How to find intersections, zeros, maximum and minimum points on the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II
On the fx-CG50: MENU → 5 (GRAPH) … SHIFT F5 (G-SOLVE).On the TI-84 Plus CE: 2nd trace (CALC).On the TI-Nspire CX II: menu → Analyze Graph → Zero.
Draw a graph, then read off roots, turning points and where two curves meet with G-Solve / CALC / Analyze Graph. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
The most-used GDC skill in AI and AA Paper 2: roots of a model, the maximum profit, the times two particles meet, the vertex of a quadratic model. Write down what you did ("intersection of y = f(x) and y = 2 using GDC") and give coordinates to 3 s.f.
Practise it: SL AI applications of functions · SL AA graphing functions · SL AI optimisation · HL AA polynomials.
Which IB course and paper
| Course | GDC papers | What you use it for |
|---|---|---|
| AA SL | Paper 2 (Paper 1 is calculator-free) | Intersections, zeros and turning points on Paper 2; trig equations in an interval. |
| AA HL | Papers 2 and 3 (Paper 1 is calculator-free) | As SL, plus inequalities and rational functions solved graphically. |
| AI SL | Papers 1 and 2 | Zeros, maximum/minimum values and intersections of models; optimisation. |
| AI HL | Papers 1, 2 and 3 | As SL, with harder models. |
A GDC is allowed on every AI paper and on AA Papers 2 and 3; AA Paper 1 is calculator-free, so learn the method by hand as well.
Topic notes: Quadratic functions (AI SL) · Optimisation (AI SL) · Piecewise functions and modelling (AI SL) · Optimisation using calculus (AA SL) · Rational functions and asymptotes (AA SL) · Inequalities (AA HL) · Modelling with functions (AI HL).
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Graphs: intersections, zeros, max and min on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
Watch the key presses on the fx-CG5013 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- MENU 5 (GRAPH). Type Y1 = X³ − 4X + 1 (X is the X,θ,T key) and EXE; type Y2 = 2 − X and EXE.
- F6 (DRAW). If the curve is off screen, SHIFT F3 (V-WIN) to set Xmin/Xmax/Ymin/Ymax, or F2 (ZOOM) → AUTO.
- SHIFT F5 (G-SOLVE) opens the menu: F1 ROOT, F2 MAX, F3 MIN, F4 Y-ICEPT, F5 INTSECT (intersection).
- Choose ROOT (or MAX, MIN). With two graphs drawn, pick the curve with ▲ ▼ and press EXE. Press ► to jump to the next root, max or min to the right.
- For an intersection choose F5 (INTSECT), select the two graphs with EXE each time, then ► for the next one.
- F6 (▷) inside G-SOLVE gives more: Y-CAL / X-CAL (the y for a given x and the x for a given y) and ∫dx (the area under the curve between two x values).
- After G-Solve the result is kept in the X and Y variables until you solve again: go back to RUN-MATRIX and store the one you need, e.g. X → A.
- The roots, MAX and MIN are only found inside the current View Window: widen it if a turning point is off screen.
Graphs: intersections, zeros, max and min on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
Watch the key presses on the TI-84 Plus CE11 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- y=: Y1 = X^3 − 4X + 1, Y2 = 2 − X (X is the X,T,θ,n key). graph. Use zoom 6: ZStandard or window if needed.
- 2nd trace (CALC): 2: zero, 3: minimum, 4: maximum, 5: intersect.
- zero, minimum, maximum: choose the curve, then move the cursor to the left of the point and press enter (Left Bound?), to the right and enter (Right Bound?), then near it and enter (Guess?).
- intersect: enter on the first curve, enter on the second, move near the intersection and enter (Guess?). Repeat for each intersection.
- The answer is the X= and Y= at the bottom of the screen; they are stored in X and Y until the next calculation.
- Maximum/minimum use a numerical method: the last digit can differ slightly (e.g. 1.1547 vs 1.1546993). Round to 3 s.f.
