Texas Instruments TI-84 Plus Manual

The Most Used Keys

Setting up the TI-84 Calculator

• Turn the calculator on
  • Press:
• Turn the calculator off:
• Press:
• Set the mode for use:
  • Press:
  • Setup should look like the image below:
    
  • Use the arrow keys to move around and set the mode the way you want.
    
    Start with:
    • 2^nd line: NORMAL
    • 3^rd line indicates the number of digits to display when doing calculations. Initially, use FLOAT
    • 4^th line: DEGREE
    • 5^th line: FUNCTION
    • 8^th line: REAL
    • Other lines: as shown in the illustration.
  • Get out of the mode screen by pressing:
    • Note that you can get out of any screen and back to the standard calculator screen by pressing:
• Reset the graphing window by pressing 6:ZStandard.
• Reset any previously entered functions for graphing by pressing
  • If more than one function is shown, move to each one using the up and down arrow keys and press
• Clear previously entered lines that may be in the display or above it:
  • Press 3:Clear Entries ( key is over the + sign).
  • Press to blank-out the display if you prefer a clean display.

Tips for Using the TI-84 Plus

• and are your friends. They will get you out of most situations you don't know how to get out of otherwise.
• To break or interrupt an extended calculation or program in a loop, press and hold for several seconds un l the error message [Break] appears on the screen.
• Use lots of parentheses in complex inputs. It's be er to use too many parentheses than not enough. It is a common error for students to enter expressions without enough parentheses to allow the TI-84 to provide the answer desired by the student.
• Clearing the contents on a line or the screen
  • Press: once to clear a line. Press twice to clear the en re screen.
• Delete a character.
  • Move the cursor over the character with the arrow keys and press
• Insert a character.
  • Move the cursor over a character with the arrow keys and press
  • The new character will be inserted before the one the cursor was on.
• Use memory to store values you will use o en. You can give values one-letter variable names by pressing "value you want stored" followed by a one-le er variable name.
  • To use the stored variable, simply refer to it by name.
    
• When using variables, be careful that they are what you entered and not what a prior user entered. If you want to zero out a variable, store 0 in the variable name.
• To use information previously entered, you may use the arrow keys to move up or down to the previous expression or value and press the enter key at any point in a calculation.
  • See the screen sequence below, in which I calculate areas for a couple of circles, then mul ply the first of those by 7 to get the volume of a cylinder.
  • Notice in the second screen that I have moved up from the line I was working in to capture the result of a previous calculation, which is highlighted in blue by the calculator.
  • Press the enter key after highlighting what you want to use in order to enter it in the calculator, and continue working.
    
• To use the answer in the previous line, press (see below left) or simply start typing an opera on or function key and the TI-84 will assume you wanted to start the expression with the previous answer (see below right).
  
• The absolute value function is located at: NUM 1: abs(
• If you define variables for the tasks at hand, you might want to zero them out at the end of your session. For example, if you used the variable Z, zero it out by pressing: 0 at the end of your session.
• When using the trace function to find a specific point in a graph, remember that the x- and y-values produced may not be very accurate. If accuracy is paramount, use the calculation key (over the trace key) to find your point.
• After completing a graphing problem, you may want to reset the graphing window by pressing 6:ZStandard
• Some things that the TI-84 can't do or does poorly:
  • Show vertical asymptoses.
  • Show holes in graphs.
  • Show precise points on a graph at most x-values.
  • Show precise x- or y-values on a graph.
• If you use a TI-84 that was last used by another student, follow the setup instructions, before using it.
• When dealing with angles, be careful to set the 4^th line of the mode screen to either RADIAN or DEGREE, depending on your goal.
  • When graphing Trig functions, you will want to use RADIAN.
  • When doing problems with angles, you will probably want to use DEGREE.
  • You may have to move back and forth during a test or other exercise. Be careful.

TI-84 Factory Reset

• A factory reset will delete all information entered in the TI-84 and restore it to the settings it had when new.
• This is not recommended except in extreme circumstances.
• Press: 7:Reset... 1:All Memory 2:Reset
• In general, it's not a good idea to mess with the memory of the TI-84.

