Chapter 17: Activities

You will enter two parabolas in the Y= editor and draw them in the ZDecimal window.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the Y= editor.
   Y=
2. Type the first quadratic into Y₁. The X,T,θ,n key types the variable X without needing ALPHA. Press ENTER afterwards to move down to Y₂.
   2X,T,θ,nx²−11X,T,θ,n+14ENTER
3. Type the second quadratic into Y₂.
   2X,T,θ,nx²−6X,T,θ,n+5
4. Press ZOOM and choose 4:ZDecimal. This window is centered on the origin and each pixel is 0.1 wide, which is perfect for this problem.
   ZOOM4
5. Look at the two curves. The first parabola crosses the x-axis, so 2x² − 11x + 14 = 0 has real solutions (near x = 2 and x = 3.5). The second one never touches the axis, so 2x² − 6x + 5 = 0 has no real solutions. They must be complex numbers.

You will store the coefficients, build one root with the fraction template, then copy that entry and change + to − to get the other root.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Store the coefficients on one line: 2→A:⁻11→B:14→C. Use STO► for the arrow, ALPHA: for each colon, and (−) (not the subtract key) for the negative sign in −11.
   2STO►ALPHAAALPHA:(−)11STO►ALPHABALPHA:14STO►ALPHACENTER
2. Open the FRAC shortcut menu with ALPHAF1 (that key's green legend is F1).
   ALPHAF1
3. Choose 1:n/d. A stacked fraction template appears and the cursor waits in the numerator.
   1
4. Type the numerator, ⁻B+√(B²−4AC). Pressing 2nd√ gives the square-root template. Everything you type next stays under the root sign.
   (−)ALPHAB+2nd√ALPHABx²−4ALPHAAALPHAC
5. Press ▶︎ once to leave the square root, and once more to jump from the numerator down into the denominator. Then type 2A.
   ▶︎▶︎2ALPHAA
6. Press ENTER to evaluate. The answer 7/2 is shown as a fraction because the fraction template was used.
   ENTER
7. To get the other root you do not need to retype the formula. Press ▲︎ twice to highlight the formula you entered (first the answer, then the entry), and press ENTER to copy it to the entry line.
   ▲︎▲︎ENTER
8. Move the cursor left with ◀︎ until it sits on the + sign in the formula (13 presses here), then press − to replace the + with a minus.
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎−
9. Press ENTER. The second solution is 2. The roots of 2x² − 11x + 14 are 2 and 7/2 (3.5). The graph agreed: the parabola crosses the x-axis at both of these.
   ENTER

Calculate the first root again and convert 7/2 into 3.5.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Build the root again: open the n/d template, type ⁻B+√(B²−4AC), move into the denominator and type 2A, then press ENTER.
   ALPHAF11(−)ALPHAB+2nd√ALPHABx²−4ALPHAAALPHAC▶︎▶︎2ALPHAAENTER
2. Open the FRAC shortcut menu again and select 4:►F◄►D. It is pasted on a new line and works on the last answer.
   ALPHAF14
3. Press ENTER to convert 7/2 to its decimal form, 3.5.
   ENTER

You will switch to a+bi mode, enter the formula with the divide key, and then copy and edit it for the second root.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the MODE screen and move down six rows to the last row (Real a+bi re^θi). Press ▶︎ to highlight a+bi, then ENTER to select it.
   MODE▼︎▼︎▼︎▼︎▼︎▼︎▶︎ENTER
2. Press 2ndQUIT (QUIT) to return to the home screen, and CLEAR to wipe it.
   2ndQUITCLEAR
3. Store the new coefficients: 2→A:⁻6→B:5→C.
   2STO►ALPHAAALPHA:(−)6STO►ALPHABALPHA:5STO►ALPHACENTER
4. Type the formula in classic style, with ordinary parentheses and the divide key instead of the fraction template: (⁻B+√(B²−4AC))÷(2A). Press ▶︎ once after the radicand to step out of the square-root template.
   ((−)ALPHAB+2nd√ALPHABx²−4ALPHAAALPHAC▶︎)÷(2ALPHAA)
5. Press ENTER. The first solution is 1.5 + 0.5i.
   ENTER
6. Highlight the formula you just evaluated with ▲︎ twice and press ENTER to paste it.
   ▲︎▲︎ENTER
7. Press ◀︎ until the cursor is on the + (15 presses), then press − to replace it.
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎−
8. Press ENTER to get the other solution, 1.5 − 0.5i. The two solutions are complex conjugates of each other, exactly as the picture of the graph (no x-intercepts) suggested.
   ENTER

Enter V as a function of X so the calculator can tabulate and graph it.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press Y= to open the function editor, where graphable functions are stored.
   Y=
2. Type (20−2X) and then open the fraction template (ALPHAF1 then 1:n/d) for the 25÷2 part.
   (20−2X,T,θ,n)(ALPHAF11
3. Type 25 on top, press ▶︎ to drop to the denominator, type 2, and press ▶︎ again to leave the fraction.
   25▶︎2▶︎
4. Finish with −X) and then the last factor, X. Press ENTER to store the function. Check it: Y₁(3.68) is the volume at X = 3.68.
   −X,T,θ,n)X,T,θ,nENTER

