TI-84 Plus cookbook

See also the generic guide on graphing calculators in AA.

IB makes 25 to 35% of Paper 2 about technology. The goal is to do things correctly and efficiently.

Here are the most useful and easy to remember procedures for the IB Analysis and Approaches (math AA) course using TI-84 Plus (CE). You should also refer to the official manual for your model. The first three sections are must-read: principles, setup, and overview.

This is calculator screen text. This is a ti-84 button. This is the right arrow: ▶.

For anyone using a different model, you can do nearly everything on this page and maybe a bit more. Here are some considerations

• Use the catalog/book on your calculator if you are looking for a similar built-in function but have no idea where it is.
• Xres is unique to TI 84 Plus. Graphing speed is not an issue in other models that I have tried, so long as you do not graph integrals.
• You are mostly likely not able to graph inverse functions or even vertical lines.
• What does each ctrl [button] combo do on TI Nspire? Also for Nspire always use explicit multiplication signs, because I don’t know which genius allowed every letter to be a function.

Contents

• Principles
• Setup
  • apps
  • exam mode
  • recommended settings
• Overview
  • storage
  • data entry techniques
  • Operators and built-ins
  • ways to solve
• Solver
• System of linear equations
  • solving the system
  • using the solution
• Lists
  • generate sequence
  • evaluate a function over a sequence
  • find common element of two lists
• Table of values
• Curve sketching
  • evaluate at one value, and find $y$-intercept
  • change function display style of a function
  • graph function
  • zero, min, max
  • point of inflexion
  • vertical asymptote
  • horizontal asymptote
  • limit at a point
  • piecewise functions
  • compositions and transformations of functions
  • inverse relations, areas, etc.
• Non-linear equations and inequalities
  • intersections of functions
  • inequalities
• Composite functions and calculus
  • absolute value function
  • evaluate Y2 at where Y1 fulfills some condition
  • tangent line
  • definite integral evaluated over a given interval
  • maximize the definite integral
• Regression
  • entering the data
  • the regression line $y$ on $x$
  • the regression line $x$ on $y$
  • Pearson’s correlation coefficient
  • intersection of the two regression lines
  • find intersection given the two equations
• Frequency distributions and discrete random variables
  • all data known or summarized
    • entering the data (midpoints and frequency)
    • mean, variance, quartiles
    • convert from frequency to cumulative frequency
    • convert cumulative frequency to frequency
  • some data missing
• Binomial distributions
  • find $p$ given PMF or CDF
  • find $n$ or $x$ given PMF or CDF
• Normal distributions
  • Find unknown parameters
    • optional method using solver
    • must-know method using standard normal
  • Evaluate probability
  • CDF over finite interval with one unknown parameter
• Finance
• Complex numbers (HL)
  • bearings
• Euler’s method (HL)
• More examples
  • solver vs graphing vs ???
    • solver
    • graphing
    • iterative method
  • May 2025 TZ2 P2 AA HL 10f / AA SL 8f

Principles

There are key approaches to using your graphic display calculators (GDC).

1. Show your work using standard math notation and not calculator notation. IB accepts writing down the equation (if not given), a rough graph sketch, and a table of values as showing work. Do just enough work by hand so you can enter it into your calculator.
2. In over 95% of the time, how well you use the calculator matters more than calculator speed and functionality. Explore your calculator well in advance of mocks and exams.
3. Where possible, leverage past expressions and results stored on calculator. Enter data once, correctly, in a way that suit a variety of purposes.
4. There are multiple methods or paths that get the same answer.
5. You will not need to estimate or guess, except to set certain parameters or an initial guess.
6. The calculator processes what you entered, not what you think you entered.
7. Check your solution before moving on to the next question.

Setup

The latest operating system is required for IB exams: TI 84 Plus (v 2.55 from 2011) , TI 84 Plus Silver Edition (v 2.55 from 2011)  TI 84 Plus CE (v 5.8.4 from Oct 2025) . You need to download the TI Connect CE software to your computer, also available from above links, and connect your handheld calculator using a mini USB to USB-A (or to USB-C) cord. This software is backwards compatible with TI 84 Plus.

You can view your OS from 2nd mem " + 1:About.

apps

The TI-84 Plus comes with several apps. Finance (for compound interest calculations) cannot be deleted. Here are the other permitted TI-84 Plus apps on IB math exams. Download links are provided in case they are deleted or missing.

For CE user guides, see the non-CE links above. The catalog is also availalble in various other languages, and their corresponding TI-84 Plus apps are all permitted on IB exams.

Both CtlgHelp and Transfrm requires one-time activation, after entering into exam mode.

In any menu, typing the number or, in case of a letter, A‑lock alpha followed by the letter will select that option. Or, use the arrow keys to highlight a command and use mem " + to view syntax information. In syntax screen, press stat plot f1 y= for any variations.

The Transfrm App alters the available graphing styles as well as settings in 2nd format f3 zoom. Transfrm can be turned off by opening the running app and choose the appropriate option. At this time, I do not feel Transfrm to be necessary but you are encouraged to try it out yourself. Transfrm can speed up certain brute-force guessing and checking.

exam mode

How to make TI-84 Plus suitable for IB examination use:

1. Go to 2ndmem " +. Select 7:Reset...
2. Under RAM, select 1:All RAM. entry solve enter. Select 2:Reset to confirm.
3. Repeat step 1. Under ARCHIVE (using ▶ arrow), select 1:Vars... Then, select 2:Reset to confirm.
4. Repeat step 1. In 2:Mem Mgmt/Delete..., under A:Apps..., delete each disallowed app using ins del.

Ensure that your invigilator performs the correct steps, and does not delete approved apps using 2:Apps... or 3:Both... or ALL memory. Ensure that they do not reset your calculator, which would delete all apps!

recommended settings

After going into exam mode, under quit mode, make these changes:

1. Select SIMUL and not SEQUENTIAL. It speeds up graphing.
2. In ANSWERS, select DEC for decimal answers.
3. Set STAT DIAGNOSTICS to ON (to enable Pearson correlation coefficient).
4. (optional) Change LANGUAGE, if needed.
5. Under tblset f2 window, set Xres to 3 to speed up graphing.
6. (non-CE models) Activate angle B apps CtlgHelp

All calculations on this page assume these settings.

