IB Maths AA HL Topic 2 — Functions Paper 1 & 2 ~6 min read

Intersecting Graphs

Two graphs intersect where they share an (x, y) point — and that point’s x-coordinate is automatically a solution to the equation f(x) = g(x). This single idea turns nasty equations like 2^x = 3 − x (which can’t be solved by ordinary algebra) into easy GDC questions: just plot both sides, hit “intersect”, and read off the answer. The same trick counts how many solutions an equation has — it’s the number of intersection points.

📘 What you need to know

Intersection ⟺ shared point: a point (a, b) lies on both graphs precisely when f(a) = b = g(a).
Solving an equation by intersection: solutions to f(x) = g(x) are the x-coordinates of the points where the two graphs cross.
Special case: solutions to f(x) = k are where y = f(x) crosses the horizontal line y = k.
Number of solutions = number of intersection points. Useful when the question only asks “how many” without needing values.
GDC workflow: plot both graphs in the same window → use the “intersect” function → read off the coordinates.
Algebra still works when both functions are simple — but for mixed exponentials, logs, or trig, the GDC is essential.
Make sure your window is big enough to capture all the intersections — a wrongly-zoomed plot will miss some.

What does intersection mean?

The fundamental link (a, b) is an intersection of y = f(x) and y = g(x)
⟺ f(a) = g(a) = b

So finding intersections and solving simultaneous equations are the same task, just dressed up in different language. The x-coordinates of the meeting points are the values of x that satisfy both equations at once.

Intersection of two graphs — same point on both curves

Two key uses

Solve f(x) = k

plot y = f(x) and y = k

x-coordinates where the curve crosses the horizontal line are the solutions

Solve f(x) = g(x)

plot both graphs together

x-coordinates of intersection points are the solutions

🤔 Why is this so useful?

Equations mixing different function types — like an exponential equal to a polynomial, or a log equal to a trig — usually have no algebraic method. Graphing both sides and reading off the intersections is often the only practical way to solve them.

Counting solutions without finding them

If a question asks “how many real solutions does f(x) = k have?”, you don’t need to find the actual solutions — just count the intersection points between y = f(x) and y = k.

Sliding-line trick: imagine the horizontal line y = k sliding up and down. The number of times it crosses the curve at each height tells you the number of solutions of f(x) = k for that k.

Algebra or GDC — when to use which

Use algebra when…

both functions are polynomial or simple

e.g. line meets parabola → quadratic equation; line meets line → linear

Use GDC when…

mixed function types

exponential meets line, trig meets cubic, log meets polynomial — algebra is impractical

Worked examples

WE 1

Find the intersection of two lines (algebra)

Find the point of intersection of y = 2x + 1 and y = −x + 7.

Step 1: Set the y-values equal 2x + 1 = −x + 7 Step 2: Solve for x 3x = 6 → x = 2 Step 3: Substitute back to find y y = 2(2) + 1 = 5 intersection at (2, 5) always check by substituting into the OTHER equation: −2 + 7 = 5 ✓

WE 2

Line meets parabola — find both intersection points

Find the points of intersection of y = x² − 4 and y = 3x.

Step 1: Equate x² − 4 = 3x x² − 3x − 4 = 0 Step 2: Factor (x − 4)(x + 1) = 0 x = 4 or x = −1 Step 3: Find y at each x x = 4 → y = 3(4) = 12 x = −1 → y = 3(−1) = −3 intersections: (4, 12) and (−1, −3) two solutions to the quadratic ⟹ two intersection points

WE 3

Count the number of real solutions

Use a graph to determine the number of real solutions of x³ − 4x = 5.

Step 1: Set up the two graphs y = x³ − 4x and y = 5 Step 2: Plot the cubic on GDC cubic has local max at x ≈ −1.15, height ≈ 3.08 local min at x ≈ 1.15, height ≈ −3.08 Step 3: Slide the horizontal line y = 5 across y = 5 sits ABOVE the local max (3.08), so it cuts the cubic only on the rising right tail 1 real solution if y = 5 had been below 3.08 (and above −3.08), there would have been 3 real solutions

WE 4

Use GDC for a non-algebraic equation

Solve 2^x = 3 − x by graphing both sides.

Step 1: Plot y = 2ˣ and y = 3 − x on the same axes 2ˣ is increasing exponential; 3 − x is a decreasing line they cross exactly once Step 2: Use GDC’s intersect function intersection at x = 1, y = 2 Step 3: Verify directly 2¹ = 2 ✓ and 3 − 1 = 2 ✓ x = 1 no algebraic method works here — exponentials and polynomials don’t combine to give clean equations

WE 5

Find an intersection using the GDC

Find, to 3 sf, the coordinates of the intersection of f(x) = x² − 1 and g(x) = 2/x for x > 0.

Step 1: Plot both graphs on the GDC parabola passes through (0, −1); reciprocal y = 2/x for x > 0 is decreasing Step 2: Use intersect function x ≈ 1.521, y ≈ 1.314 Step 3: Round to 3 sf intersection ≈ (1.52, 1.31) algebraically: x² − 1 = 2/x → x³ − x − 2 = 0; this cubic has no rational roots, so the GDC is the only practical route

WE 6

Use intersection to find an unknown coefficient

The line y = kx + 1 passes through the point of intersection of y = x² and y = 4 in the first quadrant. Find the value of k.

Step 1: Find the intersection of y = x² and y = 4 x² = 4 → x = ±2 first quadrant: x = 2, y = 4 → point (2, 4) Step 2: Substitute (2, 4) into y = kx + 1 4 = 2k + 1 2k = 3 k = 3/2 three graphs meeting at one point is a classic exam set-up — find the easy intersection first, then use it as a constraint on the third

💡 Top tips

Set the equations equal when finding an intersection algebraically. f(x) = g(x) is the equation to solve.
For mixed function types, go straight to the GDC. Plot both, use the intersect function, read off the answer.
Check the window covers all visible intersections. Zoom out if needed.
“How many solutions” questions only need the count of intersection points — no need to find values.
For f(x) = k, plot the curve and the horizontal line — solutions are the x-coordinates where they meet.
Always check by substitution after solving algebraically — both equations should be satisfied.
State coordinates in pairs (x, y), not just x-values, when “find the point of intersection” is asked.

⚠ Common mistakes

Reporting only the x-coordinate when the question asks for the point of intersection. The answer is a coordinate pair.
Missing intersections off-screen on the GDC. Always zoom out before concluding.
Trying algebra on equations that have no closed-form solution. Reach for the GDC when the equation involves exp + poly, log + poly, or trig + poly mixes.
Confusing “no intersection” with “no solution”. Two graphs that don’t cross mean the equation f(x) = g(x) has no real solutions — that’s a valid finding, just say it.
Sign errors when equating: x² − 4 = 3x rearranges to x² − 3x − 4 = 0 (subtract 3x), not x² + 3x − 4 = 0.
Forgetting to give 3 sf when the GDC is involved and the question demands accuracy. Read the question carefully.
Treating tangent points as two intersections. A tangent meets the curve at one point (with double multiplicity in algebra, but counted as ONE intersection on the graph).

And that closes the Functions Toolkit. You’ve now got every tool the IB tests on functions — language, composition, inverses, special structures (odd/even/periodic/self-inverse), graphing, and intersections. The next section moves into transformations of graphs — taking a known function and shifting, stretching, or reflecting it to build new ones. Many exam questions in later topics rely on these moves, so the patterns coming up are worth a careful read.

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