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the TABLE SETUP screen.
   2ndTBLSET
2. Accept TblStart=0 by pressing ENTER. The cursor moves to ΔTbl.
   ENTER
3. Set the increment to 1. Leave Indpnt: Auto and Depend: Auto alone so the calculator fills the table in for you.
   1ENTER
4. Display the table with 2ndTABLE (TABLE). The bottom line reminds you that + changes ΔTbl.
   2ndTABLE
5. Compare the Y₁ column as X grows: the volume rises to a peak and then falls. The biggest value in view occurs for X around 4, between 3 and 5 (Y₁(3) = 399, Y₁(4) = 408 and Y₁(5) = 375).
6. Hold ▼︎ to scroll the table (here 11 presses) until a negative Y₁ appears. The box has no volume once the cuts eat up the sheet, so X cannot be larger than 10 or so. Between X = 10 and X = 11 the sign changes.
   ▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎

Narrow the search from steps of 1, to 0.1, to 0.01, and read full precision from the table.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. You are looking at the table after scrolling down to X = 11. Open TABLE SETUP again.
   2ndTBLSET
2. TblStart has changed to the first X that was displayed, so the table would continue from where you scrolled to. Go back to the table and press ▲︎ twice to scroll back to X = 3.
   2ndTABLE▲︎▲︎
3. Press +. A ΔTbl= prompt appears on the entry line. Type .1 and press ENTER.
   +.1ENTER
4. Scroll down eight rows. The table now steps by 0.1 and ends at X = 3.8. In this view the largest Y₁ is 410.26 at X = 3.7, so the maximum lies between 3.6 and 3.8.
   ▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎
5. Zoom in again: open TABLE SETUP and start the table at 3.6 with steps of 0.01.
   2ndTBLSET3.6ENTER.01ENTER
6. Show the table.
   2ndTABLE
7. Scroll with ▼︎ (7 presses) to X = 3.67, then press ▶︎ to move the cursor into the Y₁ column. The bottom line shows Y₁ in full precision: 410.261226.
   ▼︎▼︎▼︎▼︎▼︎▼︎▼︎▶︎
8. Press ▼︎ to see the next row. At X = 3.68 the volume is 410.264064.
   ▼︎
9. Two more rows. Y₁(3.69) = 410.262318 and Y₁(3.7) = 410.256. So the table says the maximum is at X = 3.68, which is as good as you can measure with a ruler marked in hundredths of a centimetre.
   ▼︎▼︎

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the window editor. The cursor starts on Xmin.
   WINDOW
2. Type 0 for Xmin and press ENTER.
   0ENTER
3. Type the expression 20÷2 for Xmax and press ENTER. The calculator evaluates it and stores 10.
   20÷2ENTER
4. Press ENTER to accept Xscl=1.
   ENTER
5. Finish the rest of the window: Ymin=0, Ymax=500, Yscl=100 and Xres=1, pressing ENTER after each.
   0ENTER500ENTER100ENTER1ENTER

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press GRAPH to draw the selected function in the window you set.
   GRAPH
2. Press an arrow key and the free-moving cursor appears in the middle of the screen, with its X and Y coordinates on the bottom line. Move it toward the top of the hill: 12 presses of ◀︎ and 19 of ▲︎. Each press moves one pixel, and X and Y update continuously.
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎
3. Now press TRACE. The cursor jumps onto the function. The function being traced is named in the top-left corner.
   TRACE
4. Press ◀︎ and ▶︎ to trace one X pixel at a time. Each step evaluates Y₁ at the new X. Watch Y rise as you move left.
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎
5. You can also tell TRACE exactly where to go. Start typing a number and an X= prompt appears at the bottom left. Enter your estimate, 3.8, and press ENTER.
   3.8ENTER
6. Use ◀︎ and ▶︎ to hunt for the highest Y. This is the maximum of Y₁ for the X pixel values. The true maximum may lie between two pixels, which is why you also tried the table and are about to use CALC.
   ◀︎

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Draw the graph.
   GRAPH
2. Press ZOOM to open the ZOOM menu. Like every TI-84 Plus menu, you can choose an item by its number, or by moving down to it and pressing ENTER.
   ZOOM
3. Press 2 for 2:Zoom In. The graph returns, and the cursor shape tells you a ZOOM instruction is waiting for you.
   2
4. Move the cursor near the maximum (12 presses of ◀︎ and 19 of ▲︎), then press ENTER. The new window is centered on the cursor.
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎▲︎ENTER
5. Both Xmax−Xmin and Ymax−Ymin shrank by a factor of 4, the default zoom factors: the width went from 10 to 2.5 and the height from 500 to 125. Press WINDOW to see the new window values.
   WINDOW
6. To get back to the previous graph, choose 1:ZPrevious from the ZOOM MEMORY menu: ZOOM ▶︎ 1.
   ZOOM▶︎1

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the CALCULATE menu and pick 4:maximum.
   2ndCALC
2. The graph comes back with a Left Bound? prompt. The trace cursor starts in the middle of the screen. Press ◀︎ 15 times to move to a point left of the hump (about X = 3.4), then press ENTER.
   4◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎◀︎ENTER
3. A marker at the top of the screen shows the bound you picked, and you are now asked for the Right Bound?. Press ▶︎ 10 times, to a point right of the hump (about X = 4.5), and press ENTER.
   ▶︎▶︎▶︎▶︎▶︎▶︎▶︎▶︎▶︎▶︎ENTER
4. Finally the calculator asks Guess?. Press ◀︎ seven times to get close to the top and press ENTER. (You can also type a guess such as 3.8 and press ENTER. Typed values work for the left and right bounds too.)
   ◀︎◀︎◀︎◀︎◀︎◀︎◀︎ENTER
5. The maximum is X ≈ 3.68 and Y ≈ 410.264, the same answer the table, the free-moving cursor and TRACE were closing in on. A box cut with X ≈ 3.68 cm squares holds about 410 cm³ of volume.

