BrownMath.com→TI-83/84/89→GraphingFunctions

Updated9Nov2020(What’sNew?)

Copyright © 2001–2023 by StanBrown, BrownMath.com

**Summary:**It’s pretty easy to produce some kind of graph on the TI-83/84for a given function. This page helps you with the tricks that mightnot be obvious. You’ll be able to find asymptotes, intercepts,intersections, roots, and so on.

**Seealso:**How to Evaluate Functions withTI-83/84

How to Graph Piecewise Functions onTI-83/84

**Contents:**

- Graphing Your Function
- Common Problems
- Tuning Your Graph
- Zooming
- Adjusting the Window
- Adjusting the Grid

- Exploring Your Graph
- Domain and Asymptotes
- Function Values
- Intercepts

- Multiple Functions
- Intersection

- What’s New?

The techniques in this note will work with any function, butfor purposes of illustration, we’ll use

## Graphing Your Function

Step 1: **Clear unwanted plots.**

You need to look for any previously set plots that mightinterfere with your new one. | Press [`Y=` ] (the top left button).Look at the top ofthe screen. If any of `Plot1` `Plot2` `Plot3` is highlighted,cursor to it and press[`ENTER` ] to deactivate it. (No information is lost; youcan always go back and reactivate any plot.) To verify that you havedeactivated the plot, cursor away from it and check that it’s nothighlighted. |

(Sometimes you might want tograph more than one function on thesame axes. In this case, make sure to deactivate all the functionsyou don’t want to graph.) | Now check the lines starting with `Y1=` ,`Y2=` , and so on. If any`=` sign is highlighted, either delete the whole equationor deactivate it but leave it in memory. To delete an equation, cursorto it and press the [`CLEAR` ] button. To deactivate it withoutdeleting it,cursor to its `=` sign and press [`ENTER` ].My screen looked like this after I deactivated allold plots and functions. |

Step 2: **Enter the function.**

If your function is not already in y= form, usealgebra to transform it before proceeding.Two cautions: - For
*x*, use the [`x,T,θ,n` ] key, not the[`×` ] (times) key. - The TI-83/84 follows the standard order of operations.If there are operations on top or bottom of a fraction, you must useparentheses—for
*x*+ 2 divided by*x*− 3,you can’t just enter “x+2/x−3”.
| Cursor to one of the `Y=` lines, press[`CLEAR` ] if necessary, and enter the function. |

Check your function and correct any mistakes. For example, if yousee a star `*` in place of an `X` , youaccidentally used the times key instead of [`x,T,θ,n` ]. | Use the [`◄` ] key and overtype any mistakes.To delete any extra characters, press [ `DEL` ].If you need to insert characters, locate yellow `INS` abovethe [`DEL` ] key. Press [`2nd` `DEL` makes `INS` ] and typethe additional characters. As soon as you use a cursor key, the TI-83/84goes back to overtype mode. |

Step 3: **Display the graph.**

“Zoom Standard” is usually a goodstarting point. It selects standard parameters of −10 to +10 for x andy. | Press [`ZOOM` ] [`6` ]. |

## Common Problems

If you don’t see your function graph anywhere, your window isprobably restricted to a region of the *xy* plane the graph justdoesn’t happen to go through. Depending on the function, one of thesetechniques will work:

`ZoomFit`

is a good first try.Press [`ZOOM`

] [`0`

]. (Thanks to Marilyn Webb for thissuggestion.)See AlsoSample Statistics on TI-83/84You can try to zoom out (like goinghigher to see more of the

*xy*plane) by pressing[`ZOOM`

] [`3`

] [`ENTER`

].Finally, you can directlyadjust the window to select a specificregion.

For other problems, please seeTI-83/84 Troubleshooting.

## Tuning Your Graph

You can make lots of adjustments to improve yourview of the function graph.

### Zooming

The window is your field of view into the *xy* plane, and thereare two main ways to adjust it. This section talks aboutzooming, which is easy and covers most situations. Thenext section talks about manually adjusting thewindow parameters for complete flexibility.

