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

Updated9Nov2020(What’sNew?)

Copyright © 2001–2024 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.)You 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.To

*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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please click to donate!Because this article helps you,

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Updates and new info: https://BrownMath.com/ti83/

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