TI-84 Skills for the IB Maths HL

▶ Essential Setup & Navigation

First-Time Setup

Upgrade OS: 2nd MEM → 1:About (check version ≥ 2.55MP).

Install: PlySmlt2 (Polynomial & System Solver).

Reuse previous entry: scroll up with ▲, press ENTER.

Jump to start/end of expression: 2nd ◄ or 2nd ►.

Common Errors

ERR: DIM MISMATCH — STATPLOT is on. Go to Y=, unhighlight plots with ENTER.

ERR: WINDOW RANGE — Xmin ≥ Xmax. Fix in WINDOW.

X = 5.67E-12 means $X = 0$ (floating-point artifact).

Hard reset: 2nd MEM → 5:Reset → 2:Defaults → 2:Reset.

Fractions & Exact Values

MATH 1: ►Frac — decimal → fraction.

$0.375 \Rightarrow \tfrac{3}{8};\quad 1371/3656 \Rightarrow \tfrac{3}{8}$.

MATH 2: ►Dec — fraction → decimal.

Identifying Exact Answers

If answer looks irrational, try:

Square it → integer? → it’s $\sqrt{n}$.
Divide by $\pi$ → rational? → it involves $\pi$.
Check $\ln(\text{ans})$ → rational? → it’s $e^k$.

Example: $1.7320508\ldots \xrightarrow{x^2} 3 \Rightarrow \text{ans} = \sqrt{3}$.

Degree/Radian Mode

Check MODE — IB always uses radians unless stated. Confirm before every trig calculation.

Storing & Recalling Values

After any CALC result, press ENTER to store coordinates into variables $X$ and $Y$.

Recall $x$: X,T,θ,n or ALPHA X.

Recall $y$: ALPHA Y.

Store manually: value STO► A (or any letter).

Example: √(10) STO► T → recall with T.

Ans Variable

Ans always holds the last result. Chain calculations: √(Ans), Ans² etc.

Avoid retyping long decimals — use Ans for full precision.

WINDOW Settings

ZStandard: ZOOM 6 — sets $[-10,10] \times [-10,10]$.

ZTrig: ZOOM 7 — ideal for trig functions (in radians).

ZSquare: ZOOM 5 — equal aspect ratio (use for inverse/DrawInv).

ZoomFit: ZOOM 0 — auto-fits $y$-range for current $x$-window.

Speed up slow graphs: set Xres = 4 (or 8) in WINDOW.

Useful Custom Windows

Probability: $[-0.1,\ 1.1]\times[-0.1,\ 1.1]$.

Normal dist: $[\mu-4\sigma,\ \mu+4\sigma]$.

Trig: $[0,\ 2\pi]\times[-2,\ 2]$.

▶ Calculus

2nd CALC Menu — Core Tools

Zero	Roots of $f(x)=0$; set left & right bounds.
Minimum	Local min; set left & right bounds.
Maximum	Local max; set left & right bounds.
Intersect	Solve $f(x)=g(x)$; press ENTER×3.
dy/dx	Numerical $f'(a)$ at typed $x$-value.
∫f(x)dx	Definite integral; set lower & upper bounds.

Always use CALC — never TRACE/ZOOM to read off values. TRACE only hits pixel centres and will be wrong to 3 s.f.

Derivatives

Numerical derivative at a point:

MATH 8: nDeriv(expr, x, value)

Example: $\dfrac{d}{dx}(x^2-2x+3)\Big|_{x=4}$

nDeriv(x²-2x+3, x, 4) $\to 6$.

Or: graph → 2nd CALC 6:dy/dx → type $x$.

Tangent Line Equation

Graph function → 2nd DRAW 5:Tangent( → type $x$-value.

TI displays the equation of the tangent line.

Example at $x=1$: $y = 3x – 4$.

Second Derivative (graph)

$Y_2 = $ nDeriv(Y1,x,x) → $f'(x)$.

$Y_3 = $ nDeriv(Y2,x,x) → $f”(x)$.