Special Keys and functions

Math Key

• Provides access to additional math functions, as shown in the illustration to the right.
• Press
• Across the top are menus with various functions.
  • Move the cursor right or le with the arrow keys.
  • When you see the function that you wish to use, move the cursor up and down with the arrow keys and press:
  • There may be more functions available than are shown on the screen.
    To see them, scroll down with the arrow keys.
  • Example: move right to NUM and down to 5: int ( to insert a function that returns the integer part of whatever you are working with.
    • To get the integer portion of 8.2³, press the following without the quota on marks:
      "NUM" "5: int(" "(8.2^3)"
    • The calculator will show the illustration to the right because 8.2³ = 551.368 and int(551.368) = 551.
      

Fractions

• Converting from a fraction to a decimal.
  • Enter the fraction as a division.
• Converting from a decimal to a fraction.
  • Enter the decimal, then press: 1:→ Frac
  • The ability to do the conversion is clearly limited (we would prefer that 0.7777 return 7/9). However, this capability may be useful on occasion.
    

Finding a function Maximum or Minimum with the Math Key

• 6:fMin( expression, variable, low x-value, high x-value)
• For the example below, the function is 𝑦 = xe^x and the range of x-values over which we seek the minimum is [−5, 0].
  • To obtain the x-coordinate of the local minimum:
    6:fMin( , −5, 0)
  • To store the resulting value as the variable Z:
  • To obtain the y-coordinate of the local minimum, substitute the value of Z found above into the expression of the function:
    Press:
    
  • Note: it is not preferred to use X as the variable in this process because every me you use X after this, the TI-84 would use the value you stored as X instead of trea ng it as a variable.
  • A better approach to finding a minimum or maximum of a function is to graph the function as described below. Why? Because the range of x-values selected in the fMin function may not be where the minimum or maximum truly is. Using a graph, you can easily see the location.

Scientific Notation

• To enter a number in scientific notation, you can enter it as, for example, 1.5 × 10 14 (for 1.5 × 10¹⁴) or use the key as a shortcut:
  1.5 14 is a good way to type in 1.5 × 10¹⁴.
• It's a good idea to place numbers in scientific notation inside parentheses, regardless of which of the above methods you use.

Entering Matrices

• To enter data in a matrix, first define its name: will bring up the matrix menu.
• Move to the right at the top to EDIT. Select the matrix you want to define using the up and down arrow keys.
  Press . Matrix names are typically A, B, or C.
  
• You will next be asked to define the dimensions of the matrix. The first number is the number of rows and the second number is the number of columns. After you define the dimensions, press and presto, a matrix of the dimensions you chose will appear.
• The matrix you chose may already have numbers in it. No worries, just change them by using the arrow keys to select each position in the matrix and type the values you want in each position. Press after entering each value.
  
• The matrix is now defined. Press to get back to the main window.
• If you want to see your matrix, press , scroll up or down to the matrix you want to see, and press twice.
  • If you want to use your matrix, use this same procedure, but press only once.
  • Example: to mul ply matrix A (previously defined) by the new matrix B from the le, press: 2:[B] × 1:[A] . See the results below:
    
• Note: matrices can be added, subtracted and mul plied if they have the proper dimensions. They cannot be divided.

Complex Numbers

• Complex numbers have the form a + bi. That's exactly how they are entered in the TI-84, except it's a good idea to put parentheses around them.
  • Example: to calculate (2 +3i)⁴ , press (2+3 ) 4
  • The key is over the period.
    
• Taking a root or a fractional power of a complex number will provide the principal value only, with lots of decimals.
  

Apps on the TI-84

The TI-84 CE comes with a number of pre-loaded apps. You may want to check them out, but a full description of how to use them is beyond the scope of this document. Among the most useful apps are:

• 9:PlySmlt2 – provides a polynomial root finder and a simultaneous equation solver.
• 4:Conics – is a conics graphing app: circles, ellipses, hyperbolas, and parabolas.
• Check out the others. You may find a gem.

