TI-84 Plus CE lesson 7: Tables: checking, sign changes and sequences

A table of values is the quickest way to check an answer, compare two models, trap a root between two numbers, or find the first term of a sequence that passes a target. Two screens do all of it: TBLSET to choose the x-values, and TABLE to read them.

Lesson 7 of 12
About 55 minutes
TI-84 Plus CE (OS 5.x)
Free, no sign-in

Objective

Use TBLSET and TABLE to read values of one or two functions, zoom the step in to locate a root, and answer "smallest n such that …" questions.

By the end, students can:

set TblStart and ΔTbl in TBLSET and read the TABLE
type your own x-values with Indpnt: Ask
find a sign change, then make ΔTbl ten times smaller to trap a root to 2 decimal places
compare two models side by side with Y1 and Y2
use a step of 1 to find the smallest whole n that passes a target

Which courses this lesson suits

IB Maths AA (SL and HL): Papers 2 and 3
IB Maths AI (SL and HL): every paper
A Level Maths (UK and IAL): calculator allowed in every paper

A Level: the sign-change steps are the numerical methods skill of locating a root in an interval.

Before you start

Check mode is on FUNCTION (not SEQ or PARAMETRIC). Clear any old functions in y=.

The key presses, step by step

Every key is shown as it is printed on the calculator, like 2nd or enter. A second function (printed above a key) is 2nd then that key; a menu item is chosen by its number, or by moving to it and pressing enter.

Press y= and type Y1 = X³ − 2X − 7 (X is X,T,θ,n), then enter.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Press 2nd window (TBLSET). Set TblStart = 2 and ΔTbl = .1 (type 0.1), with enter after each. Leave Indpnt and Depend on Auto.
TblStart is the first x in the table; ΔTbl is the gap between x-values.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Press 2nd graph (TABLE). Y1 changes sign between X = 2.2 (−.752) and X = 2.3 (.567), so a root lies between them. Scroll with ▼ and ▲; highlight a value to see it in full at the bottom.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Zoom the step in. Back in TBLSET set TblStart = 2.24 and ΔTbl = .01, then TABLE again. The sign changes between 2.25 (−.1094) and 2.26 (.02318). Check the halfway value 2.255: Y1 is still negative, so the root is between 2.255 and 2.26, and it is 2.26 to 2 d.p.
Each zoom gives one more decimal place. The halfway check is what shows the rounding is right.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Your own x-values. In TBLSET choose Indpnt: Ask. Now TABLE starts empty: type any x, such as 2.255, and enter. Y1 appears next to it: −.043268625.
Ask is the fastest way to check a set of answers or fill in a table from a question with uneven x-values.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Sequences. For "the smallest n with 500 × 1.08ⁿ > 2000", type Y1 = 500 × 1.08^X, set Indpnt back to Auto, TblStart = 17 and ΔTbl = 1. In TABLE, n = 18 gives 1998 (highlight it: the full value is just under 2000) and n = 19 gives 2157.9. So n = 19.
X plays the part of n. A step of 1 gives only whole numbers, which is what "smallest n" needs. (mode SEQ is another way, with u(n) in y=; switch back to FUNCTION afterwards.)
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

Your turn: check your understanding

Try these straight after the key presses, on your own calculator, before looking at the answers.

Make a table of y = 2x² − 5 from x = −2 in steps of 0.5. What are y at x = 0.5 and at x = 1.5?
x³ + x − 5 = 0 has one root. Use a table with ΔTbl = 0.1 to find the interval of width 0.1 that contains it.
With Indpnt: Ask, find y = 2x² − 5 at x = 0.37.

Show the answers

−4.5 and −0.5
Between 1.5 and 1.6 (the sign changes there; the root is 1.52 to 3 s.f.).
−4.7262

Exam efficiency

Check an answer you found by algebra: put the equation in Y1 and look at the table at your x. It should be 0 (or very close).
For two models, put one in Y1 and one in Y2: the table shows where one overtakes the other.
Say what you saw: "f(2.25) < 0 and f(2.26) > 0, so there is a root between 2.25 and 2.26" earns the method mark.
Watch out for "greater than": if a term equals the target exactly, it does not count.