The lists are WLEFT, WRGHT, MLEFT and MRGHT (women or men, left or right hand).

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press STAT and choose 5:SetUpEditor. It is pasted on the home screen.
   STAT5
2. Type the four list names separated by commas: WLEFT,WRGHT,MLEFT,MRGHT. Every letter needs ALPHA, which the tutorial keys supply for you. Then press ENTER.
   ALPHAWALPHALALPHAEALPHAFALPHAT,ALPHAWALPHARALPHAGALPHAHALPHAT,ALPHAMALPHALALPHAEALPHAFALPHAT,ALPHAMALPHARALPHAGALPHAHALPHATENTER
3. Press STAT and choose 1:Edit. The editor now shows only your four lists, starting with WLEFT.
   STAT1
4. Type the ten left-hand scores for the women, pressing ENTER after each one to move to the next row.
   8ENTER9ENTER12ENTER11ENTER10ENTER8ENTER12ENTER7ENTER9ENTER11ENTER
5. Press ▶︎ to move to WRGHT and type the women's right-hand scores. (The 1 in the second row is an unusually low score. Remember it.)
   ▶︎4ENTER1ENTER8ENTER12ENTER11ENTER11ENTER13ENTER12ENTER11ENTER12ENTER
6. Move right again to MLEFT and type the twelve left-hand scores for the men.
   ▶︎7ENTER8ENTER7ENTER5ENTER7ENTER8ENTER11ENTER4ENTER10ENTER14ENTER13ENTER5ENTER
7. Finally, MRGHT: the men's right-hand scores.
   ▶︎12ENTER6ENTER12ENTER12ENTER7ENTER11ENTER12ENTER8ENTER12ENTER11ENTER9ENTER9ENTER

Plot 1 will show WLEFT and Plot 2 will show WRGHT.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the STAT PLOTS menu and choose 1:Plot1.
   2ndSTAT PLOT1
2. The first row is On Off. Make sure the cursor is on On (press ◀︎ to be safe) and press ENTER to turn the plot on.
   ◀︎ENTER
3. Move down to the Type row and press ▶︎ three times to the fourth icon, the modified box plot. Press ENTER to select it. Notice that the Ylist prompt is replaced by Freq, because a box plot only needs one list.
   ▼︎▶︎▶︎▶︎ENTER
4. Move down to Xlist. Press 2ndLIST to open the list NAMES menu and pick WLEFT (item 9; the lists L₁ to L₆ come first, then your lists in alphabetical order).
   ▼︎2ndLIST9
5. Now define Plot 2 the same way. Quit this screen with 2ndQUIT, open 2:Plot2, turn it on, choose the modified box plot, and use WRGHT (the last item in the NAMES menu, which you reach with 0) for the Xlist.
   2ndQUIT2ndSTAT PLOT2◀︎ENTER▼︎▶︎▶︎▶︎ENTER▼︎2ndLIST0
6. Make sure no functions are selected in the Y= editor (there are none on a fresh calculator, so the graph will show only the plots). Next press WINDOW and set Xscl=1 and Yscl=0. A Yscl of 0 turns off the y-axis tick marks, which would only clutter the box plots.
   2ndQUITWINDOW▼︎▼︎1ENTER▼︎▼︎0ENTER
7. Press ZOOM 9 (9:ZoomStat). The calculator fits the window to your data and draws both box plots.
   ZOOM9
8. Press TRACE. The cursor lands on the median of the first plot. The values are Med = 9.5.
   TRACE
9. Press ◀︎ to step to the first quartile, Q1 = 8.
   ◀︎
10. One more press of ◀︎ reaches the smallest value, minX = 7.
    ◀︎
11. Now press ▶︎ four times to walk through Q1, Med, Q3 and maxX. Q3 is 11 and the maximum is 12.
    ▶︎▶︎▶︎▶︎
12. Press ▼︎ to jump to the second plot (the women's right hand). Its median is 11, higher than 9.5 for the left hand.
    ▼︎
13. Step left with ◀︎: Q1 = 8, then minX = 4, and one more press lands on the outlier, the lone score of 1 that sits far below the rest.
    ◀︎◀︎◀︎
14. Which hand were the women more accurate with? According to the box plots, the right: its median (11) is higher than the left-hand median (9.5), even though one woman did very poorly with her right hand.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. In this setup Plot 1 already uses MLEFT and Plot 2 uses MRGHT (the men). Use ZOOM 9 to fit the window to the men's scores.
   ZOOM9
2. Press TRACE. The cursor starts on the median of the left-hand plot (Plot 1), which is 7.5. Use ◀︎ and ▶︎ to read the rest: Q1 = 6 and minX = 4.
   TRACE
3. Press ▼︎ to switch to the right-hand plot. The median is 11, and Q1 = 8.5. For the men the right hand scored clearly higher than the left (11 against 7.5), the opposite of what the earlier children's experiment found for boys.
   ▼︎
4. Now compare left hands only. Redefine Plot 1 to use WLEFT and Plot 2 to use MLEFT. (Choosing a new name from the NAMES menu replaces the old one.)
   2ndSTAT PLOT1▼︎▼︎2ndLIST92ndQUIT2ndSTAT PLOT2▼︎▼︎2ndLIST7
5. Zoom to fit and trace the plots. The women's left-hand median is 9.5, while the men's is 7.5. Women were the better left-hand guessers.
   ZOOM9TRACE
6. Finally compare right hands: Plot 1 uses WRGHT and Plot 2 uses MRGHT.
   2ndSTAT PLOT1▼︎▼︎2ndLIST02ndQUIT2ndSTAT PLOT2▼︎▼︎2ndLIST8ZOOM9TRACE
7. Trace both plots with ◀︎, ▶︎ and ▼︎. Both groups have the same median, 11, so with the right hand the two groups were about equally good. The women's box is shifted slightly lower because of that one very low score.
   ▶︎
8. In the original study, boys did not guess as well with their right hands and girls guessed equally well with either hand. That is not what these adult plots show. Is it because adults have learned to adapt, or because 22 people is too small a sample? Box plots are a great way to start a question like that, but they cannot settle it on their own.

