TI-83 Graphing Calculator Notes

Chapter 3: Texas Instruments TI-83

3.1 Getting Started with the TI-83

• 3.1.1 Basics:

  • Press the ON key to start using the TI-83.
  • Adjust display contrast by pressing 2nd, then holding the down arrow key (▼) to lighten or the up arrow key (▲) to darken.
    • An integer between 0 (lightest) and 9 (darkest) will appear in the upper right corner.
  • Turn off the calculator by pressing 2nd and then OFF to conserve battery power.
  • Check settings by pressing MODE.
    • Use arrow keys to move the cursor to a setting and press ENTER to select a new setting.
    • Initial settings should be: normal display, floating decimals, radian measure, function graphs, connected lines, sequential plotting, real numbers, and full screen display.
  • Details on alternative options will be provided later.
  • Press CLEAR to leave the MODE menu.

• 3.1.2 Editing:

  • The TI-83 can display up to eight lines at once.
  • Example: Type the sum: 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20
  • Press ENTER to see the answer (210).
  • To correct mistakes, use the arrow keys to move the cursor to the error.
    • Type over the incorrect character.
    • Press 2nd INS to insert characters (cursor changes to a blinking underline).
    • Press DEL to delete characters.
  • Technology Tip:
    • Press ▲ (up arrow key) to move quickly to the beginning of an expression.
    • Press ▼ (down arrow key) to jump to the end of that expression.
  • To edit the previous expression (even after pressing ENTER), press 2nd ENTRY to recall it.
    • Press 2nd ENTRY repeatedly to access earlier entries.
  • Technology Tip: Use editing to simplify evaluating a formula for different variable values.
    • Formula for balance in an investment account: P = (1 + \frac{r}{n})^{nt}, where:
      • P = principal.
      • r = rate of interest (as a decimal).
      • n = number of times interest is compounded each year.
      • t = number of years.
    • Example: $5000 in the account, 8.5% interest compounded annually.
      • Balance after 3 years:
        • Keystrokes: 5000 (1+.085) ^ 3 ENTER
          • Balance: $6386.45
      • Balance after 5 years:
        • Keystrokes: 2nd ENTRY ◄ 5 ENTER
          • Balance: $7518.28
      • Balance after 10 years:
        • Keystrokes: 2nd ENTRY ◄ 10 ENTER
          • Balance: $11,304.92
    • To change the interest rate to 7.5% after 5 years, press 2nd ENTRY DEL 5 ◄ 7 ENTER.

• 3.1.3 Key Functions:

  • Most keys offer access to multiple functions.
  • The primary function is indicated on the key itself.
  • To access the second function (left above a key), press 2nd (cursor changes to a blinking *) and then the key.
    • Example: Calculate \sqrt{25} by pressing 2nd √ 25 ENTER.
  • Technology Tip: TI-83 automatically places a left parenthesis after many functions (LOG, 2nd 10x, LN, 2nd ex, SIN, COS, TAN, 2nd √).
    • A matching right parenthesis is not needed if no other calculations follow.
  • To use a letter or character printed to the right above a key, press ALPHA (cursor changes to a blinking A) and then the key.
    • Example: Use the letter K by pressing ALPHA K.
  • For multiple letters in a row, press 2nd A-LOCK (like CAPS LOCK on a computer).
    • Press ALPHA again to restore keys to their primary functions.

• 3.1.4 Order of Operations:

  • TI-83 follows standard algebraic rules.
  • Calculations are performed from left to right, outwards from inner parentheses.
  • Order: powers/roots, then multiplications/divisions, then additions/subtractions.
  • Examples:
    • 7 - 5 x 3 ENTER results in -8
    • (7 - 5) x 3 ENTER results in 6
    • 120 - 10^2 : 120 - 10 x^2 ENTER results in 20.
    • (120 - 10)^2: (120 - 10) x^2 ENTER results in 12100.
    • 24 \div 2^3: 24 ÷ 2 ^ 3 ENTER results in 3.
    • (24 \div 2)^3: (24 ÷ 2) ^ 3 ENTER results in 1728.
    • (7 - (-5)) \times (-3): (7 - (-) 5) × (-) 3 ENTER results in -36.