Set Xres=8 in WINDOW — slow otherwise.

Definite Integrals

Method 1 (Graph): Graph $f(x)$ → 2nd CALC 7:∫f(x)dx → lower bound → upper bound.

Method 2 (Home screen): MATH 9:fnInt(expr, x, a, b).

Method 1 preferred — more reliable.

Area between curves: $\displaystyle\int_a^b |f(x)-g(x)|\,dx$.

Type Y1 = |Y1orig – Y2orig| then integrate, or split at intersections.

Solve Integral Equations

Solve $\displaystyle\int_0^m \frac{dx}{2x+3}=1$:

Y4 = fnInt(1/(2x+3),x,0,x), Y5 = 1.

Intersect → $m \approx 9.58$. Set Xres=4.

Inflection Points & Optimisation

Find inflection of $f(x)$:

Graph $f$ in $Y_1$.
Graph $f’ = $ nDeriv(Y1,x,x) in $Y_2$.
Find Min/Max of $Y_2$ using CALC → gives inflection $x$.
Or find Zero of $Y_3 = \text{nDeriv}(Y_2,x,x)$.
On home screen: type $X$ to confirm, then Y1(X) for $y$-coord.

Example $y=x^3+2x-4$: inflection $\approx(-2/3,\ 3.26)$.

Distance Optimisation

Minimise $d^2$ (not $d$) — avoids square roots.

Point on $f(x)=x^2$ nearest $A(6,0)$:

Y = (x-6)²+(x²)²; CALC Min → $a\approx1.33$.

Euler’s Method for DEs

Given $\dfrac{dy}{dx}=f(x,y),\ y(x_0)=y_0,\ \text{step }h$:

$x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h\cdot f(x_n,y_n)$

$(x_n,\ y_n)$

$f(x_n,y_n)=\dfrac{dy}{dx}$

$\Delta y = h\cdot f(x_n,y_n)$

$(x_{n+1},\ y_{n+1})$

On TI-84 using TABLE: Store $x_0$ in $A$, $y_0$ in $B$, step $h$ in $H$.

In Y= set: $Y_1 = f(x,y)$ where $y =$ previous $Y$ value (manual iteration).

Alternatively use the sequence mode: MODE → SEQ, set $u(n)=u(n-1)+h\cdot f(n\text{th }x,u(n-1))$.

▶ Algebra & Functions

Solving Equations

Graphical (recommended): Set both sides as $Y_1$ and $Y_2$, find intersection.

Root-finding: Rearrange to $f(x)=0$, graph, use CALC 2:Zero.

PlySmlt2: Polynomial Roots

APPS → PlySmlt2 → 1:Poly Root Finder

Solves any polynomial (real + complex roots) up to degree 10.

Example: $3x^3-2x+1=0 \Rightarrow x=-1$ (only real root).

PlySmlt2: Simultaneous Equations

APPS → PlySmlt2 → 2:Simult. Eq. Solver

Up to 10 equations, 10 unknowns.

Example: $2x+3y=5,\ 3x+5y=7 \Rightarrow x=4,\ y=-1$.

Matrix Operations

Access: 2nd MATRIX → EDIT to enter; NAMES to recall.

Solve $AX=B$: $X = [A]^{-1}[B]$.

Inverse: [A]^{-1} ENTER.

Determinant: MATH → det([A]).

Transpose: MATRIX → MATH → T (or T superscript).

rref( — Row Reduction

MATRIX → MATH → rref(

Enter augmented matrix $[A|B]$. Output gives reduced form.

Example: system with $\lambda=5$ →
$x=2-3z,\ y=1+z,\ z=z$.

Piecewise Functions & Domain

Restrict domain (divide by boolean):

$Y_1 = (3x^2-4)/(x \ge 0)$

Piecewise (multiply-and-add):

$f(x)=\begin{cases}x+3&x\le0\\3&0<x\le2\\2x-1&x>2\end{cases}$

$Y_1=(x{+}3)(x{\le}0)+3(x{>}0)(x{\le}2)+(2x{-}1)(x{>}2)$

Logical operators: 2nd TEST; AND/OR: 2nd TEST LOGIC.