Graphing a function – Tutorial

• Keys:
  • To enter the function you want to graph.
  • To change the x- and y- limits on the screen, and to set the resolution of the function being graphed.
  • To zoom in or out on a portion of the graph.
  • To move the cursor along the graph to an interesting point.
  • To graph the function that you defined on the screen
• Example: Graph the function: 𝑦 = ln(𝑥²) + 2
  • )+2
  • The calculator may show something like:
    
  • If your calculator does not show the graph above, change the settings in to be as shown below. Make sure you use the key at the bo om of the calculator to enter a negative number.
    • Scroll up and down the fields shown to enter the values you want.
    • Don't change any entries below the Xres line.
    • When you are finished, press to see what the graph looks like.
      
  • You may Notice that something interesting is happening to the graph when x = 0. To see what is happening better, change the settings for x and y to be the following, and press
    
  • Well, that's not much be er. Something is clearly going on at 𝑥 = 0, and we have reached a limit to what the calculator can do.
    • We can understand, without the calculator, what is going on by subs tu ng 𝑥 = 0 into the equation 𝑦 = ln(𝑥²) + 2. The problem is that ln(0²) = ln(0) is not defined. There is no natural log of zero, and the graph approaches −∞ on both sides of zero. Unfortunately, the calculator cannot show that, so we have to use our math brains instead.
  • Here are some things it can't do or does poorly:
    • Show ver cal asymptotes.
    • Show holes in graphs.
    • Show precise points on a graph at most x-values.
    • Show precise x- or y-values on a graph.
• In order to determine precise values we use the calc function, which is over the key. That is, will allow us to determine precise values.
  • Example:
    • Reset your window settings: MEMORY 1:ZPrevious
    • To determine 𝑦 when 𝑥 = 2, press 1:value
    • The calculator will show the screen below with the cursor blinking after X=.
      
    • Type the number 2 and press
    • The calculator will show the value of y at the bo om of the screen, the function used at the top of the screen, and a point on the graph, as shown below.
      
    • It's a good idea to check to make sure the correct function is displayed and that the point on the graph makes sense. The y-value shown at the bo om of the screen is what you were looking for.
• Locating key points on the graph
  • There will be mes when you want to get a be er idea where a key point on a graph is. In this case you have two choices. If you know something special about a point, e.g., it is a minimum, a maximum, or a zero of the function, you can use ; otherwise you can the graph to obtain an approximate x- and y- value at the key point.
  • Example: Using
    • Let's start be redefining our function. To get rid of the function we previously defined and enter a new one:
      ×
    • This function is 𝑦 = xe². It's a weird function, but those are the ones that teach the most about graphing.
    • Adjust your window settings for x- and y- as follows and re-graph. Your settings and graph should look something like this:
    • Now, this is an interesting graph. Let's try to find the minimum with a trace. With the graph s ll displayed, press
    • A point cursor, x- and y-values, and the function definition will appear on the graph. Move the cursor un l it looks like it is at the lowest point of the graph. As you move the cursor, look at the changing y-values and stop the cursor at that point.
      
    • It appears that the lowest value of the function occurs at the point (−1.030303, −0.367714). Your teacher will probably ask you to round these values because we know the graph is not accurate to six decimals.
    • Example: Using
      • To get a more accurate es mate of the lowest point on the graph, we will use the calculator's calculation capability.
      • Press 3:minimum. Again, a point cursor, x- and y-values, and the function definition will appear on the graph.
      • Hold down the le arrow key to move the cursor so that it is clearly to the le of the minimum point. Press
      • Next, hold down the right arrow key to move the cursor so that it is clearly to the right of the minimum point. Press twice.
      • The calculator will find the minimum point and display its coordinates at the bo om of the screen:
        
      • The point identified, (−0.999999, −0.367879), is a much more accurate es mate of the value of the function's local minimum. For those who are interested, the actual value can be determined with Calculus to be (−1, −1/𝑒) ≈ (−1, −0.367879), to six decimals. So you can see how much more accurate this approach is.
      • Notice that, even with this approach, the x-value determined by the calculator is not accurate to six decimal places. However, any rounding at all will give an accurate value of 𝑥 = −1.
      • If you look at the menu under , you will see that this approach also works in finding graph zeros, maxima, and the intersection of two function graphs.
    • Suppose you want to look a li le closer at the part of the graph where the minimum exists.
      • Press 2:Zoom In to zoom in. You will see a flashing point cursor. This cursor will be the center of the zoomed-in graph once you press
      • If you want the center to be somewhere else, move the cursor to that location and then press
      • In the illustration below, I moved the cursor to be near the minimum point before zooming in, though this is of little value in this situation.
        