Here’s a summary of the zooming techniques you’re likely touse:

You’ve already met standard zoom, which is[

`ZOOM`

] [`6`

]. It’s a good starting point formost graphs.You’ve also met zoom fit, which is[

`ZOOM`

] [`0`

]. It slides the view field up ordown to bring the function graph into view, and it may also stretchor shrink the graph vertically.To

*zoom out*, getting a larger field of view with less detail, press[`ZOOM`

] [`3`

] [`ENTER`

].You’ll see the graph again,with a blinking*zoom cursor*. You can press[`ENTER`

] again to zoom out even further.(Video) How to Graph with the TI-83 and TI-84 CalculatorTo

*zoom in*, focusing in on a part of the graph withmore detail, press [`ZOOM`

] [`2`

] but don’tpress [`ENTER`

] yet. The graph redisplays with a blinking*zoom cursor*in the middle of the screen. Use the arrow keysto move the zoom cursor to the part of the graph you want to focus on,and then press [`ENTER`

]. After the graph redisplays, youstill have a blinking zoom cursor and you can move it again and press[`ENTER`

] for even more detail.Because this article helps you,

please click to donate!Because this article helps you,

please donate at

BrownMath.com/donate.Your viewing window is rectangular, not square.When your

*x*and*y*axes have the same numerical settingsthe graph is actually stretched by 50% horizontally.If you want a plot where the*x*and*y*axes are to the same scale, press[`ZOOM`

] [`5`

] for*square zoom*.

There are still more variations on zooming. Some long winterevening, you can read about them in the manual.

### Adjusting the Window

You may want to adjust the window parameters to see more of thegraph, to focus in on just one part, or to get more or fewer tickmarks. If so, press [`WINDOW`

].

`Xmin`

and`Xmax`

are the left and rightedges of the window.`Xscl`

controls the spacing of tickmarks on the*x*axis. For instance,`Xscl=2`

putstick marks every 2 units on the*x*axis. A bigger`Xscl`

spaces the tick marks farther apart, and a smaller`Xscl`

places them closer together.`Ymin`

and`Ymax`

are thebottom and top edges of the window.`Yscl`

spaces thetick marks on the*y*axis.`Xres`

is a number 1–8 inclusive. With 1,the default, the calculator will find the*y*value at*x*-valuescorresponding to every pixel along the*x*axis. With 2, thecalculation occurs every 2 pixels, and so on. Higher values drawgraphs faster, but fine details may be lost. My advice is, just leavethis at 1.

Color TI-84s have two additional windowparameters:

`Δx`

is the*x*distance between the centers ofadjacent pixels. The calculator determines this automatically from`Xmin`

and`Xmax`

, so you don’t need to mess with it.However, if you do change it, the calculator will then determine`Xmax`

from`Xmin`

and`Δx`

.`TraceStep`

is the step size when you press ◄or ► while tracing along a graph. By default it’s twice thevalue of`Δ`

, but you can change it if you wantto.

To **blow up a part of the graph for a more detailed view**,increase `Xmin`

or `Ymin`

or both,or reduce `Xmax`

or `Ymax`

. Then press[`GRAPH`

].

If you want to see more of the *x**y* plane, compressed to asmaller scale, reduce `Xmin`

and/or `Ymin`

,or increase `Xmax`

or `Ymax`

. Then press[`GRAPH`

].

The graph windows shown in your textbook may have smallnumbers printed at the four edges. To make your graphing window looklike the one in the textbook, press [`WINDOW`

] and use the numbers at left and rightedges for `Xmin`

and `Xmax`

, the number atthe bottom edge for `Ymin`

, and the number at the top edgefor `Ymax`

.

### Adjusting the Grid

The *grid* is the dots (dots or lines,in color TI-84s) over the whole window that line up tothe tick marks on the axes, kind of like graph paper. The grid helpsyou see the coordinates of points on the graph.

