10 Easy Steps to Draw a Tangent on a TI-84

Drawing a tangent line to a graphing function on a TI-84 calculator is a simple process that can be completed in just a few steps. This technique is particularly useful for determining the slope of a curve at a specific point or for creating tangent lines to circles or other conic sections. By following the steps outlined in this guide, you will be able to confidently construct tangent lines using your TI-84 calculator. Transitioning to the next paragraph, we will delve deeper into the first step of the drawing process, which involves setting up the calculator and entering the function.

To begin, ensure that your TI-84 calculator is in the “Func” mode. This mode can be accessed by pressing the “Mode” button and selecting “Func” from the list of options. Once in the “Func” mode, you can enter the function you want to draw. To do this, press the “Y=” button and use the arrow keys to navigate to the appropriate line. Enter the function using the calculator’s keypad, and press “Enter” to save the function. Transitioning to the next paragraph, we will discuss the process of finding the tangent point, which is a crucial step in drawing the tangent line.

After entering the function, you need to identify the point at which you want to draw the tangent line. This point is known as the tangent point. To find the tangent point, use the arrow keys to move the cursor to the desired location on the graph. Once the cursor is positioned at the tangent point, press the “Trace” button. This will display the coordinates of the tangent point in the bottom-left corner of the screen. Transitioning to the next paragraph, we will provide a step-by-step guide on how to complete the drawing process.

Navigating the TI-84 Calculator Interface

The Texas Instruments TI-84 calculator is a powerful graphing calculator with a wide range of capabilities. It can be used for a variety of mathematical and scientific calculations, as well as for programming and data analysis. The TI-84 has a user-friendly interface that makes it easy to navigate and use.

The TI-84’s interface is divided into several main sections: the homescreen, the menu bar, the status line, the soft keys, and the arrow keys. The homescreen is where you enter and execute commands. The menu bar contains a list of menus that provide access to different functions and settings. The status line displays information about the current state of the calculator, such as the current mode and the battery level. The soft keys are located below the screen and their functions change depending on the current context. The arrow keys are used to navigate through menus and to move the cursor on the homescreen.

Keys and Functions

The TI-84 calculator has a variety of keys that can be used to enter commands and perform calculations. The following table lists some of the most commonly used keys and their functions:

Key	Function
ON	Turns the calculator on or off.
2nd	Accesses the secondary function of a key (e.g., the SIN key also accesses the COS key).
DEL	Deletes the last entered character or command.
CLEAR	Clears the homescreen.
ENTER	Executes the current command or moves the cursor to the next line.
ALPHA	Accesses the alphabetic characters (e.g., the A key also accesses the a character).
X-VAR	Accesses the variable screen, where you can view and edit variables.
STAT	Accesses the statistics menu, where you can perform statistical calculations.
MATH	Accesses the math menu, where you can perform a variety of mathematical calculations.
PRGM	Accesses the program menu, where you can create and edit programs.

Entering the Tangent Function

The TI-84 calculator has a built-in tangent function that can be used to find the tangent of an angle. To enter the tangent function, follow these steps:

1. Press the “Y=” button to access the equation editor.
2. Use the arrow keys to move the cursor to the line where you want to enter the equation.
3. Press the “tan(” button to insert the tangent function into the equation.
4. Enter the angle for which you want to find the tangent.
5. Press the “Enter” button to evaluate the equation.

For example, to find the tangent of 30 degrees, you would enter the following equation into the calculator:

“`
tan(30)
“`

The calculator would then display the result, which is 0.5773502692.

### Additional Details for Entering the Tangent Function

* You can also enter the tangent function using the “Math” menu. To do this, press the “2nd” button followed by the “Math” button. Then, use the arrow keys to navigate to the “Trig” submenu and select the “tan(” function.

* The tangent function can be used to find the tangent of an angle in either degrees or radians. To specify the units, use the “angle” button located in the “Math” menu.

* The tangent function is undefined for angles that are multiples of 90 degrees (pi/2 radians). If you try to evaluate the tangent function for one of these angles, the calculator will display an error message.

Specifying the Graph Window

To start graphing tangents using the TI-84, you need to set up the graph window appropriately. This involves defining the ranges for the x and y axes, as well as the scaling and aspect ratio.

Setting the X and Y Ranges

The x and y ranges determine the visible portion of the coordinate plane. To set these ranges, use the Window menu:

Setting	Description
Xmin	Leftmost visible x-value
Xmax	Rightmost visible x-value
Ymin	Bottommost visible y-value
Ymax	Topmost visible y-value

For example, to view the portion of the coordinate plane from x = -5 to x = 5 and y = -3 to y = 3, you would set the following ranges:

Xmin = -5
Xmax = 5
Ymin = -3
Ymax = 3

Setting the Scaling and Aspect Ratio

The scaling and aspect ratio determine how the graph is displayed. The Xscl and Yscl settings control the unit spacing for the x and y axes, respectively. A larger value for Xscl or Yscl will make the graph more zoomed out, while a smaller value will zoom in.