Some important mode settings that you may choose to change mid-exam.

MATHPRINT vs CLASSIC: CLASSIC means all math are written inline, with more brackets, no nice-looking fractions or integrals, etc. MATHPRINT allows nicer looking math in many places, at the cost of having long expressions requiring to scroll with arrow keys to read.

RADIAN vs DEGREE: It is recommended that you first finish all questions mentioning degrees ($\degree$) in DEGREE mode, before switching to RADIAN for the remaining questions.

Overview

This section outlines some essential one-step procedures. Lengthier calculations will often make use of these steps.

storage

Pretty much every result can be stored and retrieved. You should make use of these options to avoid re-entering long decimals into the calculator.

If a few minutes passed since you last used a particular stored value, double check its value before using.

The automatic storages are erased once a new value is available. Note that X is a variable when graphing, but elsewhere it is the X coordinate of the last point you looked at.

When you leave the main calculation page and come back, Ans may have changed if you performed a calculation or trace.

The rcl X sto> button can store the left side into the compatible A‑lock alpha letter, list, or matrix.

data entry techniques

Overwrite mode is the default, with the blinking box. Insert mode will add before the blinking underscore.

Operators and built-ins

round( and int( are especially helpful as Y1(int(X)) turns any discrete function into a continuous one. This allows using discrete functions in numerical solver and while graphing.

ways to solve

In graphing, finding zeros can be easier to zoom into than finding intersections. Express LHS = RHS as LHS - RHS = 0 is useful when the ranges span a large interval.

Numeric solver is prone to the risk of finding the wrong root or missing correct roots. Always check if the answer makes sense. It does not necessarily find the nearest root to your guess. You cannot copy an expression from solver, so best that you define it in a function in case you need it later. All in all, graphing is preferred, but the solver allows you to quickly obtain the root if you know for sure there is only one root. Solver is also more useful when you cannot pin the unknown down to a finite interval, such as when you only know a lower bound but not an upper bound.

Table should be used when the unknown can only take on integers or other discrete values. Examples include finding $n$ for binomial expansion or binomial distribution. Use graphing or solver if the unknown can take on real numbers, even if you need the smallest or largest integer under some condition.

Solver

The numeric solver may be able to solve one-variable equations over the real numbers (ie any decimal). The solution found is sensitive to the initial guess, but may not necessarily return the nearest root, let alone a root.

find a real solution to an one-variable equation

Example: Find $t \in \mathbb R^+$ such that

1. Ensure the variable can take on an interval of real numbers. For example, the sum of the first $n$ terms of a geometric series allows for non-integer $n$, and it means that solver can be used. This is not the case for $\binom nr$, the binomial coefficient, which can only take on $n, r \in \mathbb N, r \leq n$ on the calculator.

2. Identify any facts you know about the solution, such as a lower bound, upper bound, or a rough estimate of its value. This may use statements given in the problem statement or involve some light algebra work.

3. If the expression is long, ie you’d rather not having to enter it repeatedly, first store it in stat plot f1 y=. This is especially important for non-CE users as the solver only allows inline math.

   

4. Follow the screen in the numeric solver from test A math ▲. Enter the equation. You can use A‑lock alpha calc f4 trace to retrieve functions. Non-CE would enter 0=… while CE enters left side and right side of an equation separately, as E1 and E2.

   

   

5. Enter the initial guess, and change the interval to search over as necessary. If applicable, enter the known values of the constants.

   In this particular question, it did not really matter what we put for X which is a constant when solving for T, and it does not matter here as it is only a dummy variable.

6. Highlight the desired variable to solve for and press A‑lock alpha entry solve enter, or in CE versions, table f5 graph for SOLVE.

   

   Unknowns in integrals take a long time to solve. The solution will be marked with a dot so as to separate it from a guess.

   

7. Change the initial guess or interval to find a different root. If wanting a large root, guess a large number.

8. When you leave the solver, the root will be automatically saved (as an A‑lock alpha letter) for use in further calculations.

   

System of linear equations

Example: Paraphrasing AA Specimen SL P2 Q9

Consider $b, p, q \in \mathbb R$, and the functions

The function $g$ passes through the points $(3 , 2.5)$ and $(6 , 15.1)$.

solving the system

a) Find the value of $p$ and the value of $q$. [5]

We first save $f(x)$ in Y₁. We can define Y₂ as the part of $g$ without the parameters, ie $\displaystyle \sin\left(\frac{2\pi}{9}(x-3.75)\right)$

We obtain the system

In angle B apps PlySmlt2 2.SIMULTANEOUS EQUATION SOLVER. Choose 2 by 2 system, and use radians.

Choose table f5 graph (NEXT), then enter the coefficients.

Press table f5 graph to solve.

$p = 8.4, q = 6.7\qed$

If you want, you can store the system matrix in [A] for later use. In this case we can just remember the solution, but in general you should store the solution in matrix [B]. Matrices are in 2nd matrix D x⁻¹.

using the solution

b) Find the value of $x$ for which the functions have the greatest difference. [2]

Method 1:

To find the max difference, we can use fMax.

[B](2,1) denotes the second row and first column of matrix B. The solutions aren’t always memorable numbers like $6.7$

You can also graph to find maximum, if you prefer.

Max difference at $x \approx 1.64 \qed$

Method 2:

To graph a function, we hide the other functions, set the XMin, Xmax and ZoomFit. We then go to 2nd calc f4 trace to find the maximum. Graphical approaches are discussed in depth in #Curve sketching.

Max difference at $x \approx 1.64 \qed$

Lists

A list is a column of values. They are typically used in statistics, but can also be used for sequences with an explicit formula (as opposed to recursive formulas).

Lists are defined using list stat 1:EDIT. Use 2nd alongside numbers 1 through 6 for L1 to L6.

generate sequence

1. In 2nd list stat, ▶ arrow for OPS, go to 5:seq(.
2. Follow the menu to input the sequence with some general expression involving a variable.