Composite Functions

Graph $f(g(x))$: set Y3 = Y1(Y2(X)).

Access $Y_n$ names with VARS → Y-VARS → 1:Function.

Logarithms & Exponentials

Change of base: $\log_a x = \dfrac{\ln x}{\ln a}$

ALPHA F2 → 5:logBASE( for direct $\log_a x$.

Example: $\log_2 3$ → logBASE(3,2) $\approx 1.585$.

Or use: log(3)/log(2).

Graphing Inverse Functions

Put $f(x)$ in Y1. Quit to home screen.

2nd DRAW → 8:DrawInv( Y1.

Use ZOOM 5:ZSquare for correct aspect ratio.

TI does not check if $f$ passes HLT — you must check.

Solve Exponential Equations

Solve $3=2^x$: graph $Y_1=2^x,\ Y_2=3$; find intersection → $x\approx1.585$. Or use logBASE(3,2).

TABLE (2nd GRAPH)

Setup: 2nd TBLSET — set TblStart and ΔTbl.

Then 2nd TABLE to view values.

Investment problem: \$5000 at 6.3% exceeds \$10 000 after $n$ years?

Y3 = 5000(1.063)^x; TblStart=0, ΔTbl=1.

Scroll until $Y_3 > 10000$ → $n=12$.

Factoring via Roots

TI can’t factor symbolically. Graph polynomial, find zeros.

$10x^3-9x^2-13x+6$: zeros at $0.4,1.5,-1$.

→ $10(x-0.4)(x-1.5)(x+1)$. Multiply out to verify.

Parametric Mode

Switch: MODE → PAR. Enter Y= for $X_{1T}$ and $Y_{1T}$.

Set WINDOW: Tmin, Tmax, Tstep.

Nearest point on line to origin:

Line: $x=1-\lambda,\ y=2-3\lambda,\ z=2$.

Type $d^2 = (1-x)^2+(2-3x)^2+4$ in $Y=$ (function mode, $x=\lambda$).

CALC Min → $\lambda\approx0.7$. Substitute back.

Parametric Derivatives

$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$. Use dy/dx in CALC menu while in PAR mode.

▶ Complex Numbers

Complex Number Operations

TI operates in a+bi mode: MODE → a+bi.

Enter $i$: 2nd i.

Example: $(3+2i)(1-i)$ — type directly and press ENTER.

MATH CPX menu:

1:conj( — complex conjugate $\bar{z}$
2:real( — $\text{Re}(z)$
3:imag( — $\text{Im}(z)$
4:angle( — $\arg(z)$ in current angle mode
5:abs( — $|z|$

Argument, Modulus & Polar Form

$|z| = $ abs(z); $\arg(z) = $ angle(z) (radians by default).

$z = r(\cos\theta + i\sin\theta)$ — convert manually using real( and imag(.

Find exact b using angle(

Given $z=(b+i)^2,\ \arg(z)=60°$. Find exact $b$.

Graph $Y=$ angle((X+i)²)-60 (degree mode).
Find zero → $b\approx 1.732$.
Square: $1.732^2\approx3 \Rightarrow b=\sqrt{3}$.

De Moivre / Powers

Compute $z^n$ directly: type $(a+bi)^n$ in a+bi mode. Use ►Frac or abs/angle to interpret result.

Roots of Unity & Polynomials

Find all roots of $z^n=w$: use PlySmlt2 Poly Root Finder.

Switch to a+bi mode first to see complex roots.

Example: $z^3-1=0$: enter coefficients [1,0,0,−1] → roots $1,\ -\tfrac{1}{2}\pm\tfrac{\sqrt{3}}{2}i$.

Verify roots by substituting back: (a+bi)^3 should equal the RHS.