        
      • Notice that the button has an op on to zoom out, as well as other op ons. One very useful op on is 0:ZoomFit, for which the calculator does its best to fit the most useful part of the graph in the display window.
      • Rest the window x- and y-values to those shown below. Then, press 0:ZoomFit to display the window on the right.
        
      • From here, you could choose to zoom in on the minimum using the technique described above, if that is what you are interested in. In any case, this gives you a helpful look at more of the graph.
• You may graph mutiple functions if you like, simply by pressing and entering as many graphs as you like. However, note that the more graphs you show, the more complex the resulting display will be.

Some Problems to Try

Problem 1:

Problem 2: Which two functions below are inverses of each other?

Solution: There are multiple ways to do this, let's graph them.

• The TI-84 will graph these functions using the colors in the screen.
• We want two functions that reflect over the line 𝑦 = 𝑥.
• Here's the trick we want to use. The two functions that are inverses of each other will o en intersect the line 𝑦 = 𝑥 (pink) at the same point.
• The red and gray lines clearly intersect the pink line at the same point, so they are inverses. Looking back at the screen, we see that these two functions are 𝒈(𝒙) and 𝒉(𝒙), which is our answer.
  

Problem 3: Given that 𝑦 = 𝑥³− 𝑥² − 9𝑥 + 9, find the zeros and give the multiplicity of each.
Solution: We can do this by factoring, but let's practice our graphing.

• First, clear all of the functions in the screen by using the up and down arrows and the button.
• Next,
• We can see on the graph that zeros appear to be at x= - 3, 1, 3 and, in fact these are what we are looking for.
  
• Further, a cubic equation had at most 3 real solu ons and since we have found them, there are no more.
• One more thing. The multiplicities of each zero must be odd because the function passes through the x-axis at each zero. Therefore, the multiplicities of each zero must be 1.
• Note that if a function has an even multiplicity at a zero, the curve will bounce off the x-axis and not pass through it.

Problem 4: Find all values of 𝑥 for which ln 𝑥 = e^x .

Solution: This is very difficult algebraically, and tailor-made for graphing.

• Let's graph both functions.
  ) ln 𝑥 is set up as Y1
  e^x is set up as Y2
  
• Ntice that the two graphs never intersect, so there are no values of 𝒙 for which 𝐥𝐧 𝒙 = e^x. The graph is the proof of this.
  

Problem 5: Solve the rational equation:

Solution: We can do this problem purely algebraically, but it's instructive to look at its graph.

• First, clear all of the functions in the screen by using the up and down arrows and the button.
• Next, ( −4) ÷ ( +5)
• Let's look at the graph, at 𝑓(𝑥), and exercise our brains.
  
• This function has a ver cal asymptote where the denominator is zero, so at 𝑥 = −5.
• 𝑓(𝑥) has a horizontal asymptote at 𝑥 = 1 because the fraction divides out to 1 if we ignore the constants in both the numerator and denominator. Good trick, huh!
• So, the function is never below zero on its le side, and is below zero on its right side from the ver cal asymptote un l it intersects with the x-axis. We know the 𝑓(𝑥) = 0 at 𝑥 = 4 because that makes the numerator zero, so the Solution is all values of 𝑥 between −5 and 4.
• We can write the Solution in set notation as {𝒙 | −𝟓 < 𝒙 < 𝟒}.
• We can write the Solution in interval notation as (−𝟓, 𝟒). Note that this is arrange of 𝑥-values and not a coordinate.
• Note that the le endpoint of the range of x-values is not included in the Solution because it lies on a ver cal asymptote. The right endpoint of the range of x-values is not included in the Solution because at 𝑥 = 4, 𝑓(𝑥) = 0 and we want 𝑓(𝑥) < 0.

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