• 3.1.5 Algebraic Expressions and Memory:

  • Calculator can evaluate expressions after assigning values to variables.
  • Example: Store 200 in memory location N by pressing 200 STO ALPHA N ENTER.
  • Evaluate \frac{N(N+1)}{2} by typing ALPHA N (ALPHA N+1) ÷ 2 ENTER}.
  • For N=200, \frac{N(N+1)}{2} = 20100
  • Reveal the content of the memory location by just entering the letter and pressing ENTER.

• 3.1.6 Repeated Operations with ANS:

  • The result of the last calculation is stored in ANS.
  • Example: 30 + 15 ENTER (45 is displayed). Then press 2nd ANS ÷ 9 ENTER to get 5.
  • For functions like division, press the function key after the argument.
    • Instead of 2nd ANS ÷ 9, press ÷ 9.
  • Works for: +, -, ×, /, x², x-1
  • Example: Calculate earnings at $5.85 per hour.
    • 8-hour day: 5.85 x 8 ENTER results in $46.80
    • 5-day week: x 5 ENTER results in $234.
    • 52-week year: x 52 ENTER results in $12,168.

• 3.1.7 The MATH Menu:

  • Operators and functions are available from keys or via the 2nd key.
  • Direct access to arithmetic operations, trigonometric functions, exponential and logarithmic functions, and the constant \pi.
  • TI-83 requires the argument of a function after the function.
    • Example: \sqrt{16} is 2nd √ 16 ENTER.
  • Examples:
    • \sqrt{3^2 + 4^2}: Key strokes: 2nd √(3x²+4x²) ENTER -> Display: 5
    • 2 + \frac{1}{3}: Key strokes: 2 + 3x-1 ENTER -> Display: 2.333333333
    • log 200: Key strokes: LOG 200 ENTER -> Display: 2.301029996
    • 2.34 \times 10^5: Key strokes: 2.34 x 2nd 10 ^ 5 ENTER -> Display: 234000
  • Additional math operations are available in the MATH menu.
    • Calculate \sqrt[3]{7} by pressing MATH then either 4[$\sqrt[3]{}$] or ▼ ▼ ▼ ENTER; finally press 7: Display 1.912931183.
  • Factorial (!): product of integers from 1 up to the given integer.
    • 4! = 1 \times 2 \times 3 \times 4 = 24
    • Calculate 4! by pressing 4 MATH 4 [!] ENTER or 4 MATH ▼▼▼ ENTER ENTER.
    • Sub-menus can be selected using or .

• Complex Numbers:

  • TI-83 can perform calculations with complex numbers.
  • Enter the imaginary unit i by pressing 2nd i.
  • Example: Divide 2 + 3i by 4 – 2i: ( 2 + 3 2nd i) ÷ ( 4 - 2 2nd i) ENTER. Result: 0.1 + 0.8i.
  • Find the complex conjugate of 4 + 5i: MATH ►► ENTER 4 + 5 2nd i ENTER. Result: 4 - 5i.

3.2 Functions and Graphs.

• 3.2.1 Evaluating Functions:

  • Example: Monthly salary of $1975 + 10% commission.

  • W = 1975 + 0.10x, where x = sales in dollars.

  • Access the function editing screen by pressing Y=.

    • Enter the expression by pressing 1975 + .10 X,T,θ,n.
      • The X,T,θ,n key enters the variable X without using ALPHA.
    • Press 2nd QUIT to return to the main calculations screen.
    • Assign the value 2230 to x by these keystrokes : 2230 STO X,T,θ,n.
    • Evaluate Y_1 by pressing VARS ► 1 [Function] 1[Y_1] ENTER.

  • Values can be directly evaluated in a function: example evaluate Y_1(2230) by pressing VARS ► 1[Function] 1[Y_1] (2230) ENTER.
    To simplify the steps to change a previously entered command make sure to use: 2nd ENTRY

  • Create Table of Values for the function: Press 2nd TBLSET to set up the table.

    • Move the blinking cursor onto Ask beside Indpnt:, then press ENTER.
    • Now to View the Table Press 2nd TABLE to enter the Values
    • You can enter additional values and the calculator will automatically complete the table with the corresponding values of Y_1.
    • Press 2nd QUIT to leave the TABLE screen.

  • Technology Tip: TI-83 does not require multiplication to be expressed between variables, so xxx means x^3. It is often easier to press two or three x's together than to search for the square key or the powers key.