Tip: Radian vs Degree for arg

angle( returns result in current MODE. IB HL usually wants radians. Double-check MODE before using angle(.

Verifying Complex Results

Always verify by:

Substituting root back into the original equation.
Checking $\text{real}(\text{result}) \approx 0$ and $\text{imag}(\text{result}) \approx 0$.

For $z = re^{i\theta}$: $r = $ abs(z), $\theta = $ angle(z).

Check: r*(cos(θ)+i*sin(θ)) should equal z.

Conjugate Root Theorem check

If $a+bi$ is a root of a real polynomial, so is $a-bi$. Confirm with conj(root) and re-substitute.

▶ Statistics & Probability

Lists & Descriptive Statistics

Enter data: STAT → EDIT → 1:Edit → type into L1.

Clear list: cursor to list name → CLEAR ENTER (not DEL — DEL removes the list name).

Restore L1–L6: STAT → 5:SetUpEditor ENTER.

1-Var Stats

STAT → CALC → 1:1-Var Stats L1

With frequencies: 1-Var Stats L1, L2.

Gives: $\bar{x}$ (mean), $\sigma_x$ (population SD), $S_x$ (sample SD), $n$, $\Sigma x$, $\Sigma x^2$, min, max.

Do NOT use 1-Var Stats for Median, Q1, Q3 — read them from the 5-number summary (↓ to scroll).

2-Var Stats

STAT → CALC → 2:2-Var Stats L1, L2 — gives $\bar{x},\ \bar{y},\ \Sigma xy$, etc.

Regression (Bivariate Analysis)

Enter $x$ in L1, $y$ in L2.

STAT → CALC → 4:LinReg(ax+b) L1,L2,Y1

Storing to Y1 lets you graph the regression line immediately.

Output: $a$ (slope), $b$ (intercept), $r$ (correlation), $r^2$.

Turning on Diagnostics (r, r²)

2nd CATALOG → DiagnosticOn → ENTER ENTER.

Must do this once per reset — otherwise $r$ and $r^2$ are hidden.

Other Regression Types

STAT → CALC menu contains:

ExpReg $y=ab^x$
PwrReg $y=ax^b$
QuadReg $y=ax^2+bx+c$
LnReg $y=a+b\ln x$

Normal Distribution

2nd DISTR menu:

normalcdf(lower, upper, μ, σ)

For $P(X<a)$: lower = $-10^{99}$ (use (-) 1 EE 99 or $-E99$).

For $P(X>a)$: upper = $10^{99}$.

Example: $\mu=20,\sigma=3,\ P(19\le x\le 23)$: normalcdf(19,23,20,3) $\approx 0.4719$.

invNorm — Finding Cutoff Values

invNorm(area to LEFT, μ, σ)

Example: $P(X<d)=0.05,\ \mu=20,\sigma=3$:

invNorm(0.05, 20, 3) $\approx 15.1$.

For $P(X>d)=0.05$: use invNorm(0.95, μ, σ).

Find μ and σ Simultaneously

Use $\dfrac{x-\mu}{\sigma}=\texttt{invNorm}(p,0,1)$. Set up two equations, graph in $Y=$ with $y=\mu,\ x=\sigma$, find intersection.

Example: $P(X<84)=0.15,\ P(X>95)=0.1 \Rightarrow \mu\approx88.9,\ \sigma\approx4.75$.

Discrete Distributions

Binomial $X\sim B(n,p)$:

binomPDF(n, p, k) → $P(X=k)$.

binomCDF(n, p, k) → $P(X\le k)$.

For $P(X\ge k)$: $1 – $ binomCDF(n,p,k-1).

For $P(a\le X\le b)$: binomCDF(n,p,b) – binomCDF(n,p,a-1).

Poisson $X\sim \text{Po}(\lambda)$:

poissonPDF(λ, k) → $P(X=k)$.

poissonCDF(λ, k) → $P(X\le k)$.