Graphs: intersections, zeros, max and min on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS (the CX II CAS is a different model: check the exam rules before using one).
- Add a Graphs page: ctrl doc → Add Graphs. Type f1(x) = x^3 − 4x + 1, enter; tab to f2(x) = 2 − x, enter.
- menu → Window/Zoom to adjust the view if needed.
- menu → Analyze Graph → Zero, Minimum, Maximum or Intersection.
- Choose the graph, click to the left of the point (lower bound) and to the right (upper bound). The point and its coordinates are labelled on the graph.
- For an intersection click on each graph first, then give the lower and upper bounds.
- Hover over the labelled coordinates and press ctrl menu → Store to save them to a variable.
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Worked example
f(x) = x³ − 4x + 1. Find (a) the zeros of f, (b) the local maximum and minimum, (c) where y = f(x) meets y = 2 − x.
Answers (checked with Python)
(a) x = −2.11491, 0.254102, 1.86081. (b) Maximum (−1.15470, 4.07920); minimum (1.15470, −2.07920). (c) x = −1.53209, −0.347296 and 1.87939 (the last one at (1.87939, 0.120615)).
Now try it on an exam-style question
Use the keystrokes above on IB-style functions and models questions. Free to try, no sign-up.
Common calculator mistakes
- The turning point is off the screen, so MAX finds nothing or an end point: widen the View Window first.
- Picking the wrong curve when two are drawn: check which equation is highlighted at the top of the screen before pressing EXE.
- Giving only x when the question asks for the coordinates (or only y when it asks for the maximum value).
- Trig graphs in degree mode when the question is in radians (or the other way round): the graph looks flat or squashed.
- Reading values off a trace instead of using G-Solve/CALC: tracing only lands on pixel positions and is not accurate enough.
- An answer read off the graph without saying so: write "using GDC" and the equation you solved, so the method mark is available.
- Angle mode: calculus with trig functions (and most AA trig) needs radians; bearings and triangle questions use degrees. Check the mode before the first trig question of the paper.
- Accuracy: unless the question says otherwise, IB markschemes want answers exact or to 3 significant figures. Keep the full calculator value for later parts and round only the answer you write down (x = 2.094551… is written 2.09, not 2.1 or 2.0946).
If you get an error
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:INVALID DIM, ERR:SYNTAX
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Exponential and power models need positive data. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:INVALID DIM TI's knowledge base: one or more stat plots are turned on, but the lists they use are empty. Check: A stat plot left on from a statistics question, so the error appears when you graph a function. Fix: Turn off the stat plots you are not using, or put the data back in the lists.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Type f(x) = x^3 − 4x + 1 and y = 2 − x.
- Click a curve: the zeros, turning points and intersections appear as grey points. Click one to see its coordinates.
- Drag the view or use the wrench to set the axes; Desmos keeps finding the points as you zoom.
Open desmos.com/calculator (free, in any browser) and type:
f(x)=x^{3}-4x+1y=2-x
Questions
Can I use a calculator for graphs: intersections, zeros, max and min in the exam?
In IB Maths you can use one approved GDC on every AI paper and on AA Papers 2 and 3, in exam mode. AA Paper 1 is calculator-free, so learn the method by hand as well.
What are the fx-CG50 keystrokes for graphs: intersections, zeros, max and min?
MENU 5 (GRAPH). Type Y1 = X³ − 4X + 1 (X is the X,θ,T key) and EXE; type Y2 = 2 − X and EXE. Then F6 (DRAW). If the curve is off screen, SHIFT F3 (V-WIN) to set Xmin/Xmax/Ymin/Ymax, or F2 (ZOOM) → AUTO. Then SHIFT F5 (G-SOLVE) opens the menu: F1 ROOT, F2 MAX, F3 MIN, F4 Y-ICEPT, F5 INTSECT (intersection). The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer exact or to 3 significant figures unless the question says otherwise. Keep full accuracy in later parts.
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the models shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.