Chapter 6: Sequence Graphing

TI-84 Plus Graphing Mode Similaritiesp. 103

The steps for defining a sequence graph are close to the steps for defining a function graph. Y= editing, graph styles, selecting and deselecting equations, TRACE, ZOOM, Smart Graph and the FORMAT settings all behave the way they do in Func mode, with the differences described below.

Setting Sequence Graphing Modep. 103

Press MODE and select Seq in the graphing-mode row. You must do this before you enter sequence window variables and before you enter sequence functions, because the Y= editor and the WINDOW editor change to their sequence versions.

TI-84 Plus Sequence Functions u, v, and wp. 103

The calculator has three sequence functions you can type: u, v and w. They are the 2nd functions of the 7, 8 and 9 keys: 2ndu gives u, 2ndv gives v and 2ndw gives w.

A sequence can be defined in terms of:

• the independent variable n alone, as in u(n)=2n
• the previous term of the same sequence, u(n−1)
• the term before the previous one, u(n−2)
• the previous term (or the one before it) of another sequence, for example u(n−1) used inside the definition of v(n)

Displaying the Sequence Y= Editorp. 103

With Seq selected, press Y= to open the sequence Y= editor. It has an nMin= line followed by a pair of lines for each sequence: u(n)= for the rule and u(nMin)= for the initial value (and likewise for v and w).

• nMin is the first n value that is evaluated. It is the same variable as nMin in the WINDOW editor: change it in one place and the other updates too.
• u(nMin), v(nMin) and w(nMin) hold the starting values of the corresponding sequences.

Selecting Graph Stylesp. 104

The icon to the left of u(n), v(n) and w(n) shows that sequence’s graph style. In Seq mode the default is dot, which shows the separate (discrete) values of the sequence. Line and thick are also available. Graph styles are ignored in Web axes format.

Selecting and Deselecting Sequence Functionsp. 104

Only selected sequences are graphed. A sequence is selected when the = signs of both its u(n)= and u(nMin)= lines are highlighted.

To change a sequence’s status, put the cursor on the = sign of its name line and press ENTER. The status changes for both the rule u(n) and its initial value u(nMin).

Defining and Editing a Sequence Functionp. 104

Define or edit a sequence exactly as you would a function in Y=. The independent variable is n. In Seq mode you can type it with X,T,θ,n. The guidebook also mentions pasting n from the CATALOG (2ndCATALOG), but X,T,θ,n is simpler and works everywhere.

The names u, v and w come from 2ndu, 2ndv and 2ndw. You must type every character of a term such as u(n−1) yourself: 2ndu ( X,T,θ,n − 1 ).

Sequences are evaluated only at consecutive whole numbers. n is always a run of consecutive integers that starts at 0 or at any positive whole number (nMin).

Nonrecursive Sequencesp. 105

In a nonrecursive sequence the nth term depends only on n. No other term is needed, so you can compute u(5) directly without knowing u(1) to u(4).

For example, u(n)=2n gives the sequence 2, 4, 6, 8, 10, … for n = 1, 2, 3, 4, 5, …. You can leave the initial value u(nMin) blank for a nonrecursive sequence.

Recursive Sequencesp. 105

In a recursive sequence each term is defined from the previous term, u(n−1), or from the two before it, u(n−2). It may also use n itself, as in u(n)=u(n−1)+n. To find u(5) you must first compute u(1), u(2), u(3) and u(4).

Example: u(n)=2u(n−1) with the initial value u(nMin)=1 doubles every step and returns 1, 2, 4, 8, 16, ….

A recursive sequence refers to terms that do not exist yet, so it needs starting values:

• If every term is built from the previous term, u(n−1), give one initial value in u(nMin).
• If terms are built from the one before the previous term, u(n−2), give two initial values in u(nMin) as a list in braces with a comma between them, for example {0,1}. For the guidebook’s example the first term is 0 and the second is 1.

Setting Window Variablesp. 106

Press WINDOW to see the window variables for sequence graphing. These defaults apply in both Radian and Degree mode:

WINDOW
nMin=1
nMax=10
PlotStart=1
PlotStep=1
Xmin=⁻10
Xmax=10
Xscl=1

• nMin is the first n that is evaluated; it also appears in the Y= editor. The sequence is evaluated at u(nMin), u(nMin+1), u(nMin+2), …, u(nMax).
• nMax is the last n that is evaluated.
• PlotStart is the number of the first term that is plotted. With PlotStart=5, terms 1 to 4 are still evaluated (a recursive sequence needs them) but they are not drawn.
• PlotStep is the n increment used for plotting only. It does not change how the sequence is evaluated: with PlotStep=2 every term is still computed, but only every other one is plotted.

Setting the Graph Formatp. 107

Press 2ndFORMAT to open the FORMAT screen. The settings are the same ones used by the other graphing modes; the choice of axes on the top line exists only in Seq mode.

Setting Axes Formatp. 107

Sequence graphing offers five axes formats. They decide which quantity is plotted horizontally and which vertically:

To pick one, open FORMAT, move the cursor to the setting you want with the arrow keys, and press ENTER.

Displaying a Sequence Graphp. 107

Press GRAPH to plot the selected sequences. As the graph is drawn, the values of X, Y and n are updated. Smart Graph works for sequences just as it does for functions: if nothing has changed since the last time the graph was drawn, it is shown again right away instead of being replotted.

Free-Moving Cursorp. 107

The free-moving cursor works as it does in Func mode: press an arrow key after GRAPH and the cursor moves one pixel at a time.