Each step types one line of the program. Menu items are pasted from the menus, never typed letter by letter.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the PRGM menu, move to the NEW tab with ▶︎ ▶︎, and choose 1:Create New.
   PRGM▶︎▶︎1
2. Type the name SIERPINS, then press ENTER. The program editor opens with an empty first line.
   2ndA-LOCKSIERPINSENTER
3. Line 1: FnOff comes from VARS Y-VARS 4:On/Off… 2:FnOff. Then type a colon with ALPHA: and paste ClrDraw from 2ndDRAW (DRAW) 1:ClrDraw.
   VARS▶︎42ALPHA:2ndDRAW1ENTER
4. Line 2: PlotsOff is in the STAT PLOTS menu, 2ndSTAT PLOT 4:PlotsOff.
   2ndSTAT PLOT4ENTER
5. Line 3: AxesOff. Open the CATALOG with 2ndCATALOG, press ▼︎ 11 times to reach AxesOff and press ENTER to paste it (then ENTER again for the next line).
   2ndCATALOG▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎ENTERENTER
6. Line 4: 0→Xmin:1→Xmax. The window variables live under VARS 1:Window… (1:Xmin, 2:Xmax).
   0STO►VARS11ALPHA:1STO►VARS12ENTER
7. Line 5: 0→Ymin:1→Ymax (items 4 and 5 of the same menu).
   0STO►VARS14ALPHA:1STO►VARS15ENTER
8. Line 6: rand→X:rand→Y. rand is item 1 of the MATH PRB menu: press MATH ◀︎ to jump to the PRB tab, then 1.
   MATH◀︎1STO►ALPHAXALPHA:MATH◀︎1STO►ALPHAYENTER
9. Line 7: the loop. For( is item 4 of the PRGM CTL menu. Type K,1,3000).
   PRGM4ALPHAK,1,3000)ENTER
10. Line 8: rand→N.
    MATH◀︎1STO►ALPHANENTER
11. Line 9: If N≤1/3. If is PRGM 1, and ≤ is 2ndTEST (TEST) 6.
    PRGM1ALPHAN2ndTEST61÷3ENTER
12. Line 10: Then is PRGM 2.
    PRGM2ENTER
13. Line 11: .5X→X.
    .5ALPHAXSTO►ALPHAXENTER
14. Line 12: .5Y→Y.
    .5ALPHAYSTO►ALPHAYENTER
15. Line 13: End is PRGM 7.
    PRGM7ENTER
16. Line 14: If 1/3<N and N≤2/3. The word and is in the TEST LOGIC menu: 2ndTEST ▶︎ 1:and.
    PRGM11÷32ndTEST5ALPHAN2ndTEST▶︎1ALPHAN2ndTEST62÷3ENTER
17. Line 15: Then.
    PRGM2ENTER
18. Line 16: .5(.5+X)→X.
    .5(.5+ALPHAX)STO►ALPHAXENTER
19. Line 17: .5(1+Y)→Y.
    .5(1+ALPHAY)STO►ALPHAYENTER
20. Line 18: End.
    PRGM7ENTER
21. Line 19: If 2/3<N.
    PRGM12÷32ndTEST5ALPHANENTER
22. Line 20: Then.
    PRGM2ENTER
23. Line 21: .5(1+X)→X.
    .5(1+ALPHAX)STO►ALPHAXENTER
24. Line 22: .5Y→Y.
    .5ALPHAYSTO►ALPHAYENTER
25. Line 23: End.
    PRGM7ENTER
26. Line 24: Pt-On(X,Y) draws a point. It is on the POINTS tab of the DRAW menu: 2ndDRAW ▶︎ 1:Pt-On(.
    2ndDRAW▶︎1ALPHAX,ALPHAY)ENTER
27. Line 25: End closes the For( loop.
    PRGM7ENTER
28. Line 26: StorePic 6 is on the STO tab of the DRAW menu: 2ndDRAW ▶︎ ▶︎ 1:StorePic. Type 6 after it.
    2ndDRAW▶︎▶︎16
29. The program is finished. Leave the editor with 2ndQUIT (QUIT). Now run it: press PRGM and choose SIERPINS from the EXEC menu (item 1). It is pasted on the home screen as prgmSIERPINS.
    2ndQUITPRGM1
30. Press ENTER to run it. The calculator plots 3000 points, which takes a moment, and the finished triangle stays on the graph screen.
    ENTER
31. Press CLEAR to return to the home screen, which reports Done.
    CLEAR
32. The picture was saved in Pic6. Bring it back with RecallPic 6: 2ndDRAW ▶︎ ▶︎ 2:RecallPic, then 6 and ENTER. You will see the Sierpinski triangle.
    2ndDRAW▶︎▶︎26ENTER
33. The program turned the graph axes off. To put them back, open the FORMAT screen with 2ndFORMAT, press ▼︎ three times to the AxesOn AxesOff row, and press ENTER to choose AxesOn.
    2ndFORMAT▼︎▼︎▼︎ENTER