If you have a black&white TI-83/84, and you see a lot of horizontal lines runningacross the graph, it means your `Xscl`

is **way** toosmall, and the tick marks are running together in lines.Similarly, `Yscl`

is the number of *y* units betweentick marks. A bunch of vertical lines means your `Yscl`

is too small. Press [`WINDOW`

] and fix either of theseproblems.

To turn the grid on or off: | Locate yellow `FORMAT` above the [`ZOOM` ]key. Press [`2nd` `ZOOM` makes `FORMAT` ].Cursor to the desired `GridOn` or`GridOff` setting, and press [`ENTER` ] to lockit in.Then press [ `GRAPH` ] to return to your graph. |

Color TI-84s can present the grid as dots orlines. On the [`2nd`

`ZOOM`

*makes* `FORMAT`

] screen, you can choose`GridOff`

, `GridDot`

, or `GridLine`

, andyou can also assign a color to the grid.

## Exploring Your Graph

### Domain and Asymptotes

First off, just look at the shape of the graph.A **vertical asymptote** should stick out like a sore thumb, such as*x*=3 with this function. (Confirm vertical asymptotes bychecking the function definition. Putting *x*=3 in thefunction definition makes the denominator equal zero, which tells youthat you have an asymptote.)

Color TI-84s have the ability to detect asymptotes:press [`2nd`

`ZOOM`

*makes* `FORMAT`

] and change `Detect`

`Asymptotes`

to `On`

. That often creates a more realistic picture of thegraph, as in this case, but it can also make it harder to see anasymptote. Here are both versions:

The **domain** certainly excludes any *x* values where there arevertical asymptotes. But additional values may also be excluded, evenif they’re not so obvious in the graph. For instance, the graph of*f*(*x*)= (*x*³+1)/(*x*+1) looks like a simple parabola, butthe domain does not include *x*=−1.

**Horizontal asymptotes** are usually obvious.But sometimes an apparent asymptote really isn’t one, just lookslike it because your field of view is too small or too large.Always do some algebra work to confirm the asymptotes.This function seems to have *y*=1 as a horizontal asymptoteas *x* gets very small or very large, and in fact from the functiondefinition you can see that that’s true.

### Function Values

While displaying your graph, press [`TRACE`

] and thenthe *x* value you’re interested in. The TI-83/84 will move thecursor to that point on the graph, and will display the corresponding*y* value at the bottom.

The *x* value must be within the current viewingwindow. If you get the message `ERR:INVALID`

, press[`1`

] for `Quit`

. Thenadjust your viewing window and try again.

### Intercepts

You can trace along the graph to find any intercept. The interceptsof a graph are where it crosses or touches an axis:

x intercept | where graph crosses or touches x axis | because y = 0 |

y intercept | where graph crosses or touches y axis | because x = 0 |

Most oftenit’s the *x* intercepts you’re interested in, because the*x* intercepts of the graph *y*=*f*(*x*) are the solutionsto the equation *f*(*x*)=0, also known as the zeroes of thefunction.

To find ** x intercepts:**You could naïvely press [

`TRACE`

] and cursor left andright, zooming in to make a closer approximation.But it’s much easier to make the TI-83/84 find the intercept for you.Locate an x intercept by eye. For instance, this graph seemsto have an x intercept somewhere between x=−3 andx=−1. | Locate yellow `CALC` above the [`TRACE` ]key. Press [`2nd` `TRACE` makes `CALC` ] [`2` ]. (You select`2:zero` because the x intercepts are zeroesof the function.) |

Enter the left and right bounds. | [`(-)` ] 3 [`ENTER` ] [`(-)` ] 1 [`ENTER` ]There’s no need to make a guess; just press [ `ENTER` ]again. |

Two cautions with *x* intercepts:

- Since the TI-83/84 does approximations, you must always check theTI-83/84 answer in the function definition to make sure that
*y*comes outexactly 0. - When you find
*x*intercepts, make sure to find all of them. Thisparticular function has only one in its entire domain, but with otherfunctions you may have to look for additional*x*intercepts outside theviewing area.

