Key Idea: When equations are too complex to solve by hand, use your GDC to find the answer graphically. The key skill is setting up the problem correctly, choosing the right tool, then reading and interpreting the answer.
Three things IB tests on this topic:
Find intersections
- Enter both sides as Y₁ and Y₂
- Use 5:intersect
- Read the x-value or coordinate
Find roots
- Enter f(x) as Y₁
- Use 2:zero
- Run it once per root
Interpret meaning
- Intersection = equal values
- Root = output becomes zero
- Explain in context
📐 Two methods — same idea
| Intersection method | Root method | |
|---|---|---|
| Equation form | f(x) = g(x) | f(x) − g(x) = 0 |
| What you graph | Both functions separately | One combined function |
| What you look for | Where the graphs cross | Where the graph crosses the x-axis |
| GDC action | 5:intersect | 2:zero |
| Answer type | x-value or coordinate | x-value |
Solving f(x) = g(x) is the same as finding the roots of f(x) − g(x) = 0. On the GDC, intersections are often easier because you can enter both sides separately.
✏️ Worked examples
Simultaneous equations — one non-linear
Solve y = x² − 2x and y = x + 4.
Step by step:
Enter Y₁ = x² − 2x and Y₂ = x + 4 on the GDC.
Adjust the window so both crossings are visible.
Use 5:intersect near each crossing.
The intersections have x-values −1 and 4.
x = −1 or x = 4
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Find roots of a cubic
Find all roots of f(x) = x³ − 4x + 1.
Step by step:
Enter Y₁ = x³ − 4x + 1.
Graph the function and check how many times it crosses the x-axis.
Use 2:zero once around each crossing.
The roots are approximately −2.11, 0.25, and 1.86.
x ≈ −2.11, 0.25, 1.86
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Context — equal cost
Two cost functions are C₁ = 50 + 2x and C₂ = 100 + x. Find where they cost the same.
Step by step:
Enter Y₁ = 50 + 2x and Y₂ = 100 + x.
Use 5:intersect to find where the two cost graphs meet.
The intersection is (50, 150).
Interpret both coordinates.
Equal cost when x = 50. Both cost $150.
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Sketch the graph or describe your GDC setup so your method is clear. Window settings matter — if you cannot see the crossing or root, zoom out or manually set axes. How many roots? A quadratic can have up to 2 real roots. A cubic can have up to 3 real roots. The graph tells you what to expect. For interpretation, do not just write a coordinate. Explain what x and y mean in the context.
IB-style question [5 marks]
Solve the simultaneous equations y = x² − 5 and y = 2x + 3 using technology.
Step by step:
The solutions are where the parabola and the line intersect, so set the two expressions equal.
Move everything to one side to see the quadratic you are solving.
On the GDC, graph Y₁ = x² − 5 and Y₂ = 2x + 3 and use 5:intersect at each crossing — there are two.
Find each y-value from the line y = 2x + 3.
The graphs meet at (−2, −1) and (4, 11), so x = −2 or x = 4.