This is a longer program. Each step enters one line. Items from the CATALOG are pasted by moving down to them; everything else is in the menus noted in the step.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Create a new program: PRGM ▶︎ ▶︎ 1:Create New.
   PRGM▶︎▶︎1
2. Name it GUESS and press ENTER.
   2ndA-LOCKGUESSENTER
3. Line 1: PlotsOff :Func. PlotsOff is 2ndSTAT PLOT 4. Func is in the CATALOG. Press 2ndCATALOG, then ALPHAF to jump to the entries starting with F, and ▼︎ 16 times to Func.
   2ndSTAT PLOT4ALPHA:2ndCATALOGALPHAF▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎▼︎ENTERENTER
4. Line 2: FnOff :Radian. Paste FnOff from VARS Y-VARS 4:On/Off… 2:FnOff. In the CATALOG, ALPHAR jumps to Radian.
   VARS▶︎42ALPHA:2ndCATALOGALPHARENTERENTER
5. Line 3: ClrHome is in the I/O menu: PRGM ▶︎ 8:ClrHome.
   PRGM▶︎8ENTER
6. Line 4: "Asin(BX)"→Y1. Quotation marks come from ALPHA". The secret function is stored as text. Paste Y₁ with the YVAR shortcut menu, ALPHAF4 1.
   ALPHA"ALPHAASINALPHABX,T,θ,n)ALPHA"STO►ALPHAF41ENTER
7. Line 5: "Csin(DX)"→Y2.
   ALPHA"ALPHACSINALPHADX,T,θ,n)ALPHA"STO►ALPHAF42ENTER
8. Line 6: GraphStyle(1,1). GraphStyle( is item H of the PRGM CTL menu. The first number is the function, the second is the style (1 = line).
   PRGMALPHAH1,1)ENTER
9. Line 7: GraphStyle(2,5). Style 5 is path.
   PRGMALPHAH2,5)ENTER
10. Line 8: FnOff 2 hides Y₂ for now.
    VARS▶︎422ENTER
11. Line 9: randInt(1,10)→A. randInt( is item 5 of the MATH PRB menu (MATH ◀︎ 5).
    MATH◀︎51,10)STO►ALPHAAENTER
12. Line 10: randInt(1,10)→B.
    MATH◀︎51,10)STO►ALPHABENTER
13. Line 11: 0→C:0→D.
    0STO►ALPHACALPHA:0STO►ALPHADENTER
14. Line 12: -2π→Xmin. The π symbol is 2ndπ and the minus is the negation key (−). Xmin is VARS 1:Window… 1:Xmin.
    (−)22ndπSTO►VARS11ENTER
15. Line 13: 2π→Xmax.
    22ndπSTO►VARS12ENTER
16. Line 14: π/2→Xscl.
    2ndπ÷2STO►VARS13ENTER
17. Line 15: -10→Ymin.
    (−)10STO►VARS14ENTER
18. Line 16: 10→Ymax.
    10STO►VARS15ENTER
19. Line 17: 1→Yscl.
    1STO►VARS16ENTER
20. Line 18: DispGraph is in the I/O menu: PRGM ▶︎ 4.
    PRGM▶︎4ENTER
21. Line 19: Pause is PRGM 8 in the CTL menu.
    PRGM8ENTER
22. Line 20: FnOn 2 turns Y₂ back on (VARS Y-VARS 4:On/Off… 1:FnOn).
    VARS▶︎412ENTER
23. Line 21: Lbl Z is PRGM 9. Type the label Z.
    PRGM9ALPHAZENTER
24. Line 22: Prompt C,D is PRGM ▶︎ 2.
    PRGM▶︎2ALPHAC,ALPHADENTER
25. Line 23: DispGraph.
    PRGM▶︎4ENTER
26. Line 24: Pause.
    PRGM8ENTER
27. Line 25: If C=A. The equals sign is 2ndTEST (TEST) 1.
    PRGM1ALPHAC2ndTEST1ALPHAAENTER
28. Line 26: Text(1,1,"C IS OK"). Text( is 2ndDRAW (DRAW) item 0. Spaces inside the quotation marks are typed with ALPHA␣. The first two numbers are the pixel row and column where the text appears.
    2ndDRAW01,1,ALPHA"ALPHACALPHA␣ALPHAIALPHASALPHA␣ALPHAOALPHAKALPHA")ENTER
29. Line 27: If C≠A. The not-equal sign is 2ndTEST 2.
    PRGM1ALPHAC2ndTEST2ALPHAAENTER
30. Line 28: Text(1,1,"C IS WRONG").
    2ndDRAW01,1,ALPHA"ALPHACALPHA␣ALPHAIALPHASALPHA␣ALPHAWALPHARALPHAOALPHANALPHAGALPHA")ENTER
31. Line 29: If D=B.
    PRGM1ALPHAD2ndTEST1ALPHABENTER
32. Line 30: Text(1,50,"D IS OK") (the column is 50 so the text appears on the right half of the screen).
    2ndDRAW01,50,ALPHA"ALPHADALPHA␣ALPHAIALPHASALPHA␣ALPHAOALPHAKALPHA")ENTER
33. Line 31: If D≠B.
    PRGM1ALPHAD2ndTEST2ALPHABENTER
34. Line 32: Text(1,50,"D IS WRONG").
    2ndDRAW01,50,ALPHA"ALPHADALPHA␣ALPHAIALPHASALPHA␣ALPHAWALPHARALPHAOALPHANALPHAGALPHA")ENTER
35. Line 33: DispGraph.
    PRGM▶︎4ENTER
36. Line 34: Pause.
    PRGM8ENTER
37. Line 35: If C=A and D=B. and is 2ndTEST ▶︎ 1.
    PRGM1ALPHAC2ndTEST1ALPHAA2ndTEST▶︎1ALPHAD2ndTEST1ALPHABENTER
38. Line 36: Stop is PRGM F.
    PRGMALPHAFENTER
39. Line 37: Goto Z is PRGM 0. Type the label Z.
    PRGM0ALPHAZ
40. Quit the editor with 2ndQUIT and run the program: press PRGM and choose GUESS.
    2ndQUITPRGM1
41. Press ENTER. The program picks A and B, sets the window and shows the secret curve A sin(BX). It then waits at Pause. Study the picture: the height of the highest peak is A, and the number of full waves between x = 0 and x = 2π (the right half of the screen) is B.
    ENTER
42. Press ENTER to continue. Your guess is turned on and the program asks C=?. Make a guess of C = 5 and press ENTER. Then it asks D=?. Guess D = 3 and press ENTER.
    ENTER5ENTER3ENTER
43. The program draws both curves and pauses. Press ENTER. Verdicts such as C IS WRONG or D IS OK are printed on the graph, and it pauses again.
    ENTER
44. Press ENTER once more. Unless you were lucky, you go back to Lbl Z and the C=? prompt. For the demonstration, cheat: answer with the secret variables themselves. Type A and press ENTER for C, then B and ENTER for D.
    ENTERALPHAAENTERALPHABENTER
45. Two more presses of ENTER (for the graph and the verdict). Both coefficients now match, C IS OK and D IS OK appear, and Stop ends the program. Run it again any time for a new random puzzle.
    ENTERENTER

