IB Maths AI HLFurther Functions & GraphsPaper 2GDC intersect, simultaneous~7 min read
Intersecting Graphs
Where two graphs meet on the GDC is where the equation f(x) = g(x) is satisfied — the x-coordinates of the intersections are the solutions. For AI HL Paper 2, this turns nearly every “solve” question into “plot, then press intersect”.
📘 What you need to know
Plot both graphs on the GDC, then use the intersect tool to find each intersection’s coordinates.
The coordinates of intersection are the simultaneous solutions of the two equations.
To solve f(x) = a: plot y = f(x) and the horizontal line y = a. The x-coordinates of the intersections are the solutions.
To solve f(x) = g(x): plot both curves and read off the x-coordinates.
The number of intersections equals the number of real solutions — quick way to answer “how many solutions” questions without finding each one.
Adjust the GDC’s viewing window so every intersection is visible — missing one loses the mark.
Finding intersections with the GDC
Two graphs intersect where their y-values agree for the same x. Numerically that’s a simultaneous solution: the x– and y-coordinates of the intersection satisfy both equations at once. The GDC’s intersect function does the heavy lifting — plot the two graphs, select the curves, tap the intersection, and the coordinates appear.
Watch the window: graphs can cross more than once. Always scroll out or change the scale until you’re confident you’ve found every intersection.
Solving equations from intersections
Any equation in x can be solved by rewriting it as LHS = RHS and finding where the two sides intersect when plotted. Solve f(x) = a by plotting y = f(x) against the horizontal line y = a. Solve f(x) = g(x) by plotting both and reading off the x-coordinates where they meet. Either way, the solutions to the equation are exactly the x-coordinates of intersection.
The solutions of f(x) = a or f(x) = g(x) are the x-coordinates where the two graphs meet — not the y-coordinates.
Solving equations graphicallysolve f(x) = a: plot y = f(x) and y = a; solutions = x-coordinates of intersection
solve f(x) = g(x): plot both; solutions = x-coordinates of intersection
Counting the number of solutions
Sometimes the question asks only for the number of solutions, not the values themselves. The answer is simply the number of intersection points visible on the graph. Make sure the viewing window is wide enough to include every intersection — a transcendental equation like 2x = 3x can have two solutions you’d miss with a default window.
🧠Recipe — solving an equation using graphs
Rearrange the equation into the form f(x) = g(x) (or f(x) = a if one side is constant).
Ploty = f(x) and y = g(x) on the GDC.
Adjust the window until all intersections are visible.
Use the intersect tool for each intersection point.
Read off the x-coordinates as the solutions; if asked, state the y-coordinates too for the intersection points.
Worked examples
WE 1
Intersection of two lines
Find the point of intersection of y = 2x + 1 and y = −x + 7.
Find the points of intersection of y = x2 − 3x + 2 and y = x − 1.
set f(x) = g(x)x² − 3x + 2 = x − 1x² − 4x + 3 = 0factor(x − 1)(x − 3) = 0 ⇒ x = 1 or x = 3find y-values from the line (simpler)x = 1: y = 0x = 3: y = 2intersections: (1, 0) and (3, 2)
WE 3
Cubic meets a line
Find the coordinates of the points where f(x) = x3 − 4x and g(x) = 2x intersect, to 2 d.p.
set f = gx³ − 4x = 2xx³ − 6x = 0factorx(x² − 6) = 0 ⇒ x = 0, ±√6numerical valuesx ≈ 0, ±2.45y = 2x: 0, ±4.90(−2.45, −4.90), (0, 0), (2.45, 4.90)a cubic and a line can meet up to 3 times.
WE 4
Number of solutions
Write down the number of real solutions to the equation 2x = 3x.
plot y = 2ⁿ and y = 3xat x = 0: 2⁰ = 1, 3(0) = 0 ⇒ exponential aboveat x = 1: 2¹ = 2, 3 ⇒ line aboveat x = 4: 2⁴ = 16, 12 ⇒ exponential abovethe graphs cross twice in (0, 4)2 real solutionsthe GDC gives x ≈ 0.46 and x ≈ 3.31.
WE 5
Solve a transcendental equation
Solve ex = 5 − x, giving your answer to 2 d.p.
plot y = eⁿ and y = 5 − xeⁿ is always increasing, 5 − x always decreasingthey meet at exactly one pointuse the GDC intersect toolx ≈ 1.3066x ≈ 1.31no algebraic solution exists for this equation; the GDC is essential.
WE 6
Multi-part: sketch, count, intersect, solve
Let f(x) = x2 − 2x − 3 and g(x) = 1 − x. (a) State the vertex and intercepts of f. (b) How many solutions does f(x) = 0 have? (c) Find the intersections of f and g to 2 d.p. (d) Hence solve x2 − 2x − 3 = 1 − x.
(a) vertex: x = 2/2 = 1, f(1) = 1 − 2 − 3 = −4vertex (1, −4); roots x = −1, 3; y-int (0, −3)(b) f(x) = 0 means the parabola meets the x-axis2 solutions (the two x-intercepts)(c) f = g: x² − 2x − 3 = 1 − xx² − x − 4 = 0x = (1 ± √17)/2 ≈ −1.56 or 2.56y = 1 − x: 2.56 or −1.56(−1.56, 2.56) and (2.56, −1.56) · (d) x ≈ −1.56, 2.56
💡 Top tips
Rearrange first: the equation 3x = x2 + 1 becomes “intersect y = 3x and y = x2 + 1″ — don’t move things around inside the GDC.
Zoom out and in to confirm you’ve found every intersection.
The x-coordinate is the solution to the equation; the (x, y) pair is the intersection point.
Use the intersect tool, not the trace tool — trace is less accurate.
For transcendental equations (ex, ln, sin, etc.), graphical solution is usually the only practical method.
âš Common mistakes
Giving the y-coordinate as the solution when asked for the solution to an equation — it’s the x-coordinate.
Missing intersections because the default GDC window doesn’t show them all.
Reading values to too few decimal places — AI HL usually wants 2 d.p. or 3 s.f.
Forgetting that a tangent intersection (curves touching) still counts as a solution.
Confusing “intersect” with “zero” on the GDC menu — zero finds x-intercepts of one curve, intersect finds where two curves meet.
Next up: Quadratic Functions & Graphs — the parabola family in detail, with its vertex, axis of symmetry and root behaviour.
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