IB Maths AA HLTopic 2 — FunctionsPaper 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 2x = 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
Find the point of intersection of y = 2x + 1 and y = −x + 7.
Step 1: Set the y-values equal2x + 1 = −x + 7Step 2: Solve for x3x = 6 → x = 2Step 3: Substitute back to find yy = 2(2) + 1 = 5intersection 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: Equatex² − 4 = 3xx² − 3x − 4 = 0Step 2: Factor(x − 4)(x + 1) = 0x = 4 or x = −1Step 3: Find y at each xx = 4 → y = 3(4) = 12x = −1 → y = 3(−1) = −3intersections: (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 graphsy = x³ − 4x and y = 5Step 2: Plot the cubic on GDCcubic has local max at x ≈ −1.15, height ≈ 3.08local min at x ≈ 1.15, height ≈ −3.08Step 3: Slide the horizontal line y = 5 acrossy = 5 sits ABOVE the local max (3.08), so it cuts the cubic only on the rising right tail1 real solutionif 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 2x = 3 − x by graphing both sides.
Step 1: Plot y = 2ˣ and y = 3 − x on the same axes2ˣ is increasing exponential; 3 − x is a decreasing linethey cross exactly onceStep 2: Use GDC’s intersect functionintersection at x = 1, y = 2Step 3: Verify directly2¹ = 2 ✓ and 3 − 1 = 2 ✓x = 1no 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 GDCparabola passes through (0, −1); reciprocal y = 2/x for x > 0 is decreasingStep 2: Use intersect functionx ≈ 1.521, y ≈ 1.314Step 3: Round to 3 sfintersection ≈ (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 = 4x² = 4 → x = ±2first quadrant: x = 2, y = 4 → point (2, 4)Step 2: Substitute (2, 4) into y = kx + 14 = 2k + 12k = 3k = 3/2three 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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