Both the integral and the derivative templates are on the MATH menu (MATH 9:fnInt( and 8:nDeriv() and on the FUNC shortcut menu (ALPHAF2).

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press WINDOW and set Xmin=.01, Xmax=10, Xscl=1, Ymin=-1.5, Ymax=2.5, Yscl=1 and Xres=3. A larger Xres makes the graph faster to draw, which matters because every point needs a numerical integral.
   WINDOW.01ENTER10ENTER1ENTER(−)1.5ENTER2.5ENTER1ENTER3ENTER
2. In the Y= editor, press MATH 9 to paste the integral template with empty boxes. Fill in the boxes one by one, moving with ▶︎: the lower limit is 1 and the upper limit is X, the function is 1/T, and the variable of integration is T. (The last box contains X already, so erase it with DEL and type T.)
   Y=MATH91▶︎X,T,θ,n▶︎1÷ALPHAT▶︎DELALPHATENTER
3. In Y₂ type ln(X). Y₁ keeps the line style. Give Y₂ the path style: press 2nd◀︎ to jump to the start of the line, ◀︎ twice to reach the style icon, and ENTER four times.
   LNX,T,θ,n)2nd◀︎◀︎◀︎ENTERENTERENTERENTER
4. Press TRACE. The cursor starts on Y₁. The values of X and Y are shown at the bottom, and the integral Y is almost exactly ln(X).
   TRACE
5. Press ▼︎ to switch to Y₂, ln(X), at the same X. Use ◀︎ and ▶︎ to move around, and press ▲︎ and ▼︎ to compare the two functions at several values of X. They agree to many digits: the integral really is ln(x).
   ▼︎
6. Now check the derivative. In the Y= editor, switch off Y₁ and Y₂ (highlight each = with ◀︎ and press ENTER), and move down to Y₃.
   Y=◀︎ENTER▼︎ENTER▼︎
7. In Y₃ paste the derivative template with MATH 8: nDeriv(. Put Y₁ in the function box (use ALPHAF4 1) and X in the box for the point at which to differentiate. Y₃ is the numerical derivative of the integral.
   MATH8ALPHAF41▶︎X,T,θ,nENTER
8. In Y₄ type the function 1/X.
   1÷X,T,θ,n
9. Make Y₄ thick so the two graphs can be told apart: 2nd◀︎, ◀︎ twice, and one press of ENTER.
   2nd◀︎◀︎◀︎ENTER
10. Press TRACE and use the arrow keys to compare Y₃ and Y₄. They match: the derivative of the integral of 1/t is 1/x.
    TRACE

