IB Maths AI SL Topic 2 — Further Functions & Graphs Paper 1 & 2 GDC-essential ~7 min read

Intersecting Graphs

When two curves cross, the x-values where they meet are the solutions to the equation f(x) = g(x). That single idea turns every “solve” problem into a graph problem: enter both functions on your GDC, use the intersect tool, and read off the coordinates. The number of intersections IS the number of solutions — so this method handles equations that can’t be solved algebraically (like e^x = 4 − x) without breaking a sweat.

📘 What you need to know

Intersection = solution: where y = f(x) and y = g(x) cross, the x-coordinate is a solution of f(x) = g(x). The pair (x, y) is also the simultaneous-equations solution.
To solve f(x) = a: plot y = f(x) and the horizontal line y = a; the intersection x-values are the solutions.
To solve f(x) = g(x): plot both curves; the intersection x-values are the solutions.
Number of intersections = number of real solutions. If the question only asks “how many?”, you don’t need exact values — just count crossings on the GDC screen.
Use the GDC intersect tool: don’t read off the screen by eye. The intersect command gives exact (to many decimal places) x and y.
Adjust the viewing window: a too-small window can hide an intersection. Zoom out until both tails of each curve are clearly visible before counting.

Intersections as solutions

If two curves meet at the point (a, b), then f(a) = g(a) = b. The x-coordinate is the solution to the equation; the y-coordinate is the common function value there.

Intersections as solutions curves y = f(x) and y = g(x) cross at (a, b)
⇔ x = a solves f(x) = g(x)

The parabola and line meet at two points; both x-coordinates (1 and 3) solve x² − 3x + 2 = x − 1. The y-coordinate gives the common function value at that x.

Two key uses

Solving f(x) = a

To solve any equation of the form f(x) = a for a constant a: plot y = f(x) and the horizontal line y = a. The x-coordinates of every intersection are solutions. This is particularly handy for cubics, exponentials, and other functions where algebra would be painful.

Counting solutions (no values needed)

If a question asks “how many real solutions”, you don’t need to find them — just count intersection points on the GDC. Three crossings means three solutions. This is the fastest way to handle “for what values of a…” questions: shift the horizontal line up and down mentally and watch how many times it crosses.

Don’t forget: most algebraic equations can be turned into a “where do these graphs intersect?” question. x³ = 5x + 2 is the same as plotting y = x³ and y = 5x + 2 and finding crossings. Even ugly transcendental equations work.

🧭 Recipe — finding intersections on a GDC

Enter both functions as Y₁ and Y₂ on the GDC. For an equation f(x) = a, the second function is just the constant y = a.
Set a sensible viewing window: zoom out until both curves are visible across their full shape — including all potential intersection regions. A “missed” intersection is the biggest pitfall.
Use the intersect tool: select the two curves and a guess close to the intersection. The GDC returns exact x and y.
Repeat for every intersection: each crossing needs its own intersect command with a guess near that point. Don’t stop at one if the curves cross twice or more.
Report the answer in context: the x-coordinate(s) solve f(x) = g(x); the (x, y) pairs are the intersection points themselves.

Worked examples

WE 1

Linear and quadratic — find intersection points

Find the coordinates of the points where y = x² − 3x + 2 and y = x − 1 intersect.

Step 1 — plot both on GDC, use intersect tool two crossings: at x = 1 and x = 3 Step 2 — algebraic check: set f(x) = g(x) x² − 3x + 2 = x − 1 x² − 4x + 3 = 0 (x − 1)(x − 3) = 0 x = 1 or x = 3 Step 3 — find y for each (use either equation) x = 1: y = 1 − 1 = 0 x = 3: y = 3 − 1 = 2 (1, 0) and (3, 2) use the SIMPLER function to compute y — here the line y = x − 1 is faster than substituting into the parabola. Same answer either way.

WE 2

Solve f(x) = a using intersections

The function f is defined by f(x) = x² − 4x + 3. Use a graphical method to solve f(x) = 8.

Step 1 — set up: two graphs plot Y₁ = x² − 4x + 3 plot Y₂ = 8 (horizontal line) Step 2 — intersect tool on GDC crossings at x = −1 and x = 5 Step 3 — verify f(−1) = 1 + 4 + 3 = 8 ✓ f(5) = 25 − 20 + 3 = 8 ✓ x = −1 or x = 5 solving f(x) = c graphically: just add the horizontal line y = c. Works for ANY function, however nasty — no algebra needed.