Finding the ** y intercept** is even easier:press [

`TRACE`

]0 and read off the *y*intercept.

This *y* intercept looks like it’s about −2/3, and by plugging*x*=0 in the function definition you see that the interceptis exactly −2/3.

## Multiple Functions

You can plot multiple functions on the same screen. Simply press[`Y=`

] and enter the second function next to`Y2=`

. Press [`GRAPH`

] to see the two graphstogether.

To select which function to trace along, press[`▲`

] or [`▼`

]. The upper left cornershows which function you’re tracing.

### Intersection

When you graph multiple functions on the same set of axes, youcan have the TI-83/84 tell you where the graphs intersect. This isequivalent to **solving a system of equations graphically.**

The naïve approach is to trace along one graph until itcrosses the other, but again you can do better. We’ll illustrate byfinding the intersections of*y*=(6/5)*x*− 8 with the function we’ve already graphed.

Graph both functions on the same set of axes. Zoom out ifnecessary to find all solutions. | Press [`2nd` `TRACE` makes `CALC` ] [`5` ].You’ll be prompted `Firstcurve?` If necessary,press [`▲` ] or [`▼` ] to select one ofthe curves you’re interested in. Press [`ENTER` ].You’ll be prompted `Secondcurve?` If necessary,press [`▲` ] or [`▼` ] to select theother curve you’re interested in. Press [`ENTER` ]. |

Eyeball an approximate solution. For instance, in this graphthere seems to be a solution around x=2. | When prompted `Guess?` , enter your guess. In thiscase, since your guess is 2 you should press 2[`ENTER` ]. |

Repeat for any other solutions. |

As always, you should confirm apparent solutions bysubstituting in both equations. The TI-83/84 uses a method of successiveapproximations, which may create an ugly decimal when in fact there’san exact solution as a fraction or radical.

## What’s New?

**9 Nov 2020**: Added an example of asymptote detection incolor TI-84s.Updated for grid properties of color TI-84s here,here, andhere.

Explained the meaning of Xres,and the meanings of the two extra windowparameters for color T-84s.

Supplied missing words in the instructionsfor adjusting the grid boundaries.

Converted HTML 4.01 to HTML5, and italicized variablenames.

- (intervening changes suppressed)
**5 Aug 2007**: New article and workbook.

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## FAQs

### Can you graph functions on TI 84? ›

Graphing a Function

**Press WINDOW to input window settings.** Press GRAPH to display the graph. To evaluate the function at a given value of x while the graph is displayed, press TRACE and then input the given value of x . Then press ENTER .

**How do you graph numbers on a TI 84 calculator? ›**

On the TI-83 and TI-84, this is done by **going to the function screen by pressing the “Y=” button and entering the function into one of the lines.** **After the function has been entered, press the “GRAPH” button**, and the calculator will draw the graph for you.

**How do calculators know the answer so fast? ›**

Calculators (and computers) **combine inputs using electronic components called logic gates**. As the name implies, a logic gate acts as a barrier in an electronic circuit; it takes in two electric currents, compares them and sends out a new current based on what it finds.

**How do calculators know the answer to everything? ›**

A calculator doesn't think the way your brain thinks. **It processes numbers in binary code, where a combination of “0's” and “1's” are used to represent any number**. So in the case of the example above, “2” would become “0010,” “4” would become “0100,” and so on.

**How do you graph a table on a TI 83? ›**

**Graphing, Tables, and More!**

- Press MODE. ...
- To make sure all statistical plots and graphs are turned off before you start your work, press Y=. ...
- To enter a function, press Y=. ...
- Set up the viewing area. ...
- Before we graph the function, press 2
^{nd}FORMAT to see how the graphing window is set up. ... - Now press GRAPH. ...
- Press TRACE.

**Do you need a graphing calculator for functions? ›**

Beginning with first-year algebra, it's appropriate to start using a graphing calculator, even though **it is rarely required**. Concepts such as basic function graphing, polynomials, quadratics, and inequalities are better visualized when students can both write out the equations and use an electronic input.