   

3. The sequence can be rcl X sto→ in a list, eg in L1.

   

   Alternatively you can go to list stat 1:Edit..., ▲ arrow to highlight L1 and enter the seq( there. Press entry solve enter to confirm.

   

   

There is also a dedicated SEQ mode for defining sequences, instead of the FUNC or FUNCTION mode that we have always been using. It is a different way that you can look into if you are interested.

evaluate a function over a sequence

The previous section defined a list in L1. Lists can be used in calculations from 2nd followed by the list number.

Along the same idea, we can evaluate a function over a list in the above two ways. To find log base 10 of L1, we can either enter it in the main screen:

.. or we can go to list stat :Edit, go to L2, ▲ arrow and write the function to L2=:

find common element of two lists

Example: Paraphrased from AA M22 P3 TZ1 HL 1e [5]

A triangular number has the form $\displaystyle P_3(n) = \frac{n(n+1)}{2}$.

A pentagonal number has the form $\displaystyle P_5(n) = \frac{n(3n-1)}{2}$.

By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$, that is both a triangular number and a pentagonal number.

The markscheme gave several solutions that reduce the search space, nevertheless, here is an approach that does not require algebraic analysis.

1. Store the first $50$ terms of the two sequences in L1 and L2.

   

   

   

2. We use 2nd list stat ▶ arrow, under OPS tab, choice 9:augment( to concatenate (combine) two lists. Enter augment(L1, L2)→L3

3. Sort this list in ascending order using 2nd list stat ▶ arrow, under OPS tab, choice 1:sortA(. Enter sortA(L3).

   

4. Go to list stat 1:Edit to go through list L3 until we see a duplicate.

   

   

   Alternatively, you can use 2nd list stat OPS option 7:ΔList( to find the difference between consecutive values. Save 7:ΔList( to L4 and find the first 0.

   

   

The first number that is both triangular and pentagonal is $210 \qed$

Table of values

Some functions only permit integers, examples include $n$ in binomial distribution and binomial coefficients. To “graph” these functions, replace all occurences of X with int(X), which is found from test A math NUM 5:int(.

If $n$ is small, you may choose to forego graphing and directly look for the value from 2nd table f5 graph. Table settings, including starting value and step size, can be changed in 2nd tblset f2 window.

See an example #find n or x given PMF or CDF here.

The differences between table and lists are that

• tables only allow equally-spaced increments
• tables can keep going in either direction, but lists require a specified minimum and maximum
• tables are indexed by X; lists are indexed incrementally, i.e. first element is L1(1)
• functions of lists require you to separately make a entry for the input, eg using seq(, but table you can set the inputs directly
• lists have many useful operations not available to table

Curve sketching

Functions are defined using stat plot f1 y=. Functions can be retrived to use in A‑lock alpha calc f4 trace. In addition to graphing, functions can be used elsewhere such as on main screen, lists or numeric solver. You should always save the function in stat plot f1 y= before using.

$x$ value of the last cursor movement, trace, zero, intersection, min/max calculation is stored in X, which can be accessed either from link X,T,θ,n or A‑lock alpha rcl X sto→. Similarly the $y$ value is stored in A‑lock alpha L1 Y 1. You can access these values on the main calculation screen, where X switches from being a variable to a constant.

We will go through several functions in this section. Here is the first one.

Example: Consider the function

We begin with storing the function in Y1.

evaluate at one value, and find $y$-intercept

a) Find $f\left(0.3\right)$

This function has no $y$-intercept. But you could have used F1(0) to find it, if it had existed.

change function display style of a function

In stat plot f1 y=, use arrow keys to highlight the = and entry solve enter to toggle showing and hiding the graph. Highlight the line in front of Y₁ to change display style. Options include solid, dotted, shade above/below, among others. CE version allows changing color.

graph function

1. In stat plot f1 y=, hide the functions you do not need.
2. Enter the function in an available slot.
3. In tblset f2 window change Xmin and Xmax if known. They are typically given as a domain in the question. Make a guess if either is unknown. In most cases, you want Xres = 3, but it can mess up vertical asymptotes and other rapid changes.
4. (optional) change Xscl (tick mark spacing) to a suitable number. This is useful when you are graphing of a large area. The default is 1, which means a tick mark on $x$-axis at the integers.
5. IF you have a very good idea of the Y values (eg questions asks about $y = 4$), change Ymin and Ymax as well. Then table f5 graph.
   OTHERWISE, in format f3 zoom, choose 0:ZoomFit to automatically scale Y. The function will be graphed.
   Unless otherwise stated, all graphs on this page only set Xmin and Xmax then format f3 zoom 0:ZoomFit.
6. (optional) off on to stop graphing.

This does not work well for graphs with vertical asymptotes. In those cases you should choose Ymin and Ymax yourself.

Also if the given

Furthermore, if you want to view graph over multiple orders of magnitude, you can define Y2 = log(Y1) or Y2 = log(X), Y3 = Y1(Y2) to make Y or X axis logarithmic. See #solver vs graphing vs ??? for example of a log-log plot.

This example started in #curve sketching

As there is a vertical asymptote near $x = 0$, we can do a ZoomFit with Xmin:0.1 and Xmax:3

This is a cubic with a horizontal exponential transformation. It is not something you need to know for exams. But it can be helpful to know to understand the function.

Press calc f4 trace to show a blinking point. Entering 0.3 entry solve enter will evaluate the current function at $x=0.3$. The ◀ and ▶ arrows move along the current function. The ▲ and ▼ arrows change between the plotted functions.

Furthermore, knowing that $f(0.3) < 0$, we can change our Xmin:0.3 and redo the ZoomFit.

This shows that choosing the right X to display in conjunction with ZoomFit makes graphing very easy.

zero, min, max

The main graphical calculations are in 2nd calc f4 trace.

In this subsection, we are focused on finding a zero, a local maximum, and a local minimum. You have two other max and min functions on the calculator. The more useful one is test A math 6:fMin( and 7:fMax(, which is essentially the same as the ones in 2nd calc f4 trace except that without a graph to specify the boundaries you are more likely to find the global maximum and minimum. You also have similar operations max( and min( in the NUM tab of test A math. They are more for lists and discrete values.

