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

Solving Equations Graphically

Some equations just can’t be solved with algebra. Mix an exponential with a polynomial, or a log with a trig function, and there’s no rearrangement that gets you a clean answer. The fix is simple: plot both sides on your GDC and read off the intersection points. The x-coordinates are your solutions. This is the standard approach on Paper 2 (calculator paper) — fast, reliable, and worth full marks if you present it properly.

📘 What you need to know

Method 1: To solve f(x) = g(x), plot both y = f(x) and y = g(x). The x-coordinates of intersection points are the solutions.
Method 2: Plot y = f(x) − g(x) and find its roots (where it crosses the x-axis). These are the solutions.
Special case: f(x) = k — plot y = f(x) and the horizontal line y = k; intersection x-coordinates are the solutions.
Number of solutions = number of intersection points. Useful when the question only asks “how many”.
Use this when algebra fails: equations mixing exponential and polynomial, log and polynomial, trig and polynomial, etc.
Use this on Paper 2 (calculator allowed) unless the question explicitly says “solve algebraically” or asks for an “exact” answer.
Always sketch what you found as part of your working — examiners want to see the graph, not just an answer.

The two graphical methods

Both methods give the same answers — pick whichever is easier to plot:

Method 1 — Plot both sides

y = f(x) and y = g(x)

solutions = x-coords of intersection points

Method 2 — Plot the difference

y = f(x) − g(x)

solutions = roots (where curve crosses x-axis)

Same solutions, two views

For most exam questions, Method 1 is the go-to — plot both sides and use the intersect function on your GDC. Method 2 helps when the difference simplifies into a recognisable curve, or when you want a single graph with clear x-axis crossings.

When to reach for graphical solving

Three classic signs that you should switch from algebra to graph:

Sign 1 — High-degree polynomial

x⁵ − 3x + 2 = 0

no general formula for degree ≥ 5

Sign 2 — Mixed function types

e^x = x²

no rearrangement isolates x

Sign 3 — “How many solutions”

“how many roots does … have?”

just count intersection points

Paper 2 default

calculator allowed → use it

unless the question says “exact” or “algebraically”

The GDC recipe

🧭 Recipe — solving graphically with the GDC

Rearrange (if helpful): get the equation into f(x) = g(x) form, with both sides easy to plot.
Enter both functions on the GDC (typically Y₁ = LHS, Y₂ = RHS).
Choose a sensible window: zoom out or pick wide x and y ranges so all intersections are visible.
Use the intersect function for each intersection point. Read off the x-coordinates.
Round appropriately — to 3 sf unless told otherwise.
Sketch what you found — examiners expect to see the curves, intersections marked, and the x-values labelled.

“Exact” or “algebraically” → don’t use the graph. Those words rule out a GDC-only answer. If the question doesn’t say either, the graphical method is fair game on Paper 2.

Worked examples

WE 1

Solve a mixed exponential-polynomial equation

Find the solution to e^x = 4 − x², giving your answer to 3 sf.

Step 1: Plot both functions on the GDC y = eˣ — increasing exponential y = 4 − x² — downward parabola, vertex (0, 4), roots ±2 Step 2: Look for intersections curves cross at two points — one on each side of the y-axis Step 3: Use intersect function x ≈ −1.96 and x ≈ 1.06 x = −1.96 or x = 1.06 (3 sf) always check graphically that the number of intersections matches the number of solutions you give

WE 2

Solve a high-degree polynomial equation

Solve x⁵ + 2x² − 6 = 0, giving your answer to 3 sf.

Method: plot y = x⁵ + 2x² − 6 and find x-axis crossings at x = 1: 1 + 2 − 6 = −3 (negative) at x = 2: 32 + 8 − 6 = 34 (positive) → a root between 1 and 2 Use GDC zero/root function x ≈ 1.20 Check whether other real roots exist curve is generally rising → only one real root x = 1.20 (3 sf) no general formula exists for degree-5 polynomials, so graphical (or numerical) is the only practical route

WE 3

Determine the number of real solutions

How many real solutions does ln x = x − 3 have?

Step 1: Plot both functions on the GDC y = ln x — slow growth, only defined for x > 0 y = x − 3 — straight line, intercept (3, 0) Step 2: Count intersection points visually: the line cuts the curve twice once where ln x grows past the line on the rising tail once where the line catches back up further along 2 real solutions no need to find the actual values — counting is enough when the question only asks “how many”

WE 4

Use the difference method

By plotting y = 2^x − x − 5, find the solutions of 2^x = x + 5 to 3 sf.

Plot y = 2ˣ − x − 5 and find roots at x = 0: 1 − 0 − 5 = −4 (negative) at x = 3: 8 − 3 − 5 = 0 → x = 3 is an exact root! also a root for negative x where 2ˣ levels off near 0 and the line dips Use GDC zero function for the second root x ≈ −4.96 x = −4.96 or x = 3 (3 sf) the difference method gave a cleaner picture here than plotting two separate curves — the x-axis crossings are exactly the solutions

WE 5

Solve a trig-vs-linear equation

Solve cos x = x3 for −π ≤ x ≤ π, to 3 sf.

Step 1: Set GDC to radian mode Step 2: Plot y = cos x and y = x/3 on [−π, π] cos x oscillates between −1 and 1; line y = x/3 is shallow, rising Step 3: Find intersection in the given window single intersection at x ≈ 1.17 x = 1.17 (3 sf) always check the angle mode (degrees vs radians) — wrong mode gives totally wrong answers

WE 6

Solve f(x) = k graphically

Given f(x) = x³ − 4x + 1, find the values of x for which f(x) = 2, to 3 sf.

Step 1: Plot y = x³ − 4x + 1 and the horizontal line y = 2 cubic has local max around (−1.15, 4.08) and local min around (1.15, −2.08) y = 2 sits between the local max and local min, so it crosses three times Step 2: Use intersect function for each crossing x ≈ −2.11, x ≈ 0.254, x ≈ 1.86 x = −2.11, 0.254, or 1.86 (3 sf) three intersections means three solutions — slide the line up or down on the GDC to see how many solutions exist for different k values

💡 Top tips

Method 1 (plot both sides) is the default. Method 2 (plot the difference) is useful when the resulting curve is simpler.
Always sketch the graph in your working — even a rough sketch with intersection points labelled.
Check angle mode (radians vs degrees) for trig equations. The wrong mode silently destroys your answer.
Zoom out before concluding. A solution off-screen is a solution missed.
Round to 3 sf by default — that’s the IB standard accuracy unless told otherwise.
For “how many” questions, just count intersection points — no need to find values.
If a graph almost touches (looks like a tangent), check carefully — it might be a double root, or it might just miss. Use the GDC to find the minimum value of the difference function.

⚠ Common mistakes

Using the graphical method when the question wants exact answers. “Exact” or “algebraically” → algebra only.
Wrong angle mode on the GDC — radian vs degree confusion produces nonsense answers.
Missing intersections off-screen. Always check your window covers the relevant range.
Reading off the y-coordinate instead of the x-coordinate. Solutions are x-values.
Skipping the sketch in your written answer. Examiners want evidence of the method.
Rounding too early. Use the full GDC value internally, only round to 3 sf at the very end.
Confusing tangent points with separated curves. If two curves just touch, that’s still one solution — but if they almost touch and don’t, it’s zero.

Algebra and graphs are complementary tools — use algebra when the equation is clean, the GDC when it isn’t. The next note shifts to modelling with functions — the IB’s favourite type of applied question, where you’re given (or must choose) a function to describe a real situation, then use it to make predictions.

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