How to find intersections, zeros, maximum and minimum points on the Casio fx-CG50
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
Checking a sketch, the number of roots before a sign-change argument, the coordinates of a turning point you found by differentiating, and the intersection of two curves before you set up an integral for an area.
Practise it: Algebra and functions · Differentiation · Numerical methods.
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 not allowed in IB, A Level or IGCSE exams.
- 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 A 2, 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 a quadratic, the fx-991EX shows its vertex (x-Value Minimum / y-Value Minimum) as extra results.
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, degree 3, coefficients 1, 0, −4, 1, then EXE through the roots.
- Intersections: Equation → Solver with X³ − 4X + 1 = 2 − X, using different starting values to find each root.
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)).
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.
Do it in Desmos
Not in the exam. Desmos is not allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- 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.
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator and type:
f(x)=x^{3}-4x+1y=2-x
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 OS versions 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. Worked-example answers were recomputed independently in Python.