To find a zero, min, max, using 2nd calc f4 trace, the feature must be clearly identifiable on the screen such that you can easily provide lower (left) and upper (right) bounds of the $x$-value, that contain no other zero, min, or max.

As with calc f4 trace, you can use either arrow keys or directly key in the bounds. entry solve enter to confirm. When it asks for a guess, you can just use the upper bound again.

This example started in #curve sketching

b) Solve $f(x) = 0$.

In 2nd calc f4 trace, choose 2:zero.

Choose appropriate left and right bounds that surround the zero.

For guess, just entry solve enter to use the right bound. The zero is found.

$x \approx 0.382 \qed$

c) Find the local maximum.

Same procedure as above but with 2nd calc f4 trace 4:Maximum.

The local maximum is at $(0.645, 1.51)\qed$

point of inflexion

TI 84 Plus does not have a built-in POI finder, but we can make use of graphing the second derivative.

Example: (IB Practice P2 HL Q12b, suitable for SL) The curve defined by

has a point of inflexion at $\left(x_1, y_1\right)$ where $\e^{-\frac\pi2} < x_1 < \e^{\frac\pi2}$. Determine the coordinates of this point of inflexion.

Because second derivative is zero is insufficient to conclude the point is a POI, it is best that we graph.

1. Define the given function in stat plot f1 y=, such as in Y1. For this question, the calculator required an extra set of parentheses not stated in the question.
2. Define the second derivative using test A math 8:nDeriv( in another function, such as Y2. On non-CE, it is best that you give it a different thickness or color.

   

3. If needing a non-zero second derivative, define a third function with the value of the second derivative.
4. Graph the functions. Here ZoomFit would not work well because of the asymptotes. So instead of ZoomFit, Ymin=-4 and Ymax=4 were chosen, as we are focused on when second derivative is $0$.

   

5. Use 2nd calc f4 trace to find the zero (or intersection with the third function, for non-zero second derivative). Be sure to choose the second derivative function.
6. Visually check that the solution is indeed a POI.

   

7. Use the ▲ and ▼ arrow keys to find the point on the original function.

   

Final answer: $(0.629, -0.943) \qed$

We are told of a vertical asymptote near $\e^{\frac\pi2}$. Let us assume that we did not know this and try to find this instead. Suppose also we know there is one vertical asymptote between $x = 2$ and $x = 7$.

vertical asymptote

Example: The curve defined by

has a vertical asymptote $x = k, \ 2 < k < 7$. Find the vertical asymptote.

1. Start with a graph showing the asymptote. After putting in xmin:2, xmax:7, and using ZoomFit.

   

2. In format f3 zoom, select 2:Zoom In. Use arrow keys to where you think the asymptote intersects the $x$-axis. It just needs to be in the right ballpark and does not have to be precise. entry solve enter to confirm.

   

   

3. In format f3 zoom, select 0:ZoomFit. Here is the new graph.

   

4. Repeat Steps 2 and 3 as needed to get into a close enough look that each horizontal arrow movement moves the graph by less than 0.001, or until you can be certain of the vertical asymptote to three significant figures. Usually two of Step 2 followed by one of Step 3.
   After a few iterations, we see that it is between 4.8102 and 4.8106.

   

   

We can be quite confident in our final answer of $x = 4.81$. The actual answer was $x = \e^{\frac\pi2} \approx 4.81048 \dots$

horizontal asymptote

Example: Find all horizontal asymptotes of

1. Define the function in Y1. No need to graph.

   

2. In list stat 1:EDIT, choose an empty list, eg L1, ▲ arrow to see L1=.
3. In 2nd list stat, ▶ arrow for OPS, select 5:seq(. Put
   Expr:10^X Variable:X start:2 end:10 step:1

   

   

4. In the next empty list, eg L2, ▲ arrow to see L2=. In A‑lock alpha calc f4 trace, select where our function was defined, which was Y1. Enter the following L2 = Y1(L1)

   

5. If the last few values agree to three significant figures, then write down the answer as horizontal asymptote. Otherwise, increase the start and end values from Step 2.

   

6. Repeat steps 2 to 5 for Expr:-10^X

   

   

The asymptotes are $x = 1.57$ and $x = -1.57$.

limit at a point

It’s very similar to the horizontal asymptotes approach above, but we’ll use a slightly different data entry method.

Example: Evaluate $\displaystyle \lim_{x\to1} \frac{\sqrt[3]x - \sqrt[7]x}{x-1}$

1. Save the function to a function, eg Y1. The n-th root is available from test A math 5:ˣ√(.

   

2. 2nd quit mode. In 2nd list stat, ▶ for OPS, in 5:seq(. Use the following:Expr:1+(-0.1)^X
Variable:X
start:2
end:10
step:1
   rcl X sto→ this in L1. 1 because we are finding limit as x approaches 1.

   

   This leads to a sequence as follows

   

3. Y1(L1)→L2

   

4. We can use ◀ ▶ arrow keys, or list stat 1:Edit to view the sequence. The limit is $0.190 \qed$

   

piecewise functions

TI-84 Plus CE, unlikes the non-CE versions, has a built in test A math B:piecewise(. It makes entering piecewise functions much easier.

Graphing piecewise functions require inequality comparison operators under 2nd test A math. CE users can use comparisons under the CONDITIONS tab. For non-CE users, use the and operator in the LOGIC tab for a finite interval.

Example: Graph

non-CE: Y₁=sin(X)(X<0)+X(0≤X and X<2)+(eX-2+1)(X≥2)

0≤X<2 is not accepted.

CE: $\displaystyle Y_1(X) = \begin{cases} \sin X & X < 0 \\ X & 0 \leq X\;\text{ and }\; X < 2 \\ \e^{X-2} + 1 & X \geq 2 \end{cases}$

using 2nd test A math CONDITIONS, 7:≤X and X<

compositions and transformations of functions

In general Y1 is same as Y1(X), so when you use one function in another, you can usually ommit the (X).

There are many other transformations found in test A math and in dedicated buttons. These are the ones that are likely to be entered incorrectly.