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Open the Y= editor and move to Y₅ (four presses of ▼︎). (A fresh calculator has no other functions switched on.) Press MATH 9 for the integral template. In the lower-limit box type the list {−2,0,2}, using 2nd{ for the left brace and 2nd} for the right brace.
   Y=▼︎▼︎▼︎▼︎MATH92nd{(−)2,0,22nd}
2. Move to the next box, the upper limit, and type X. Move on to the function box and type T². Move on to the variable box, replace the X with T, and press ENTER.
   ▶︎X,T,θ,n▶︎ALPHATx²▶︎DELALPHATENTER
3. Press ZOOM 6 (6:ZStandard). The graphs appear gradually, because each point needs a calculation of the integral. That can take some time.
   ZOOM6
4. Press TRACE and then ▲︎ / ▼︎ to move between the three curves. They look identical, only shifted up or down by a constant.
   TRACE
5. Open the Y= editor, go below Y₅ and define Y₆ as the numerical derivative of Y₅: MATH 8 nDeriv(, function Y₅ (from ALPHAF4 5) and the point X.
   Y=▼︎▼︎▼︎▼︎▼︎MATH8ALPHAF45▶︎X,T,θ,nENTER
6. Press TRACE. Although the three graphs of Y₅ are different, they share the same derivative, which is the single parabola x² that Y₆ shows. The constant that distinguishes antiderivatives disappears when you differentiate.
   TRACE

The equation is stored as 0 = A − N·B²·sin(π/N)·cos(π/N). The solver lists the variables in the order A, N, B.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press MATH and choose B:Solver… (type ALPHAB). The equation editor opens with eqn:0= already on screen. (If you see the interactive solver editor instead, press ▲︎ to get back to the equation editor.)
   MATHALPHAB
2. Because 0= is already there, type only the rest of the formula: A−NB²sin(π/N)cos(π/N). Use 2ndπ for π, and the divide key for the fractions.
   ALPHAA−ALPHANALPHABx²SIN2ndπ÷ALPHAN)COS2ndπ÷ALPHAN)
3. Press ENTER. The interactive solver editor lists A, N and B, each equal to 0, and a bound line at the bottom.
   ENTER
4. Find the area of a square (N = 4) whose corners are 6 cm from the center (B = 6). The cursor is on A. Press ▼︎ to move to N and type 4, press ENTER to move to B, then type 6 and press ENTER.
   ▼︎4ENTER6ENTER
5. Press ▲︎ three times to put the cursor back on A, then press ALPHASOLVE (the green SOLVE legend over ENTER). The solution for A is displayed: 72 cm², which is exactly 2B².
   ▲︎▲︎▲︎ALPHASOLVE
6. Now work backward: what B do you need to get a given area with several numbers of sides? Type 200 for A, press ENTER, then 6 for N, press ENTER. The cursor is now on B. Press ALPHASOLVE to solve for B. A hexagon of area 200 needs B ≈ 8.7738.
   200ENTER6ENTERALPHASOLVE
7. Change N to 8: press ▲︎ to move to N, type 8 and ENTER, then press ALPHASOLVE (the cursor has moved down to B). For N = 8, B ≈ 8.4090.
   ▲︎8ENTERALPHASOLVE
8. Do the same for N = 9 (B ≈ 8.3152) …
   ▲︎9ENTERALPHASOLVE
9. … and for N = 10 (B ≈ 8.2494). The more sides there are, the smaller the radius needed for the same area, because the polygon gets closer to a circle.
   ▲︎10ENTERALPHASOLVE
10. Now fix B = 6 and find the area for larger and larger N. Type 6 for B and press ENTER. The circle of radius 6 has area π·6² ≈ 113.097, which is what the polygons should approach. Press ▲︎ three times to reach A, then move down to N, type 10, press ENTER, go back up to A (▲︎ twice), and solve with ALPHASOLVE.
    6ENTER▲︎▲︎▲︎▼︎10ENTER▲︎▲︎ALPHASOLVE
11. Repeat for N = 100. From A, press ▼︎, type 100, ENTER, then ▲︎ ▲︎ and ALPHASOLVE.
    ▼︎100ENTER▲︎▲︎ALPHASOLVE
12. N = 150.
    ▼︎150ENTER▲︎▲︎ALPHASOLVE
13. N = 1000.
    ▼︎1000ENTER▲︎▲︎ALPHASOLVE
14. And N = 10000. The area is 113.0973, matching πB² = 113.0973 to the digits shown. As N gets large, A approaches πB².
    ▼︎10000ENTER▲︎▲︎ALPHASOLVE

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. The variable B is still 6 from the solver (this tutorial stores it again for you). Press WINDOW and set Xmin=0, Xmax=200, Xscl=10, Ymin=0, Ymax=150, Yscl=10 and Xres=1.
   WINDOW0ENTER200ENTER10ENTER0ENTER150ENTER10ENTER1ENTER
2. Open the Y= editor and enter the area, using X in place of N: Y₁=X·B²·sin(π/X)·cos(π/X).
   Y=X,T,θ,nALPHABx²SIN2ndπ÷X,T,θ,n)COS2ndπ÷X,T,θ,n)ENTER
3. In Y₂ type the area of the circle, πB².
   2ndπALPHABx²
4. Give Y₂ the dot style so it looks like an asymptote: 2nd◀︎ to the start of the line, ◀︎ twice onto the style icon, and ENTER six times.
   2nd◀︎◀︎◀︎ENTERENTERENTERENTERENTERENTER
5. Press TRACE and type 100, then ENTER: for 100 sides the area is 113.02. The curve is already almost on top of the line.
   TRACE100ENTER
6. Now 150.
   150ENTER
7. And 188. As X increases, Y converges to π·6² ≈ 113.097. The line Y₂ = πB² is a horizontal asymptote of Y₁: the area of a regular N-sided polygon with radius r approaches the area of the circle, πr², as N gets large.
   188ENTER

The loan has 360 monthly payments (30 years × 12) of 800 at an annual rate of 8%.