WE 3

Count solutions without finding them

How many real solutions does the equation x³ − 3x² + 2 = −1 have?

Step 1 — plot Y₁ = x³ − 3x² + 2 and Y₂ = −1 cubic has local max y = 2 at x = 0 cubic has local min y = −2 at x = 2 Step 2 — where does y = −1 cut the curve? the line y = −1 sits BETWEEN the local min (−2) and max (2) so it slices the cubic in THREE places 3 real solutions the trick for “horizontal line vs cubic” questions: if the line lies BETWEEN the local max and local min y-values, you get 3 solutions; outside, just 1. Quick to see on a sketch.

WE 4

Non-algebraic equation — GDC essential

Find the coordinates of the point where y = e^x and y = 4 − x intersect, giving your answer to 3 s.f.

Step 1 — plot both on GDC eˣ is increasing; 4 − x is decreasing exactly ONE crossing Step 2 — intersect tool x ≈ 1.073… y ≈ e¹·⁰⁷³ ≈ 2.926… Step 3 — round to 3 s.f. (1.07, 2.93) no algebraic method exists for eˣ = 4 − x — the GDC is essential. Such “transcendental” equations are exactly what the graphical method is built for.

WE 5

Word problem — when do two investments match?

Investment A is modelled by A(t) = 100(1.10)^t and investment B by B(t) = 200(1.05)^t, where t is years from now. Find the year in which both investments have the same value, and that common value, to 3 s.f.

Step 1 — plot Y₁ = 100(1.10)ᵗ and Y₂ = 200(1.05)ᵗ at t = 0: A = 100, B = 200 (B is higher) A grows faster (10% > 5%), so it overtakes Step 2 — intersect tool t ≈ 14.9 years value ≈ 100 × 1.10¹⁴.⁹ ≈ 414 t ≈ 14.9 years, common value ≈ $414 classic “when does the faster-growing one catch up?” problem. The GDC handles it instantly; trying to solve 100(1.10)ᵗ = 200(1.05)ᵗ algebraically needs logs (which comes later).

WE 6

Parabola meeting a reciprocal

Use your GDC to find all the intersection points of y = x² − 6 and y = 5x. Give answers to 3 s.f. where needed.

Step 1 — plot both curves on GDC parabola (opens up, vertex at (0, −6)) reciprocal (two branches, vert. asymptote x = 0) Step 2 — use intersect tool for EACH crossing crossing 1: x ≈ −1.79, y ≈ −2.79 crossing 2: x = −1, y = −5 crossing 3: x ≈ 2.79, y ≈ 1.79 3 intersections: (−1.79, −2.79), (−1, −5), (2.79, 1.79) three crossings ⇒ the equation x² − 6 = 5/x has 3 real solutions. Always use the intersect tool for EACH point individually — one click only finds one intersection at a time.

💡 Top tips

Rearrange before plotting: to solve “x² + 1 = 3x“, plot y = x² + 1 and y = 3x separately — don’t combine into one curve. Two graphs are easier to read than one.
For f(x) = 0 (the roots): just plot y = f(x) and use the zero/root tool. No second curve needed.
Trial values to bracket an intersection: if the screen makes a crossing hard to spot, test a few x-values until you see f(x) − g(x) change sign — that interval contains an intersection.
Multiple intersections need multiple intersect commands: the GDC only finds one at a time, near your initial guess.
Always state the answer in the form the question wants: “solutions of an equation” need x-values only; “intersection points” need both coordinates.

⚠ Common mistakes

Missing an intersection: zooming in too tight on one crossing can hide others. Zoom out and count all crossings first; then zoom in to find each.
Reading coordinates off the screen by eye: gives at best 1 d.p. of accuracy. Always use the intersect tool for exam-level accuracy.
Giving the y-coordinate when asked for solutions: solutions of f(x) = g(x) are the x-coords, not y. Re-read the question before writing the final line.
Forgetting to plot the horizontal line for f(x) = a: the GDC won’t find the answer just by looking at y = f(x). You need both y = f(x) AND y = a.
Confusing “number of intersections” with “value of intersection”: “how many?” wants a count; “find them” wants coordinates. Different answer formats.

Up next: Quadratic Functions & Graphs. With general graphing technique in hand, the next four notes drill down into specific shape families, starting with quadratics — the most-tested function type in AI SL. You’ll learn the standard form, the axis of symmetry formula, how to read off the vertex, and the link between the discriminant and the number of x-intercepts.

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