Worked exam-style example

Question. A town's population is modelled by P = 1200 × 0.94^t and a second town's by Q = 300 + 25t, where t is the number of years from now. (a) Find the first whole year in which Q is greater than P. (b) Use a table to find, to 1 decimal place, when the two populations are equal.

(a) Y1 = 1200 × 0.94^X, Y2 = 300 + 25X. TBLSET: TblStart = 0, ΔTbl = 1. TABLE and scroll down: at X = 11, Y1 = 607.56 and Y2 = 575; at X = 12, Y1 = 571.1 and Y2 = 600. So the first whole year is t = 12.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
(b) TBLSET: TblStart = 11.4, ΔTbl = .1. At X = 11.5, Y1 = 589.05 is still above Y2 = 587.5; at X = 11.6, Y1 = 585.42 is below Y2 = 590. So they are equal between 11.5 and 11.6. The Solver (lesson 6) gives t = 11.52522319, so t = 11.5 to 1 d.p.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

What to write down

(a) t = 11: P = 608 > Q = 575; t = 12: P = 571 < Q = 600. So in year 12.

(b) P − Q changes sign between t = 11.5 and t = 11.6; t = 11.5 years (1 d.p.).

Practice

Do these on your own calculator. Write the answers to 3 significant figures unless the task says otherwise.

Show that e^x − 3x = 0 has a root between 1.51 and 1.52, and give the root to 2 decimal places.
Find the smallest whole number n for which 3 × 1.2ⁿ > 100.
With Indpnt: Ask, find g(x) = 5 ÷ (x² + 1) at x = 0.3, 1.7 and 4.2.
The arithmetic series 4 + 7 + 10 + … has Sₙ = n(8 + 3(n − 1)) ÷ 2. Find the smallest n for which Sₙ > 1000.
A student says x = 1.8 is a solution of x³ − 2x − 3 = 0. Use a table to check.

Show the practice answers

f(1.51) = −0.00327 < 0 and f(1.52) = 0.0122 > 0, so there is a root between them; the root is 1.51 (2 d.p.).
n = 20 (n = 19 gives 95.8; n = 20 gives 115)
4.59, 1.29 and 0.268
n = 26: S₂₅ = 1000, which is not more than 1000, and S₂₆ = 1079.
f(1.8) = −0.768, not 0, so x = 1.8 is not a solution.

Common mistakes

Leaving Indpnt on Ask and wondering why the table is empty: switch back to Auto.
A step that is too big jumps over two roots close together: if the graph shows more roots than the table, make ΔTbl smaller.
Rounding a root from the sign change alone: check the halfway value before writing 2 d.p.
Missing the strict inequality: S₂₅ = 1000 is not greater than 1000.
Calculator in SEQ mode: Y1 is missing from y=. Set mode back to FUNCTION.

1-minute quick check

Answer these without looking back. Most need no calculation: they check that the key presses have stuck.

Which keys open TBLSET?
Which keys open the TABLE?
What does ΔTbl set?
How do you type your own x-values?
What do you need to see in the table to say there is a root?

Show the quick-check answers

2nd window
2nd graph
The step between x-values.
TBLSET → Indpnt: Ask.
A change of sign in Y between two x-values (for a continuous function).

Teaching this lesson (about 55 minutes)

0–5 min	Starter: without a calculator, is x³ − 2x − 7 positive or negative at x = 2 and at x = 3? What does that tell you?
5–15 min	Steps 1–3: Y1, TBLSET, TABLE and the first sign change.
15–22 min	Step 4: zooming ΔTbl in and the halfway check. Your turn task 2.
22–30 min	Steps 5–6: Indpnt Ask and sequences. Your turn tasks 1 and 3.
30–40 min	Worked example: the two towns.
40–52 min	Practice tasks 1–5; check the answers.
52–55 min	1-minute quick check.

Students need their own TI-84 Plus CE. With a projector, the calculator-skills guides below have a Show on the board view and an animated walkthrough of the key presses.