Using test A math 8:nDeriv( and 9:fnInt(, or 2nd calc f4 trace we can find first, second, and higher derivatives, but can only integrate once.

inverse relations, areas, etc.

The draw menu ( 2nd draw C prgm) constructs sketches that do not permit calculations. Useful drawings include vertical lines, tangent lines, inverse relations; and shading areas between curves.

Be sure to set the desired window settings prior to drawing.

Example Sketch $y^3 - 3y^2 + y + 1 = 2x,\, -4\leq x\leq4$.

Define Y₁=0.5(X^3-3X^2+X+1). Hide this function (for faster graphing). Go to tblset f2 window and set Xmin and Ymin to -4 and Xmax and Ymax to 4.

Under 2nd draw C prgm, use the eighth option and enter DrawInv Y₁.

Note that drawings are often erased after some zoom or window changes. Drawings can be persisted using the STO tab in the draw menu.

Non-linear equations and inequalities

intersections of functions

Example: Practice AA Paper 2 HL #5 Consider the curves $y = x^2\sin x$ and $y = -1 - \sqrt{1 + 4(x + 2)^2}$ for $-\pi \leq x\leq 0$.

a) Find the $x$-coordinates of the points of intersection of the two curves.

b) Find the area, $A$, of the region enclosed by the two curves.

a)

Using our default graphing procedure with Xmin:-π, Xmax:0 and ZoomFit yields a very clear image.

We use 2nd calc f4 trace 5:intersect. When there are more than two functions plotted, use ▲ and ▼ arrows to select the two functions. Then use ◀ and ▶ arrows or directly enter a guess.

Once the left intersection is found, save it as A‑lock alpha L; once the right intersection is found, save it as A‑lock alpha R.

The $x$-cooridinates are $-2.76$ and $-1.54$.

b)

In main screen, we can use test A math fnInt( to find an integral.

The bounded area is $1.47$. In practice, IB refers to area as “unsigned area” when the area is not contiguous. So if there are several chunks of area, you should wrap Y2-Y1 in absolute values, from A‑lock alpha tblset f2 window 1:abs(. Absolute values are not needed if there is only a single region, regardless where the functions are positioned relative to the $x$-axis.

inequalities

Example: Adapted from practice Paper 2 HL Q4

The quadratic equation $(k-1)x^2 + 2x + (2k - 3) = 0$, where $k \in \mathbb R$, has real distinct roots.

a) Find the range of possible values of $k$.

We need a positive discriminant.

Note that it’s a quadratic opening down and we want it to be above $0$, so we expect a finite interval.

We also know that it is a quadratic with roots $1$ and $\frac32$ translated up by $4$ units. So we can say set Xmin:-3, Xmax:5 for graphing.

We need to find the zeros separately (note, when solving on GDC, always complete the calculation and never stop halfway because “looks like 0.5 and 2”).

$0.500 < m < 2.00 \qed$

b) Find the range of possible value of $k$, where both distinct roots are negative.

We need both

In test A math, ▶ arrow under NUM, we have 7:max( which finds the maximum of two expressions. We need the maximum of the two expressions to be negative. Since we care about zeros, let’s set Ymin:-4, Ymax:4

Note, X is the variable $k$.

$1.50 < k < 2.00 \qed$

Composite functions and calculus

Derivatives and definite integrals are more easily inputted using MATHPRINT mode.

Use test A math options 8:nDeriv( and 9:fnInt( provide numerical derivatives and definite integrals. Functions are returned when you use X=X for the numerical derivative, or set a limit of integration to X when integrating.

Example: AA Specimen P2 HL Q5 / SL Q6

The displacement, in centimetres, of a particle from an origin, $O$, at time $t$ seconds, is given by

absolute value function

(a) Find the maximum distance of the particle from $O$. [3]

We can first save the function in Y₁, replacing $t$ with X.

Distance is $\lvert s(t) \rvert$. We can use fMax but you can also use the max from graphing.

Max distance is $16.5$ cm $\qed$

evaluate Y2 at where Y1 fulfills some condition

(b) Find the acceleration of the particle at the instant it first changes direction. [4]

The particle first changes direction at the first local min or max. And then we need the acceleration at that time. This is a more complicated calculation so we should graph just to be safe.

We want the first extremum near 1.5. We can get the $x$ value graphically as 1.5198546. On the main screen, X becomes this value (unless you move to a different point).

Then evaluate second derivatative on main screen.

On non-CE version you would get $-8.9194071$. TI changed the derivative implementation across the years.

Acceleration at first turning point is $-8.92$ cm/s² $\qed$

tangent line

Example: Consider the graph of $\displaystyle f(x) = \frac{\sin x}{\ln x}, \ 2 \leq x \leq 12$.

For some $a \neq 5$, the tangent line to $f$ at $x = a$ passes through $(5, f(5))$. Find all such tangent lines.

The tangent line, using point-gradient form, is

The tangent must also pass through $(a, f(a))$. Substitute this into the tangent line, we get

We have a one-variable equation that we can solve via graphing. Here is the function and its graph.

Having a good idea of the graph, we define a second function
Y2 = d/dx Y1 | X=X (X - 5) + Y1(5) - Y1

We ignore the root at $x = 5$ as instructed by the question. We use 2nd calc f4 trace 2:zero to find the two zeros that are greater than $5$ for possible values of $a$. Once a root is found, we save it to an A‑lock alpha letter for future use.

For the point-gradient form of a line, we need the gradient as well as a point. We can use $(5, f(5))$ for both. But we also need the two gradients.

We obtain the following two tangent lines

Unless otherwise stated in the question, IB accepts, and encourages the use of point-gradient or point-slope equations as final answers.

Here are their graphs, using all available precision stored on the GDC.

definite integral evaluated over a given interval

This example is for next two parts.

Example: A particle has the following velocity:

The displacement at $t = 0$ is $-2$.

a) Find the displacement when the particle changes direction for the third time.

The displacement function is

with $(t_0, s_0) = (0, -2)$, then this is

Velocity changes direction every time its sign change, which occurs at a zero.

Be careful to enter the function correctly. If it helps, rewrite with the appropriate number of parentheses before keying it in.