The Xres and Yres settings control the aspect ratio of the graph. A higher value for Xres or Yres will make the graph appear more pixelated, but will allow for more detail. A lower value for Xres or Yres will result in a smoother graph, but may hide some details.

For example, to graph a circle with a radius of 1, you might use the following settings:

Xmin = -2
Xmax = 2
Ymin = -2
Ymax = 2
Xscl = 1
Yscl = 1
Xres = 128
Yres = 128

Adjusting the Tangent Angle Increment

The angle increment in the tangent function can be adjusted to control the smoothness of the graph. A smaller increment will result in a smoother graph, while a larger increment will produce a more jagged graph. To adjust the angle increment, follow these steps:

Press the “MODE” button to open the MODE menu.
Scroll down to the “Angle” setting and press the “ENTER” button.
Enter the desired angle increment (in degrees) and press the “ENTER” button.

For example, to set the angle increment to 1 degree, you would enter the following value:

Angle Increment
1

Once you have adjusted the angle increment, the tangent function will use the new increment when graphing.

Additional Notes:

The angle increment can be set to any value between 0 and 360 degrees. A value of 0 degrees will result in a vertical line, while a value of 360 degrees will result in a horizontal line.

The angle increment can also be set using the “WINDOW” menu. To set the angle increment using the “WINDOW” menu, follow these steps:

Press the “WINDOW” button to open the WINDOW menu.
Scroll down to the “Angle” setting and press the “ENTER” button.
Enter the desired angle increment (in degrees) and press the “ENTER” button.

Enabling Tangent Lines
On the TI-84 calculator, you can enable tangent lines by following these steps:

1. Press the “Y=” button.

This will bring up the equation editor. Enter your function \( y = f(x) \) after the word “Y1=”.

2. Press the “2nd” button followed by the “Window” button.

This will bring up the “Format” menu. Select the “Draw” tab.

3. Turn on the “Tangent” option.

This will enable the drawing of tangent lines on the graph.

4. Set the “Tangent Lines” mode.

You can choose between “On” (draws tangent lines at all points), “Slope” (draws tangent lines with a specified slope), or “Off” (disables tangent lines).

5. Specify the tangent line parameters.

If you choose “Slope” mode, you will need to enter the desired slope of the tangent line. If you choose “On” mode, you can additionally specify the x-coordinate of the point where the tangent line is drawn using the “Tangent X” option. Here’s a breakdown of the parameters and their effects:

Tangent X: The x-coordinate of the point where the tangent line is drawn. If left blank, the tangent line is drawn at the current cursor position.
Tangent Lines (On): Draws tangent lines at all points on the graph.
Tangent Lines (Slope): Draws tangent lines with the specified slope. Enter the desired slope value.
Tangent Lines (Off): Disables the drawing of tangent lines.

Positioning the Tangent Lines

To position the tangent lines accurately, follow these steps:

1. Graph the Function

Enter the function into the TI-84 and graph it. Make sure the graph is visible.

2. Find the Tangent Point

Determine the x-coordinate of the point where the tangent line is desired. This can be done using the zero or minimum/maximum feature of the calculator.

3. Calculate the Slope

Find the slope of the tangent line at the given point using the derivative feature of the calculator.

4. Construct the Tangent Line Equation

Use the point-slope form of the line equation (y – y₁ = m(x – x₁)) to construct the equation of the tangent line, where (x₁, y₁) is the tangent point and m is the slope.

5. Equation-Solving

If necessary, solve the tangent line equation for y to get it in the form y = mx + b.

6. Tangent Line in Matrix Form

Create a matrix that represents the tangent line equation. This matrix can be used to perform various operations related to the tangent line, such as finding the intersection points with other lines or planes.

Matrix Form
[1]	[m]	[-b]
[0]	[1]	[1]

where m is the slope and b is the y-intercept of the tangent line.

7. Display the Tangent Line

Use the “Tangent” feature in the TI-84 to display the tangent line on the graph. Enter the equation of the tangent line in the “y=” section and press “Graph”.

Displaying Tangent Line Coordinates

To display the coordinates of the tangent line, follow these steps:

Find the derivative of the function at the given point.
Evaluate the derivative at the given point to find the slope of the tangent line.
Use the point-slope form of the equation of a line to write the equation of the tangent line.
Simplify the equation if possible.
Press the “CALC” button and select “2:zero”.
Enter the equation of the function and the equation of the tangent line into the calculator.
Press the “ENTER” button to find the point of intersection between the two equations.
The point of intersection is the coordinates of the tangent line.

Step	Action
1	Find the derivative of the function at the given point.
2	Evaluate the derivative at the given point to find the slope of the tangent line.
3	Use the point-slope form of the equation of a line to write the equation of the tangent line.
4	Simplify the equation if possible.
5	Press the “CALC” button and select “2:zero”.
6	Enter the equation of the function and the equation of the tangent line into the calculator.
7	Press the “ENTER” button to find the point of intersection between the two equations.
8	The point of intersection is the coordinates of the tangent line.

