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).
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
Graphical solution questions ("use your graph to solve …") and checking the turning point of a quadratic. A scientific calculator cannot draw graphs, so use its table of values instead (see the fx-991EX / fx-991CW tabs).
Practise it: IGCSE units (Unit 3: Graphs).
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).
- 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.
- 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.
Graphs: intersections, zeros, max and min on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
The fx-991EX cannot draw graphs. Use MENU 9 (Table) to see where f(x) changes sign or turns, and MENU A → Polynomial for the roots and the vertex of a quadratic or the roots of a cubic. For f(x) = g(x) use SOLVE on f(x) − g(x) = 0.
- Zeros of a polynomial: MENU, highlight Equation/Func with the cursor keys and select it, 2 (Polynomial), degree 3, coefficients 1, 0, −4, 1, then = through the roots.
- Turning points of a cubic: roots of f′(x) = 3x² − 4 = 0 give x = ±1.1547; substitute into f with CALC.
- Intersections: MENU 1, type X³ − 4X + 1 = 2 − X, SHIFT CALC with different start values (−2, 0, 2) to find each root.
- After the roots of a quadratic, the fx-991EX also shows the coordinates of its vertex.
Graphs: intersections, zeros, max and min on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
The fx-991CW cannot draw graphs. Use the Table app to spot sign changes and turning points, the Equation app (Polynomial) for polynomial roots and a quadratic's vertex, and Solver for f(x) = g(x).
- Zeros of a polynomial: HOME → Equation → Polynomial → ax³+bx²+cx+d, coefficients 1, 0, −4, 1, then EXE through the solutions.
- Intersections: Equation → Solver with X³ − 4X + 1 = 2 − X (= is SHIFT (), using different starting values (−2, 0, 2) to find each one.
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 IGCSE-style quadratic and cubic graphs 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.
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.
fx-991EX: Math ERROR, Syntax ERROR, Can't Solve
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, 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 the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation. In Matrix and Vector calculations it also appears when the sizes do not suit the operation. Check: A missing or extra bracket, or two operation signs in a row. With matrices or vectors: adding a 2 × 2 to a 3 × 3, or a product whose sizes do not match. Fix: Casio's advice: make the corrections. For matrices and vectors, check the size you set for each one.
Can't Solve SOLVE could not find a solution. Check: A typing slip in the equation, or a start value far from the root you want. Fix: Casio's advice: check the equation for mistakes, then give a start value close to the solution you expect.
fx-991CW: Math ERROR, Syntax ERROR
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, 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 the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation: it is typed in a form the calculator cannot read. Check: A missing or extra bracket, two operation signs in a row, or a function with nothing after it. Fix: Casio's advice: make the corrections and try again.
Do it in Desmos
Not in the exam. Desmos is not allowed in IGCSE exams. Use it to learn, to check an answer and to see the graph behind a question.
- 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?
On Pearson Edexcel International GCSE Maths A (4MA1), yes: Paper 1H and Paper 2H both allow a calculator with no CAS. On Cambridge IGCSE 0580 (from 2025), Papers 1 and 2 are non-calculator, and the syllabus says you should have a scientific calculator for Papers 3 and 4.
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, TI-Nspire CX II, Casio fx-991EX ClassWiz and Casio fx-991CW ClassWiz. 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 to the accuracy the question asks for (often 3 significant figures). 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.
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- 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.