We observe that each zero does cross the $x$-axis. So we need the third zero.

We save this to A‑lock alphaT.

We can evaluate the integral in the main screen. But it looks cooler from 2nd calc f4 trace 7:integral. When prompted, enter 0 for lower bound and T for upper.

The displacement is $7.29 - 2 \approx 5.29 \qed$

maximize the definite integral

This example started in #definite integral

Consider all $5$-second intervals $[a, a + 5]$ with $1\leq a \leq 5$.

b) Find

i) the time interval in which the distance travelled is maximized;

ii) the distance travelled in that interval.

The distance travelled is $\displaystyle \int_a^{a+5} \lvert v(t) \rvert \d t$.

To maximize it with respect to $a$, we need to consider the critical values, including the endpoints and where derivative with respect to $a$ is $0$.

The derivative is $\lvert v(a+5)\rvert - \lvert v(a)\rvert = 0$. Furthermore at a maximum, the derivative will change from positive to negative.

As we need to verify the values of the distance travelled over at least three different intervals, we should also define a function for distance travelled.

We are hiding the velocity and distance travelled functions to make graph easier to see.

Because maxima have derivative cross from positive to negative, we care about the zeros near $2.5$ and $4.8$. The other roots are local minima. We should save the zeros once we found them to A‑lock alpha L and R for left and right.

Now we have our candidates, $a \in \{1, 2.52, 4.77, 5\}$ and we need to check all four to find the maximum distance travelled.

We want the one corresponding to L. The time interval is $\left[2.52, 7.52\right]$. The distance travelled is $4.25 \qed$

You could have also tried test A math 7:fMax( with the integral, but on my device it took about 160 seconds to return the answer. As we cannot know in advance how long a calculation will take, it is better to do the above analysis under that time. Certainly it is easier to get intermediate marks using the above method.

Regression

The following few procedures will make use of this set of data.

Example: (Adapted from 2021 IB SPEC papers HL P2 Q4, SL P2 Q5) Given a table of values

entering the data

1. In list stat, under EDIT table, choose 1:Edit....
2. Input the $x$ values in L1, and the $y$ values in L2.

   

To use the lists: 2nd L1 Y 1 and 2nd L2 Z 2.

the regression line $y$ on $x$

1. Input the $x$ values in L1, and the $y$ values in L2.
2. In list stat, under CALC tab, choose 4:LinReg(ax+b).
3. Enter XList:L1, YList:L2. Leave FreqList blank.
4. For Store RegEQ, use A‑lock alpha calc f4 trace and select a function not in use. We can use Y1 as an example.

   

5. Highlight Calculate and press entry solve enter.
   $$y = 0.700x + 10.2 \qed$$

   

6. To estimate $y$ when $x = 26$, for instance, type in Y1(26), using A‑lock alpha calc f4 trace from an earlier step. This leads to an estimate of $28.4 \qed$

   

the regression line $x$ on $y$

Same as before, but switching $x$ and $y$.

1. Input the $x$ values in L1, and the $y$ values in L2.
2. In list stat, under CALC tab, choose 4:LinReg(ax+b). Do not choose option 8.
3. Enter XList:L2 (i.e. our $y$ values), YList:L1 (i.e. our $x$ values). Leave FreqList blank.
4. For Store RegEQ, use A‑lock alpha calc f4 trace and selects a blank function. We can use Y2 as an example.

   

5. Highlight Calculate and press entry solve enter. The regression line is $\mathbf{x = ay + b}$.
   $$x = 1.29y - 10.4 \qed$$

   

6. To estimate $x$ when $y = 29$, for instance, type in Y2(29), using A‑lock alpha calc f4 trace from an earlier step. This leads to an estimate of $27.1\qed$

   

Pearson’s correlation coefficient

1. In quit mode, ensure STAT DIAGNOSTICS is on.
2. Perform either $y$ on $x$ or $x$ on $y$ regression, and it will show $r$ along with the regression line.

intersection of the two regression lines

1. Input the $x$ values in L1, and the $y$ values in L2.
2. In list stat, under CALC tab, choose 2:2-Var Stats.
3. Enter XList:L1, YList:L2. Leave FreqList blank.

   

4. The mean $x$ is x̅. The mean $y$ is y̅. For our question, we see that the regression lines intersect at $(28.7, 30.3) \qed$

   

find intersection given the two equations

Example: May 2025 TZ1 P2 AA SL #5b / AA HL #4b

The regression line of $F$ on $A$ is found to be $F = 0.335A - 32.6$, and the regression line of $A$ on $F$ is found to be $A = 2.89F + 99.3$. Each regression line passes through the mean point. Find the mean $A$ and the mean $F$.

1. Arrange into the standard form for a system of two linear equations.
   $$\begin{align*} F& - 0.335 &A &=-32.6\\ -2.89F& + &A &= 99.3 \end{align*}$$
2. In 2nd matrix D x⁻¹ ◀, edit an empty matrix. We’ll use [A] as an example.
3. Change the size to 2 ×3.
4. Input the coefficient matrix.

   

5. In angle B apps, select PlySmlt2. Select 2:SIMULTANEOUS EQ SOLVER. Use default settings and with 2 equations and 2 unknowns. Click on NEXT.

   

6. Using calc f4 trace, load the appropriate coefficient matrix (for us it is [A]), from 2nd matrix D x⁻¹ under NAMES tab.

   

7. Press table f5 graph to solve. We see that mean $F = 20.9$, mean $A = 160.$ If you get fractions you can use table f5 graph (for F◀▶D) to get decimals.

   

   The original question was asked in context of mean arm length and mean foot length, so you should specify that in your answer.
8. The solution can be stored using calc f4 trace into a matrix for further calculations.

Frequency distributions and discrete random variables

There are enough similarities that we can use the same method to analyze them, namely 1-Var Stats.

There are also cumulative frequencies and cumulative probabilities.

all data known or summarized

We will modify the grouped data from Nov 2024 AA SL P2 TZ1 Q7. The paper viewed the age data as continuous.

