, you’ll need to use a piecewise function. A piecewise function is a function that is defined differently over different intervals of its domain.

To create a piecewise function on Desmos Graphing, follow these steps:

Click on the “Graph” button in the top left corner.
Click on the “Piecewise” tab.
Enter the different function segments into the boxes provided.
Click on the “Graph” button to see the graph of the piecewise function.

For example, to graph the piecewise function:

“`
f(x) =
{
x + 1 if x < 0
x^2 if 0 <= x < 2
x – 1 if x >= 2
}
“`

you would enter the following into the boxes provided:

Interval	Expression
x < 0	x + 1
0 <= x < 2	x^2
x >= 2	x – 1

The graph of the piecewise function would look like this:

[Image of the graph of the piecewise function]

Customizing Graph Settings

Desmos graphing calculator offers robust customization options to tailor graphs to specific needs. Here are some key settings that can be adjusted:

1. Axes

Domain: Set the starting and ending points of the x-axis.
Range: Set the starting and ending points of the y-axis.

2. Grid

Show Grid: Enable or disable the gridlines.
Grid Lines: Customize the number, spacing, and color of gridlines.

3. Appearance

Function Color: Choose the color of the graph line.
Thickness: Set the width of the graph line.
Function Labels: Add labels to the different functions.

4. Interpolation

Smooth Function (Straight Lines): Display the graph as straight line segments.
Smooth Function (Curves): Display the graph as smooth curves.

5. Point Properties

Point Mode: Enable or disable displaying individual points on the graph.
Point Style: Customize the size, shape, and color of points.

6. Advanced Options

Decimal Precision: Set the number of decimal places displayed for coordinates and values.
Angle Measure: Choose between degrees or radians for angle measurements.
Display Settings: Control the visibility of axes, tick marks, and the origin point.
Labels: Customize the labels for the axes, tick marks, and points.
Axes Bounds: Set the maximum and minimum values for the x- and y-axes.
Window: Control the size and position of the graphing window.

Adding Labels and Descriptions

Once you’ve created your piecewise function, you can add labels and descriptions to improve its clarity and organization.

Adding Labels to Variables and Functions

To add a label to a variable or function, click on the variable or function in the left-hand panel. A small pop-up window will appear. Click on the "Label" field and enter the desired label.

Adding Descriptions to Functions

To add a description to a function, click on the function in the left-hand panel and click on the "Description" field. Enter the desired description.

Adding Labels to Breakpoints

To add a label to a breakpoint, click on the breakpoint in the graph. A small pop-up window will appear. Click on the "Label" field and enter the desired label.

Wrapping Function Descriptions

If your function description is long, you can wrap it to multiple lines by pressing the Enter key.

Using Latex to Format Labels and Descriptions

You can use LaTeX to format your labels and descriptions. For example, to make the label for the variable "x" bold, you would enter x in the "Label" field.

Formatting Labels and Descriptions

You can format your labels and descriptions using HTML tags. For example, the following table shows how to use HTML tags to format the label for the variable "x":

Using Line Breaks in Descriptions

To add a line break in a description, use the HTML tag <br>. For example, the following description would add a line break after the word "Hello":

Hello
<br>
World

Sharing and Exporting the Graph

Once you’re satisfied with your piecewise function graph, you can share it with others or export it for use in other documents.

Sharing the Graph

To share your graph, click the “Share” button in the top-right corner of the screen. You can then copy the link to the graph and share it via email, social media, or a direct message.

Exporting the Graph

To export your graph, click the “Export” button in the top-right corner of the screen. You can then choose from a variety of export options, including:

PNG (image file)
SVG (vector file)
PDF (document file)

Once you’ve selected an export option, click the “Export” button to download the file.

Exporting Multiple Graphs

If you have multiple graphs open in Desmos, you can export them all at once. To do this, click the “Share” button in the top-right corner of the screen, then select “Export All”. You can then choose from the same export options as described above.

Exporting a Graph as LaTeX

If you want to export your graph as LaTeX code, click the “Export” button in the top-right corner of the screen, then select “LaTeX”. This will download a .tex file that you can open in a LaTeX editor.

HTML Tag	Description
	Bold
	Italic
	Underline
	Superscript
	Subscript

Export Option	Description
PNG	A raster image file format that is commonly used for web graphics.
SVG	A vector image file format that is scalable without losing quality.
PDF	A document file format that can contain text, graphics, and other content.
LaTeX	A typesetting system that is commonly used for creating mathematical and scientific documents.

How to Do Piecewise Function on Desmos Graphing

Piecewise functions are functions that are defined by different equations over different intervals. In Desmos, you can graph piecewise functions using the “piecewise” function. The syntax for the piecewise function is:

piecewise(condition1, expression1, condition2, expression2, ..., default expression)

The “condition” is a logical expression that determines which expression is used to evaluate the function. The “expression” is the value that the function will return if the “condition” is true. The “default expression” is the value that the function will return if none of the conditions are true.

For example, the following piecewise function defines a function that is equal to x² for x ≤ 0, and equal to x + 1 for x > 0:

piecewise(x ≤ 0, x^2, x > 0, x + 1)

You can graph this function in Desmos by entering the following into the input bar:

y = piecewise(x ≤ 0, x^2, x > 0, x + 1)