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. Press MODE and set the fixed-decimal mode to 2 decimal places: on the second row, press ▶︎ three times to reach 2 and ENTER. Leave the other settings at their defaults. Then leave the MODE screen with 2ndQUIT.
   MODE▼︎▶︎▶︎▶︎ENTER2ndQUIT
2. Press APPS and choose 1:Finance…, then 1:TVM Solver…. (Pressing ENTER twice does both.) The TVM Solver is a list of the variables in a loan.
   APPSENTERENTER
3. Enter the loan: N=360 and I%=8. Press ENTER to skip PV (this is the number you want to find), type 800 for PMT, 0 for FV (a loan is worth nothing once it is paid off), and 12 for both P/Y and C/Y (payments and compounding periods per year).
   360ENTER8ENTERENTER800ENTER0ENTER12ENTER12ENTER
4. Positive 800 shows the payment as a cash inflow (the lender receives it), which makes the payment amounts show up as positive numbers on the graph later. PMT:END is selected, because a payment is due at the end of each period.
5. Move the cursor onto the PV= line by pressing ▲︎ five times, and press ALPHASOLVE (SOLVE). The present value, the mortgage amount, is displayed in the PV= line: −109,026.80. (It is negative because the lender hands that money over.)
   ▲︎▲︎▲︎▲︎▲︎ALPHASOLVE

▶ Try this step-by-step on the online TI-84 Plus — keys light up as you go

1. The calculator is already in fixed 2-decimal mode and the TVM variables hold the loan (set up as in the previous tutorial). Press MODE and set Par and Simul.
   MODE▼︎▼︎▼︎▶︎ENTER▼︎▼︎▶︎ENTER2ndQUIT
2. Open the Y= editor and enter X₁ᴛ=T. In this mode X,T,θ,n types T. Then, for Y₁ᴛ, paste ΣPrn( from the FINANCE menu (APPS ENTER, then item 0) and complete it as ΣPrn(T,T).
   Y=X,T,θ,nENTERAPPSENTER0X,T,θ,n,X,T,θ,n)ENTER
3. For the second curve enter X₂ᴛ=T and Y₂ᴛ=ΣInt(T,T). ΣInt( is item ALPHAA in the FINANCE menu.
   X,T,θ,nENTERAPPSENTERALPHAAX,T,θ,n,X,T,θ,n)ENTER
4. For the third, enter X₃ᴛ=T and Y₃ᴛ=Y₁ᴛ+Y₂ᴛ. Paste the Y-variables from VARS Y-VARS 2:Parametric… (item 2 is Y₁ᴛ, item 4 is Y₂ᴛ).
   X,T,θ,nENTERVARS▶︎22+VARS▶︎24
5. Press WINDOW and set Tmin=1, Tmax=360, Tstep=12, Xmin=0, Xmax=360, Xscl=10, Ymin=0, Ymax=1000 and Yscl=100.
   WINDOW1ENTER360ENTER12ENTER0ENTER360ENTER10ENTER0ENTER1000ENTER100ENTER
6. Press TRACE. After the graph is drawn, type 240 and press ENTER to move the trace cursor to T = 240, the 240th payment (20 years of payments).
   TRACE240ENTER
7. The graph shows that for the 240th payment (X = 240), 358.03 of the 800 goes to principal (Y = 358.03). The sum of principal and interest (Y₃ᴛ = Y₁ᴛ + Y₂ᴛ) is always 800.
8. Press ▼︎ to move the cursor onto the interest curve (X₂ᴛ, Y₂ᴛ) and enter 240 again.
   ▼︎240ENTER
9. The graph shows that for the 240th payment, 441.97 of the 800 payment is interest (Y = 441.97). 358.03 + 441.97 = 800.
10. Check the figures on the home screen. Press 2ndQUIT, then APPS ENTER and choose 9:bal(. The balance after payment 239 is the amount owed at the start of payment 240. Type 239) and press ENTER. The balance is about −66,295.3.
    2ndQUITAPPSENTER9239)ENTER
11. One month of interest on that balance is Ans×(.08/12). Start the line with ×; the calculator inserts Ans for you. The result is −441.97, matching the graph (the sign is negative because the balance is negative).
    ×(.08÷12)ENTER
12. A closing question: at which monthly payment does the principal part first surpass the interest part? Each month the interest part of the payment is the balance times 0.08/12, so the interest drops below half of 800, that is below 400, when the balance falls under 60,000. Try payment 257: the balance after payment 256, bal(256), times .08/12 is the interest part of payment 257.
    APPSENTER9256)×(.08÷12)ENTER
13. Interest is 399.15, so principal is 800 − 399.15 = 400.85, which is larger. For payment 256 the interest part is 401.81, which is still larger than the 398.19 of principal. So the principal allocation first surpasses the interest allocation with the 257th payment.