Example: MyLife is a social media platform with 89.8 million users, all aged 12 years old and above (according to the users themselves). The following frequency table shows the number of users by age group.

Mean and variance are estimated using the midpoints, and IB accept specific values. Quantiles and percentiles are estimated using a cumulative frequency graph, and IB accept a small interval of correct values.

entering the data (midpoints and frequency)

For simpler data, such as equally spaced every 10 or 20, you should directly input the values. However as the example is more complex, here is a different way.

1. In list stat, under 1:Edit…, for L1, input the lower bounds for each interval as well as the last upper bound. In L3, enter the frequencies.

2. In L2, use ▲ arrow to highlight L2, so that you see L2=. Enter ΔList(L1)/2. The difference operator is from 2nd list stat, ▶ arrow for OPS, under 7:ΔList(. Then entry solve enter to confirm.

   

3. Delete the last entry in L1, then make L2 equal to L1 + L2. This will update L2 from the differences to the midpoints.

   

   

4. Finally if the data is discrete, subtract 0.5 from L2. This can be done along with previous step.

   

Again this could be something to keep in mind if it is not easy to calculate the midpoints.

mean, variance, quartiles

1. Assume the midpoints are in L2, and the frequencies in L3. list stat, ▶ arrow under CALC and 1:1-Var Stats, use List:L2 and FreqList:L3.

   

   Scrolling further return the quartiles.

2. In distr vars, under Vars, 5:Statistics..., under XY, we can retrieve 4.σx.
3. Squaring the standard deviation we get the variance.

   

   The mean is 38.1 (years) and variance is 268 (years squared).

convert from frequency to cumulative frequency

Suppose we have some frequencies in L3 and we want the cumulative frequencies in L4.

1. In 2nd list stat, ▶ arrow to OPS, choose 6:cumSum(. We want to write cumSum(L3)࢐L4, where ࢐ is the rcl X sto> button.

   

convert cumulative frequency to frequency

Suppose we have some cumulative frequencies in L4 and we want the frequencies in L5. As the ΔList( returns one value short, we need to add back the first value to the result.

1. In 2nd list stat, ▶ arrow to OPS, choose 7:ΔList(. We want to write ΔList(L4)࢐L5, where ࢐ is the rcl X sto> button.

   

2. In 2nd list stat, ▶ arrow to OPS, choose 9:augment(. We want to write augment({L4(1)}, L5)࢐L5

   

   This adds a list containing the first list from L4, to the front of L5.

Alternatively, you can start with a $0$ when entering the frequencies.

some data missing

Sometimes, data or a random discrete variable will have one or two unknowns. Typically, the mean (for data) or expected value (for discrete random variables) will be given. This is our focus. At HL, you will also see variance given, possibly forming a system of equations.

Example: Modified from math AI specimen Paper 1 HL Q1 / SL Q3

At the end of a school day, the Headmaster conducted a survey asking students in how many classes they had used the internet.

The data is shown in the following table

The mean number of classes in which a student used the internet is $2$.

a) Find the value of $k$.

b) Find the variance of number of classes in which a student used the internet.

We can first input the rows where both the value (number of classes) and the weight (number of students) are known in L1 and L2

Using 1-Var Stats, with L1 as List and L2 as FreqList, we obtain

The useful information are Σx=145 and n=88.

b)

We now add back $(3, 31)$ back to the lists and re-run 1-Var Stats.

We obtain standard deviation σx=1.353489111. We are also pleased to see that the mean is indeed 2.

We can retrieve this number by distr vars, under Vars, 5:Statistics, 4:σX, which we can then square to find the variance of $1.83 \qed$

IB will not give marks for using Sx.

Binomial distributions

Normal and binomial distributions are in 2nd distr vars. Assume B is a binomially distributed random variable. Here are common built in functions.

In particular, all TI-84 sum of binomial probabilities over an interval that does not include $B=0$.

You should look to remember that binomcdf always goes from 0 to the x value, inclusive, rather than memorize this chart. As there are so many cases, you can practice evaluating binomial distributions.

find $p$ given PMF or CDF

1. On answer booklet, write down the distribution in math notation, eg $X \sim \text{B}(n, p)$ where $n$ is known.
2. Set Y₁ = binomcdf(n, X, x). Set Y₂ = given probability.
3. Find their intersections graphically.

Example: From Specimen AI HL P1 #17b

The Head of Year, Mrs Smith, decides to select a student at random from the year group to read the notices in assembly. There are $80$ students in total in the year group. Mrs Smith calculates the probability of picking a male student $8$ times in the first $20$ assemblies is $0.153357$ correct to $6$ decimal places. Find the number of male students in the year group

Let $X \sim \text{B}\left(20, \frac{m}{80}\right)$. We find $m \in \mathbb Z^+$ such that $\text{P}(X = 8) \approx 0.153357$.

Entering the binompdf in Y₁:

The graph looks like

The intersections are at $X \approx 27.17,\; 37.00$

We choose the integer, $m = 37 \qed$.

find $n$ or $x$ given PMF or CDF

Example: Adapted from May 2025 AA P2 TZ3 SL #8f

$61\%$ of theme park visitors will go on a particular ride. Next Tuesday $n$ people will visit the theme park. The probability that at most 500 people will go on this ride is approximately $0.693$. Find $n$.

1. On answer booklet, write down the distribution in math notation, eg $X \sim \text{B}(n, p)$ where $p$ is known.

2. In stat plot f1 y=, input the PMF or CDF. Under test A math NUM, use 5:int( to wrap around the unknown. This turns discrete function continuous for graphing.

   

   

   Here, we know the Xmin is 500, we can estimate a maximum of 1000 for Xmax.

   

3. In 2nd tblset f2 window, set appropriate TblStart. Then view results in 2nd table f5 graph.

   

   

Work to be shown on exam for 3 marks:

$n = 809 \qed$

Normal distributions

In 2nd distr vars, we have two (useful) normal distribution commands

For 2:normalcdf( when lower or upper bounds are $-\infty$ or $\infty$, use any value that is at least $6$ standard deviations away from the mean.

On the TI-84 Plus CE, the 3:invNorm( has additional settings of LEFT, CENTER and RIGHT. Here is an explanation of what they do.