All CDF gives $P(X \le k)$. IB often asks $P(X \ge k)$ or $P(X < k)$ — adjust carefully.

Hypothesis Testing HL Stats Option

$z$-test & $t$-test (means):

STAT → TESTS menu:

1:Z-Test — known $\sigma$
2:T-Test — unknown $\sigma$
3:2-SampZTest
4:2-SampTTest
5:1-PropZTest
6:2-PropZTest

Input: Stats (enter $\bar{x}, s, n$) or Data (from list).

Output: test statistic, $p$-value. Compare $p$ to $\alpha$.

$\chi^2$ Tests:

STAT → TESTS → C: χ²-Test

Enter observed matrix in [A], expected auto-calculated to [B].

Output: $\chi^2$ statistic, $p$-value, df.

invT — Critical Values:

2nd DISTR → invT(area, df)

Example: $t_{0.025,\ 9} = $ invT(0.975, 9) $\approx 2.262$.

$\chi^2\text{cdf}(a, b, \text{df})$ for critical values.

Confidence Intervals HL Stats Option

STAT → TESTS:

7:ZInterval — CI for $\mu$, known $\sigma$
8:TInterval — CI for $\mu$, unknown $\sigma$
A:1-PropZInt — CI for proportion
B:2-PropZInt
C:χ²-Test

Input $\bar{x},\ s_x,\ n$, confidence level (e.g. 0.95).

Output: interval $(\text{lower},\ \text{upper})$.

Manual verification:

$\bar{x} \pm t^* \cdot \dfrac{s}{\sqrt{n}}$

where $t^* = $ invT((1+C)/2, n-1).

Graph Distributions to Solve for Parameters

Find $p$ if $P(X=4)=0.12,\ X\sim B(5,p)$:

Y1 = binomPDF(5,X,4) – 0.12

Window: $-0.1 < x < 1.1$. CALC Zero → $p\approx0.459$ (or 0.973).

Find $\lambda$ if $P(X=3)=0.20,\ X\sim\text{Po}(\lambda)$:

Y1 = poissonPDF(X,3) – 0.20. Find zero.

Combinatorics

Combinations: $\dbinom{n}{r} = $ n MATH PRB 3:nCr r.

Example: $\dbinom{5}{3} = $ 5 nCr 3 $= 10$.

Permutations: $P(n,r) = $ n MATH PRB 2:nPr r.

Factorial: n MATH PRB 4:!

IB booklet gives $\dbinom{n}{r}$ — use nCr on TI to verify expansion coefficients.

▶ Sequences, Series & Finance

Sequences & Series

Sequence mode: MODE → SEQ.

Define $u(n)$ in Y= as $u(n) = u(n-1)+d$ (AP) or $u(n)=r\cdot u(n-1)$ (GP).

Set $n\text{Min}$, initial value $u(\text{nMin})$.

Sum of sequence (home screen):

2nd LIST → MATH → 5:sum(

sum(seq(expr, x, start, end))

Example: $\displaystyle\sum_{k=1}^{10}k^2$: sum(seq(X²,X,1,10)) $=385$.

Partial Sums

2nd LIST → OPS → 6:cumSum( — running total of a list. Store a sequence as a list first using seq(.

Finance App (TVM Solver)

APPS → 1:Finance → 1:TVM Solver

Variables:

N — total number of payments/periods
I% — annual interest rate (as %)
PV — present value (negative if cash out)
PMT — payment per period
FV — future value
P/Y — payments per year
C/Y — compounding per year

Fill in known values, cursor to unknown, press ALPHA ENTER to solve.

Sign convention: money paid out is negative; money received is positive. Be consistent.

TABLE for Sequence Problems

Use TABLE to iterate compound interest / recursive sequences quickly.

Example: Find minimum $n$ for investment to double:

Put formula in Y=, e.g. Y1 = 5000(1.063)^x.
Set TblStart = 0, ΔTbl = 1.
Scroll until target exceeded.