• In RectGC format, moving the cursor updates X and Y; with CoordOn they are displayed.
• In PolarGC format, X, Y, R and θ are updated; with CoordOn, R and θ are displayed.

TRACEp. 108

The axes format changes how TRACE behaves.

Time, uv, vw and uw formats. TRACE moves along the sequence one PlotStep at a time. Press 2nd▶︎ or 2nd◀︎ to jump five plotted points at once. Press ▲︎ or ▼︎ to change to another selected sequence.

• A trace begins on the first selected sequence at the term number given by PlotStart, even if that point is outside the viewing window.
• Quick Zoom works in every direction. After moving the trace cursor, press ENTER ENTER to center the viewing window on the cursor’s location. The trace cursor then returns to nMin.

Web format. The trail left by the cursor helps you spot attracting and repelling behavior in a sequence. When you begin a trace, the cursor is on the x-axis at the initial value of the first selected function. See Graphing Web Plots.

Moving the Trace Cursor to Any Valid n Valuep. 108

To put the trace cursor on any valid n of the current sequence, just type the number. As soon as you type the first digit, an n= prompt with your entry appears in the bottom-left corner. You can even type an expression there. The value must be valid for the current viewing window. Press ENTER to move the cursor.

ZOOMp. 108

ZOOM operations work as in Func mode, and they change only the X variables (Xmin, Xmax, Xscl) and the Y variables (Ymin, Ymax, Yscl).

The exception is ZStandard, which also sets the sequence variables back to nMin=1, nMax=10, PlotStart=1 and PlotStep=1.

The items on the ZU tab of the VARS2 (Zoom) menu are the ZOOM MEMORY variables for sequence graphing:

CALCp. 108

The only CALC operation available in Seq mode is value. Press 2ndCALC to open the CALCULATE menu.

What value does depends on the axes format:

• Time: it shows Y (the value of u(n)) for the n you enter.
• Web: it draws the web and shows Y (the value of u(n)) for the n you enter.
• uv, vw or uw: it shows X and Y according to the axes setting. For uv, X is u(n) and Y is v(n).

Evaluating u, v, and wp. 109

Enter the sequence names with 2ndu, 2ndv and 2ndw. You can evaluate them in three ways:

• The nth value: u(5).
• A list of values: u({1,3,5}).
• A generated run of terms: u(nstart,nstop[,nstep]). nstep is optional and defaults to 1.

Graphing a Web Plotp. 109

To switch to Web axes, press 2ndFORMAT, move right to Web and press ENTER. A web plot graphs u(n) against u(n−1). It is the classic tool for studying the long-term behavior of a recursive sequence (does it converge, diverge or oscillate?) and for seeing how that behavior changes when the initial value changes.

Valid Functions for Web Plotsp. 109

In Web format a sequence either graphs incorrectly or produces an error unless it meets all three of these conditions:

• It is recursive with one level of recursion: u(n−1) is allowed, u(n−2) is not.
• It does not refer to n directly.
• It does not refer to any defined sequence other than itself.

Displaying the Graph Screenp. 109

In Web format, press GRAPH. The calculator:

• draws the reference line y=x, if AxesOn is selected;
• plots the selected sequences with u(n−1) as the independent variable.

Drawing the Webp. 110

Press TRACE to activate the trace cursor. The screen shows the sequence and the current n, X and Y, where X stands for u(n−1) and Y for u(n). Press ▶︎ repeatedly to draw the web one segment at a time, starting at nMin. In Web format the cursor follows this route:

1. It starts on the x-axis at the initial value u(nMin) (when PlotStart=1).
2. It moves vertically, up or down, to the sequence curve.
3. It moves horizontally to the reference line y=x.
4. It repeats the vertical and horizontal moves each time you press ▶︎.

Using Web Plots to Illustrate Convergencep. 110

Example: draw the sequence u(n)=−.8u(n−1)+3.6, first in Time format and then as a web, to watch it converge. Its web crosses y=x at 2, so the terms home in on 2. The guidebook leaves the initial value to a screenshot; this walkthrough uses u(nMin)=6, but you can try other values.

Graphing with uv, vw, and uwp. 111

The phase-plot axes settings show how two sequences relate to each other. To select one, press 2ndFORMAT, move right to uv, vw or uw, and press ENTER.

Example: Predator-Prey Modelp. 111

The predator-prey model estimates the populations of a predator and its prey that can coexist in balance. This example finds the balance for foxes and rabbits that start at 200 rabbits (u(nMin)) and 50 foxes (v(nMin)).

The model is R(n) = R(n−1)(1 + M − K·W(n−1)) and W(n) = W(n−1)(1 + G·R(n−1) − D). Rabbits go in as u(n) and foxes as v(n), so each rule refers to the other sequence.

Sequences and Window Variablesp. 113

If you are used to the TI-82, this table shows each TI-84 Plus sequence and sequence window variable next to its TI-82 counterpart. Note that the names move around: the TI-84 Plus nMin plays the role of the TI-82 nStart, and the TI-84 Plus PlotStart plays the role of the TI-82 nMin.

Keystroke Differences Between TI-84 Plus and TI-82p. 114

The guide also compares how sequence names and variables are typed on the two calculators. The TI-82 used different key combinations for these symbols (it wrote them as Un, Vn, Un−1 and so on). On the TI-84 Plus you type every character of a term yourself, as in this table of TI-84 Plus keystrokes.