Find unknown parameters

Example: (Adapted from AA Specimen P2 HL Q10 / SL Q8)

The length, $X$ mm, of certain species of seashell is normally distributed with mean $25$ and variance, $\sigma^2$.

The probability that $X$ is less than $24.15$ is $0.1446$.

a) (i) Find $\sigma$, the standard deviation of $X$.

optional method using solver

1. In test A math ▲ arrow numerical solver, solve normalcdf(-1000, 24.15, 25, X) = 0.1446.

   

   

2. We know standard deviation is positive so we guess a positive X and set lower bound of guesss to 0.

   

   

   The standard deviation is 0.802. As expected by IB and unless specified otherwise, use all decimals in subsequent calculations. We should rcl X sto→ to a different letter, eg S, as many calculations write to X.

   

must-know method using standard normal

IB may ask solve for both $\mu$ and $\sigma$, using a linear system of equations.

1. Find z-score, using standard normal.

   

2. Rearranging for $\sigma$,

   $$\begin{align*} z = \frac{x - \mu}{\sigma} \iff \sigma &= \frac{x - \mu}{z}\\ \sigma &= \frac{24.15 - 25}{z} \end{align*}$$

   

Evaluate probability

This example started in #Find unknown parameters

a) (ii) Hence, find the probability that a seashell selected at random has a length greater than $26$ mm.

A seashell selected at random has a length less than 26mm.

b) Find the probability that its length is between $24.15$ mm and $25$ mm.

Use normalcdf

Use conditional probability and normalcdf. We want

$\mathrm P(X > 26) \approx 0.106\qed$ Conditional probability is $0.398 \qed$

CDF over finite interval with one unknown parameter

Example: Given $X \sim \mathrm N(\mu, 25^2)$, and $\mathrm P(75 < X \leq 125) = 0.6$, find $\mu$.

Because the probability exceeds $0.5$, we know $75 < \mu < 125$. We can graph for $\mu$ and solve. Otherwise if the probability is smaller than $0.5$ using a wider X interval.

Two intersections and we just find them separately.

$\mu = 84.9, 115 \qed$

Note that the values are symmetric about the mean of $75$ and $125$.

Finance

The Finance app is available via angle B apps

The TVM solver can be used for compound interest, appreciation, depreciation and/or inflation problems. However, it is intended for annuities, eg mortgage payments, so some care is needed to adapt it for compound interest purposes.

In particular in the TVM solver, use negative amounts for deposits, and positive amounts for withdrawals. This makes more sense in annuities which are beyond the scope of AA.

Due to the sign differences, PV and FV are not fully interchangeable with the counterparts in the formula booklet.

Also, N depends on P/Y. For example

Usually you want P/Y=1 for N in number of years.

To solve for an unknown, first enter all known values in 1:TVM solver..., leaving 0 for the unknown. Then highlight the unknown field and A‑lock alpha entry solve enter to solve.

Complex numbers (HL)

In quit mode, a+bi returns calculations in Cartesian form, and e^(θi) returns in Euler form. If you want the exact value in Cartesian, then you need to be in a+bi as doing calculation in Euler then convert back to Cartesian may introduce large enough precision errors.

The imaginary unit is from 2nd 𝑖 ︰ •. Euler form can be entered from eˣ S ln

The available functions are in test A math under the CMPLX tab.

bearings

Instead of $x$-$y$ axes, we use a North-East axes and complex numbers (positive real is North, positive imaginary is East) in polar form to compute zigzag paths in bearings.

Example: Adapted from AA Specimen P2 SL Q7

Adam sets out for a hike from his camp at point A. He hikes for 3.15 km on a bearing of $035\degree$ from the camp, until he stops for a break at point B.

Adam leaves point B on a bearing of $114\degree$ and continues to hike for a distance of $4.6$ km until he reaches point C.

Adam’s friend Jacob wants to hike directly from the camp to meet Adam at point C.

a) Find the distance from the camp to point C.

b) Find the bearing that Jacob must take to point C.

Convert each part of the journey to a complex number then add them up.

The distance is $6.05$ km. The bearing at which Jacob should travel is $83.3\degree$

If the angle is negative, add $360\degree$ to convert to bearings.

IB has yet to accept this approach. But it can be a quick way to verify calculations.

Euler’s method (HL)

This method is by Dr. William J. Larson .

Example:

Given that

use a step size of $h= 0.1$, estimate $y$ when $x = 0.5$

1. Store the initial values in X and Y. Optionally we can also store the step size in H.

   

2. The A‑lock alpha 𝑖 ︰ • (colon) allows to write multiple commands before executing. Enter the following

   Y+H(dy/dx)→Y:X+H→X:{X,Y}
   where replace dy/dx with what is given in the question. This updates Y, updates X, then print the new coordinates.

   

3. Write down the coordinates.

4. Press entry solve enter until you get the desired $x$-value.

   

   

Write down all the coordinates for full marks. The final point is $(0.5, 1.62)\qed$

More examples

Here are a few more examples. I removed some additional tricky examples as they are more clever than useful. Also I feel that it’s already too long. If you have any specific questions on TI 84 Plus or other graphing calculators, feel free to reach out.

solver vs graphing vs ???

Example: Solve $\displaystyle \frac{\e^x}{\pi x^{\e+3}} = 10^{24}$.

As we will hope to solve this multiple ways, the left side is stored into Y1 for reuse.

solver

If we guess 0, then we get the root of $5.20 \times 10^{-5}$

But if we guess 1, then we get the root of $81.6$

graphing

This is very difficult to graph. This might be a first attempt, X from $0$ to $100$, ZoomFit. We can easily find the larger root. But we cannot view any details narrower than Δx in tblset f2 window so it is very easy to be misled.

Instead we can borrow the concept of log-log plots from AI HL (now that AA papers keep introducing and testing AI applications). Instead of plotting $y$ against $x$, we can plot $\log y$ against $\log x$. Suppose we want $x$ from $10^{-7}$ to $10^4$, we can enter Y3 = log(Y1(10^X)) and do a ZoomFit for X between $-7$ and $4$.