Faster than solving algebraically when $n$ must be an integer.

seq( on Home Screen

seq(expr, var, start, end, step)

Example: first 5 terms of $a_n=2n-1$:

seq(2X-1, X, 1, 5) → {1,3,5,7,9}.

IB-Allowed Calculator Apps

PlySmlt2 — Polynomial Root Finder & Simultaneous Equation Solver
Finance — TVM Solver
CtlgHelp — Catalogue Help
Language apps
CBL/CBR

Delete all others: 2nd MEM → 2:Mem Mgmt/Del → A:APPS. Cursor → DEL.

Exam Reminders

Check RADIAN mode before each paper.
Show all GDC setup — write the function entered and result.
Round final answers to 3 s.f. unless told otherwise.
Never copy a decimal from TRACE — always use CALC.
After Reset: re-enable DiagnosticOn for $r$ values.

▶ Advanced / Less Common (but HL-Tested) Techniques

Polar Graphing Mode

Switch: MODE → POL. Enter $r(\theta)$ in Y= as $r_1$.

Set WINDOW: $\theta\text{min}=0,\ \theta\text{max}=2\pi,\ \theta\text{step}=\pi/24$.

ZOOM 5:ZSquare for correct aspect ratio.

Area of polar region: $\displaystyle A=\tfrac{1}{2}\int_\alpha^\beta r^2\,d\theta$.

Graph $Y_1 = \tfrac{1}{2}r_1^2$ and use fnInt or CALC ∫.

Newton-Raphson / Numerical Methods

Newton-Raphson iteration: $x_{n+1}=x_n – \dfrac{f(x_n)}{f'(x_n)}$.

On TI: store $x_0$ in X. Then type:

X – Y1/nDeriv(Y1,X,X) STO► X

Press ENTER repeatedly to iterate. Converges to root.

Bisection check: verify sign change on either side of root with Y1(a) and Y1(b).

Higher Derivatives & Concavity

$Y_1 = f(x)$

$Y_2 = $ nDeriv(Y1,x,x) → $f'(x)$

$Y_3 = $ nDeriv(Y2,x,x) → $f”(x)$

Set Xres=8 in WINDOW (slow otherwise).

Concavity: $f”(x)>0$ → concave up; $f”(x)<0$ → concave down.

Inflection: $f”(x)=0$ and sign change → CALC Zero on $Y_3$.

Confirm: substitute $x$-value back into $Y_1(X)$ on home screen.

Domain Restrictions & Absolute Value

Absolute value: MATH → NUM → 1:abs(.

Graph $|f(x)|$: Y1 = abs(expr).

Restrict domain: divide by boolean expression.

$Y_1=(3x^2-4)/(x\ge0)$ — graphs only for $x\ge0$.

Multiple pieces:

$Y_1 = f_1(x)(x\le a) + f_2(x)(x{>}a)(x\le b) + f_3(x)(x{>}b)$

Logical: 2nd TEST; AND/OR: 2nd TEST LOGIC.

HL Paper 2 GDC Exam Strategy

Before the paper:

Confirm RADIAN mode.
Run DiagnosticOn for regression $r$.
Clear all Y= functions.
Clear STAT lists.
Check battery.

Showing GDC work (IB requirement):

State the equation/expression entered.
State the result to full precision first.
Round to 3 s.f. for final answer.
Write “by GDC” or “from GDC” where applicable.

Common HL GDC uses:

Solve equations graphically (intersect/zero).
Evaluate definite integrals.
Find stat test $p$-values.
Regression + $r^2$.
Matrix operations ($A^{-1}B$, det, rref).
Binomial/Poisson probabilities.
invNorm / invT for critical values.

Precision reminders:

Never round intermediate values — use Ans.
Final answer: 3 s.f. (unless exact or integer).
Money: 2 d.p.
Angles (degrees): 1 d.p.
Angles (radians): 3 s.f.
Probabilities: 4 d.p. or 3 s.f.

TI-84 Skills for the IB Maths HL - New Syllabus

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