Saving the Graph for Future Reference

Once you have created your tangent graph, you can save it for future reference by following these steps:

Press the “Y=” key to view the list of functions.
Scroll to the function that corresponds to your tangent graph and press the “EDIT” button.
In the “Name” field, enter a unique name for your graph, such as “Tangent_Graph”.
Press the “ENTER” button.
Press the “ZOOM” key and select the “ZStandard” option to return to the standard viewing window.
Press the “GRAPH” button to display your saved tangent graph.

Saving the Graph in a File

To save your tangent graph in a file, follow these additional steps:

Press the “CATLG” button.
Scroll to the “GraphOps” menu and select the “Store” option.
Enter a file name for your graph and press the “ENTER” button.
Your tangent graph will be saved as a “.84F” file on your calculator.

Exporting the Graph as an Image

You can also export your tangent graph as an image that you can use in other applications. To do this, follow these steps:

Press the “APPGREY” key and select the “Graph and Table I/O” option.
Select the “Analyze Graph” option.
Scroll to the “Export Graph” option and press the “ENTER” button.
Select the “PNG” file format and press the “ENTER” button.
Enter a file name for your image and press the “ENTER” button.
Your tangent graph will be exported as a “.png” file on your calculator.

Action	Keystrokes
Save graph as “Tangent_Graph”	Y= -> EDIT -> Name:Tangent_Graph -> ENTER
Save graph in “mygraph.84F” file	CATLG -> GraphOps -> Store -> mygraph.84F -> ENTER
Export graph as “mygraph.png” image	APPGREY -> Graph and Table I/O -> Analyze Graph -> Export Graph -> PNG -> mygraph.png -> ENTER

Troubleshooting Common Errors

9. The graph is not smooth or has unexpected breaks

This is likely due to an improper setting in the graphing window or the use of a discontinuous function. Here are some possible causes and solutions:

Cause	Solution
The window is too narrow	Zoom out to display a wider range of values
The function has discontinuities	Use the discontinuity function syntax (discont) to handle them
The step size is too large	Decrease the step size to plot more points
The function is not defined for all x-values	Limit the graphing window to the domain of the function
The calculator is running out of memory	Clear the memory and try again
The program has an error	Check the syntax of the program and make sure there are no logical errors
The calculator is not in the correct mode	Switch to the correct graphing mode (e.g., Func, Par)
The calculator is not up to date	Update the calculator’s firmware to the latest version
The calculator is faulty	Contact the manufacturer for repair or replacement

Setting the Window

Before you begin drawing, set the window dimensions to ensure the graph is displayed appropriately. Adjust the Xmin, Xmax, Ymin, and Ymax values to fit the range of the function and the desired tangent line.

Finding Critical Points

Identify the critical points of the function by finding where its derivative is zero or undefined. These points represent potential locations for tangent lines.

Evaluating the Derivative

At each critical point, evaluate the derivative to find the slope of the tangent line at that point. This value will help you draw the line’s direction.

Plotting the Tangent Line

Use the point-slope form of a line to plot the tangent line. The critical point is the starting point (x, y), and the slope is the value calculated in step 3.

Additional Tips and Tricks for Drawing Tangent Lines

10. Utilizing the Table Feature

Create a table of values for the function using the “tblset” command. This allows you to easily find critical points and evaluate the derivative at specific values. Enter the function expression, followed by the independent variable (e.g., “tblset y=sin(x), x, -π, π, 0.1”).

Draw the tangent lines for multiple critical points by using the “Tangent(” command multiple times in the plot statement (e.g., “plot(sin(x), -π, π) + Tangent(sin(x), -π/2) + Tangent(sin(x), π/2)”).

Find the equation of the tangent line by using the “Tangent(” command with the “equ” option (e.g., “Tangent(sin(x), π/4, equ)”). The output is the equation of the line.

Use the “horz” parameter to draw horizontal tangent lines (e.g., “plot(sin(x), -π, π) + Tangent(sin(x), -π/2, horz)”).

Draw vertical tangent lines using the “vert” parameter (e.g., “plot(sin(x), -π, π) + Tangent(sin(x), 0, vert)”).

How to Draw a Tangent on a TI-84 Graphing Calculator

A tangent is a line that touches a curve at a single point. It can be used to find the slope of the curve at that point, or to determine the equation of the curve. To draw a tangent on a TI-84 graphing calculator, follow these steps:

Enter the equation of the curve into the calculator.
Press the “Graph” button to graph the curve.
Move the cursor to the point where you want to draw the tangent.
Press the “Menu” button and select “Calc > Tangent”.
The calculator will display the equation of the tangent line.
Press the “Graph” button to graph the tangent line.