An intersection lies on both graphs: Where two graphs cross, that point is on both curves. So its x-value makes f(x) = g(x), and its y-value is the shared output. Finding intersections = solving f(x) = g(x).
IB-style question — check a meeting point
Verify that (2, 5) lies on both y = x² + 1 and y = 2x + 1.
Step by step
- Put x = 2 into the first.
- Put x = 2 into the second.
Final answer
Both give y = 5, so (2, 5) is a common point — an intersection.
Two outputs, one point: At an intersection the two functions agree: f(a) = g(a). That single shared value is the y-coordinate of the meeting point.
Graph both, then 'intersect': On Paper 2, type both functions into the GDC, graph them, and use the intersect tool to read each meeting point's coordinates. Set a window that shows all the crossings first.
[Diagram: math-graph-intersection] - Available in full study mode
Never wonder what to study next
Get a personalized daily plan based on your exam date, progress, and weak areas. We'll tell you exactly what to review each day.
Set them equal, move to one side, solve: On Paper 1, find intersections algebraically: set f(x) = g(x), bring everything to one side, and solve. Then put each x back into either function for the y-coordinate.
IB-style question — line meets parabola
Find where y = x² + 1 meets y = 2x + 1.
Step by step
- Set the two equal.
- Bring to one side.
- Factor and solve.
- Find each y (use y = 2x + 1).
Final answer
They meet at (0, 1) and (2, 5) — matching the GDC.
Don't forget the y-coordinates: Solving gives the x-values. The question usually wants points — substitute each x back to get y.
Solving f(x) = k is an intersection too: Solving f(x) = k is finding where the graph of f meets the horizontal line y = k. Setting k = 0 gives the x-intercepts (zeros).
IB-style question — meet the x-axis, then a line
For f(x) = x² − 5x + 6, find where the graph meets (a) the x-axis and (b) the line y = 2.
Step by step
- (a) Meets the x-axis: f(x) = 0.
- So the x-intercepts are…
- (b) Meets y = 2: set f(x) = 2.
- Factor and solve.
Final answer
(a) (2, 0) and (3, 0); (b) (1, 2) and (4, 2).
[Diagram: math-graph-intersection] - Available in full study mode
Stop wasting time on topics you know
Our AI identifies your weak areas and focuses your study time where it matters. No more overstudying easy topics.
'Initial' means t = 0; 'same height' means set them equal: Modelling questions hide standard skills. Initial value = substitute t = 0. When are the two models equal? = solve A(t) = B(t) — graph both on the GDC and use intersect, reading every crossing in the given interval.
[Diagram: math-graph-intersection] - Available in full study mode
IB-style question — two plant models
Over 0 ≤ t ≤ 12 weeks, Plant A (given fertilizer) has height A(t) = 5t + 8 cm and Plant B (no fertilizer) has height B(t) = 0.4t² + 12 cm. (a) Find the initial height of each plant. (b) Find the values of t when the two plants have the same height.
Step by step
- (a) Initial height = value at t = 0.
- (b) Same height means A(t) = B(t) — graph both and use the GDC 'intersect' tool.
- Read both crossings from the GDC (don't solve by hand on Paper 2).
Final answer
(a) Plant A: 8 cm, Plant B: 12 cm. (b) t ≈ 0.859 and t ≈ 11.6 weeks.
Don't stop at one crossing: A curve and a line can meet more than once — here they're equal twice. Scan the whole interval and report every solution the question asks for; missing the second crossing is the classic lost mark.