**What grade is graphing functions? ›**

**6th-8th Grade** Math: Graphing Functions - Chapter Summary.

**How do you graph a function step by step? ›**

- Step 1: Find the x- and y- intercepts. ...
- Step 2: Find at least one more point. ...
- Step 3: Plot the intercepts and point(s) found in steps 1 and 2. ...
- Step 4: Draw the graph. ...
- Step 1: Find the x- and y- intercepts.
- Step 2: Find at least one more point. ...
- Step 3: Plot the intercepts and point(s) found in steps 1 and 2.

**How do you graph an equation? ›**

To graph an equation using the slope and y-intercept, 1) Write the equation in the form y = mx + b to find the slope m and the y-intercept (0, b). 2) Next, plot the y-intercept. 3) From the y-intercept, move up or down and left or right, depending on whether the slope is positive or negative.

**What is a graph function example? ›**

Graphs of functions are graphs of equations that have been solved for y! The graph of f(x) in this example is the graph of **y = x ^{2} - 3**. It is easy to generate points on the graph. Choose a value for the first coordinate, then evaluate f at that number to find the second coordinate.

### Who is the fastest math calculator? ›

Did you know that the "World's Fastest Human Calculator", a title that you're probably hearing for the first time, is an Indian man? Born in Andhra Pradesh's Eluru in 1999, Neelakantha Bhanu Prakash "is to math what Usain Bolt is to running," at least according to the BBC.

**Who is the fastest human calculator in? ›**

'World's fastest human calculator': Neelakanta Bhanu is not your regular math nerd | Trending News,The Indian Express.

**Do calculators ever make mistakes? ›**

**Yes, calculators can be wrong**. They are nothing but simple computers and as such are only as good as their coding and will always have some limitations involving for instance rounding. Some errors can be avoided by paying attention to how you enter a problem if you are aware of the flaw in the coding.

**How do you say hello on a calculator? ›**

Here's another quintessential calculator word that's easy to learn. **07734 spells hello**. Interestingly, it wouldn't be possible to spell hello on a calculator if not for its modern form.

**What is the 73 trick? ›**

The Secret of 73

**Write the number 73 on a piece of paper, fold it up, and give it to an unsuspecting friend**. Tell your friend select a four-digit number and enter it twice into a calculator. Inform your friend that the number is divisible by 137 and ask him or her to verify using the calculator.

**How to tell if someone has secret calculator? ›**

**How to spot a fake calculator app**

- Check the app memory size. ...
- See how many calculator apps your child has on their phone. ...
- Search for the name of the calculator app on the app store. ...
- Search for “vault app” on the app store.

**What makes a calculator so smart? ›**

**Integrated circuits contain transistors that can be turned on and off with electricity to perform mathematical calculations**. The most basic calculations are addition, subtraction, multiplication, and division. The more transistors an integrated circuit has, the more advanced mathematical functions it can perform.

**Did Einstein use a calculator? ›**

Certainly. **Einstein worked before the days of electronic calculators as we know them**. In his day the term “calculator” more often implied “one who calculates.”

**How can you avoid getting wrong answers on a calculator? ›**

- Check your settings like are you in Radian instead of degree mode.
- Check if if your are in rounding numbers like you set it to two decimal places instead of floating point.
- Make sure you are inputting the equation correctly by make sure you put your parentheses in correctly your powers and other things.

**Why do I get different answers on different calculators? ›**

**Each calculator may have a different algorithm for calculating the answer**. Each calculator may be programed to calculate a different degree of accuracy. Each calculator may have a different number of digits it can display.

### Why do I keep making calculation mistakes? ›

**Not Practicing Enough**

Even though you have read or prepared well, your performance can only be measured on the basis of your practice. It is very vital that you give more time to practice and solve as many questions and doubts as possible.

**What causes math error in calculator? ›**

'Math Error', **when the calculation you entered makes mathematical sense but the result cannot be calculated**, such as attempting to divide by zero, or when the result is too large for the calculator to handle.