• 3.2.2 Functions in a Graph Window:

  • Press GRAPH to see the graph.
  • Example: Graph y=-x^3+4x:
    • Press Y= and delete anything in the lines.
    • Press (-) X,T,θ,n ^ 3 + 4 X,T,θ,n to enter the function.
    • Now press GRAPH .It will change the window with the graph of y = -x + 4x .
    • While calculating the coordinates the TI-83 indicator will display a busy indicator at the top right of the graph window.
  • Technology Tip: Split screen to see function and graph simultaneously:
    • Open the MODE menu, move the cursor down to the last line, and select Horiz screen.
  • The split screen is also useful when you need to do some calculations as you trace along a graph.
  • Restore the TI-83 to Full screen.
  • The displayed rectangular part is called a viewing rectangle.
    • You can easily change the viewing rectangle to enhance your investigation of a graph.
    • The viewing rectangle in Figure 3.13 shows the part of the graph that extends horizontally from -10 to 10 and vertically from -10 to 10.
    • Press WINDOW to see information about your viewing rectangle.
  • The variables Xmin and Xmax are the minimum and maximum x-values of the viewing rectangle; Ymin and Ymax are the minimum and maximum y-values.
  • Xscl and Yscl set the spacing between tick marks on the axes.
  • Xres sets pixel resolution (1 through 8) for function graphs.
  • Technology Tip: Small Xres values improve graph resolution but may cause the TI-83 to draw graphs more slowly.
  • Enter a new value to over-write a previous value and then press ENTER.
    • Remember that a minimum must be less than the corresponding maximum or the TI-83 will issue an error message.
    • Also, remember to use the (-) key, not - (which is subtraction), when you want to enter a negative value. Figures 3.13-14, 3.15-16, and 3.17-18 show different WINDOW screens and the corresponding viewing rectangle for each one.
  • Quickly initialize the viewing rectangle to the standard viewing rectangle (Figure 3.14), press ZOOM 6[ZStandard].
  • Quickly set the viewing rectangle to a square (Figure 3.15) press , ZOOM 5[ZSquare].
  • Sometimes you may wish to display grid points corresponding to tick marks on the axes. This and other graph format options may be changed by pressing 2nd FORMAT.
  • Use arrow keys to move the blinking cursor to GridOn; press ENTER and then GRAPH to redraw the graph.
  • In general, you'll want the grid turned off, so do that now by pressing 2nd FORMAT, use the arrow keys to move the blinking cursor to GridOff, and press ENTER and CLEAR.
  • Technology Tip: : The style of your graph can be changed by changing the icon to the left of on the screen. To change the icon press and then repeatedly to scroll through the different styles available.

• 3.2.3 Graphing Step and Piecewise-Defined Functions:

  • The greatest integer function, written ([[x]]), gives the greatest integer less than or equal to a number x.
  • On the TI-83, the greatest integer function is called int and is located under the NUM sub-menu of the MATH menu (Figure 3.5).
  • From the home screen, calculate [[6.78]] = 6 by pressing MATH 5[int(] 6.78 ENTER.
  • To graph y=[[x]], go into the Y= menu, move beside Y₁ and press CLEAR MATH 5[int] X,T,0,n GRAPH.
  • The true graph of the greatest integer function is a step graph, like the one in Figure 3.22.
  • For the graph of y=[[x]], a segment should not be drawn between every pair of successive points. You can change from Connected line to Dot graph. Select whichever graph type you require; press ENTER to put it into effect, and GRAPH to see the result.
  • Make sure to change your TI-83 back to Connected line, since most of the functions that you will be graphing should be viewed this way.
  • The TI-83 can graph piecewise-defined functions by using the options in the TEST menu (Figure 3.23) that is displayed by pressing 2nd TEST. Each TEST function returns the value 1 if the statement is true, and the value 0 if the statement is false.
  • For example, to graph the function

  f(x)=\begin{cases} x^{2}+2, & x<0 \ x-1, & x \geq 0 \end{cases}
  enter the following keystrokes: Y= (X,T,θ,n x² + 1) (X,T,θ,n 2nd TEST 5[<] 0 ) + (X,T,θ,n − 1 ) ( X,T,θ,n 2nd TEST 4[≥] 0) (Figure 3.24). Then press GRAPH to display the graph.

• 3.2.4 Graphing a Circle:

  • Use the technique for graphs that are not functions but can be