A cubic’s turning points come in pairs (a max and a min), or not at all. There’s no such thing as a cubic with exactly one turning point. The GDC’s maximum and minimum tools find them — just zoom out far enough to see whether the curve actually reverses direction.
Quick check from the curve: if the GDC shows the curve dipping down then back up (or vice versa), there are two turning points and possibly multiple roots. If it looks like a smooth S-stretch with no obvious wiggle, you may have a monotonic cubic with just one root.
Roots: 1, 2 or 3
A cubic must cross the x-axis at least once (because the tails go to y = ±∞, they must pass through 0 somewhere). It can cross up to three times. 2 roots happens only when the curve touches the axis at a turning point (a “double root”) — rare in AI SL but possible.
🧭 Recipe — sketching any cubic
- Read off the shape: a > 0 means bottom-left to top-right; a < 0 means top-left to bottom-right.
- y-intercept: read off d (or substitute x = 0).
- Roots: use the GDC’s zero/root tool. Most cubic roots are decimals — round to 3 s.f.
- Turning points: use the GDC’s maximum and minimum tools. There will be 0 or 2 — if you only see 1, zoom out and look more carefully.
- Draw & label: smooth curve through all the points, in the correct direction. Label every intercept and turning point with coordinates.
Worked examples
WE 1Full feature set — three roots, two turning points
The cubic y = x³ − 2x² − 5x + 6 is to be sketched. Find the y-intercept, all the roots, and the coordinates of both turning points (to 3 s.f. where needed).
Step 1 — y-intercept
d = 6 → (0, 6)
Step 2 — roots (GDC zero tool)
x = −2, x = 1, x = 3
(verify: cubic = (x + 2)(x − 1)(x − 3) ✓)
Step 3 — turning points (GDC max/min)
local max ≈ (−0.786, 8.21)
local min ≈ (2.12, −4.06)
y-int (0, 6) · roots −2, 1, 3 · max (−0.79, 8.21) · min (2.12, −4.06)
a > 0 so the curve enters bottom-left, climbs to the local max, dips through the y-axis to the local min, then climbs back up to top-right. Label all five points on the sketch.
WE 2Monotonic cubic — only one root
Show that the cubic y = x³ + 3x − 4 has no turning points, and find its single real root.
Step 1 — check derivative for turning points
dy/dx = 3x² + 3
3x² + 3 > 0 for ALL x (always positive)
so the curve is strictly INCREASING — no turning points
Step 2 — root (try x = 1 by inspection)
y(1) = 1 + 3 − 4 = 0 ✓
Step 3 — confirm it’s the only real root
monotonic ⇒ crosses x-axis exactly ONCE
no turning points · single root at x = 1
when a cubic’s derivative (a quadratic) has no real solutions, the cubic has no turning points and is monotonic. The discriminant of the derivative tells you instantly.
WE 3Negative leading coefficient
The function f(x) = −x³ + 3x² + 4 has two turning points. Find them and the y-intercept, and state how many real roots it has.
Step 1 — y-intercept
f(0) = 4 → (0, 4)
Step 2 — turning points (GDC max/min)
local MIN at (0, 4)
local MAX at (2, 8)
Step 3 — count roots (a < 0 so curve goes top-left to bottom-right)
local min y = 4 > 0, so curve sits ABOVE x-axis on the left
tail on the right goes to −∞, so it crosses x-axis ONCE
root (GDC): x ≈ 3.36
y-int (0, 4) · min (0, 4) · max (2, 8) · 1 real root at x ≈ 3.36
trick: for a < 0 with both turning points ABOVE the x-axis, there’s only 1 root (on the right tail). For both BELOW, also 1 root. Three roots only happen when the two turning points sit on opposite sides of the x-axis.
WE 4Factored form — expand and find features
The cubic y = (x + 1)(x − 2)(x − 4) is given. State the roots, find the y-intercept, expand to standard form, and find both turning points (to 3 s.f.).
Step 1 — roots read straight from factored form
x = −1, x = 2, x = 4
Step 2 — y-intercept (set x = 0)
y = (1)(−2)(−4) = 8 → (0, 8)
Step 3 — expand
(x − 2)(x − 4) = x² − 6x + 8
(x + 1)(x² − 6x + 8) = x³ − 5x² + 2x + 8
Step 4 — turning points (GDC)
local max ≈ (0.214, 8.21)
local min ≈ (3.12, −4.06)
roots −1, 2, 4 · y-int 8 · standard: x³ − 5x² + 2x + 8 · max (0.21, 8.21) · min (3.12, −4.06)
factored form gives roots for free but hides the turning points — you still need the GDC for those. Best practice: keep both forms handy.
WE 5Real-world — maximum volume of a box
An open-topped box has a square base of side x cm and a fixed surface area of 96 cm². Its volume (in cm³) is given by V(x) = 24x − x³/2. Use a graphical method to find the value of x that maximises the volume, and state the maximum volume.
Step 1 — identify shape: cubic with a < 0 in the x³ term
V(x) = −x³/2 + 24x (a = −0.5)
there will be a LOCAL MAX somewhere
Step 2 — plot V(x) on GDC; use maximum tool
local max at x = 4
Step 3 — max volume
V(4) = 24(4) − (64)/2 = 96 − 32 = 64
x = 4 cm · Vₜₐₓ = 64 cm³
classic AI SL optimisation: get the quantity as a cubic in one variable, then read the local max off the GDC. No calculus needed at this level.
WE 6Find a cubic from its features
A cubic has roots at x = −1, x = 2 and x = 5, and the curve passes through (0, −20). Find its equation in standard form.
Step 1 — start in factored form with unknown a
y = a(x + 1)(x − 2)(x − 5)
Step 2 — use (0, −20) to find a
−20 = a(0 + 1)(0 − 2)(0 − 5)
−20 = a × 1 × (−2) × (−5)
−20 = 10a ⇒ a = −2
Step 3 — expand to standard form
(x − 2)(x − 5) = x² − 7x + 10
(x + 1)(x² − 7x + 10) = x³ − 6x² + 3x + 10
multiply by −2: −2x³ + 12x² − 6x − 20
y = −2x³ + 12x² − 6x − 20
a cubic has four unknowns (a, b, c, d), so you need FOUR independent pieces of info to pin it down. Three roots + one point is one way; vertex info or extra points are other combinations.
💡 Top tips
- GDC first, algebra second: cubic roots and turning points rarely factor nicely. Plot, then use the zero/max/min tools.
- 0 or 2 turning points — never 1: if you can only find one, you’ve missed the other. Zoom out further.
- Don’t confuse local with global: a cubic’s tails go to ±∞, so it has NO global max or min. Always say “local max” / “local min”.
- Spotting an obvious root: try x = ±1, ±2 — if one works, you can factor out (x − that root) and reduce to a quadratic.
- Four conditions pin down a cubic: three roots + one point, or two roots + two points, or vertex info plus enough else — any combination of 4 independent facts.
⚠ Common mistakes
- Reporting only one turning point on a 2-turning-point cubic: if you find a max, there has to be a min too (and vice versa). Always look for the pair.
- Claiming the local max is the maximum: a cubic with a > 0 has a tail going to +∞, so the “max” is just a local feature, not the highest value. Always say “local”.
- Getting the direction wrong for a < 0: negative cubic goes from top-left DOWN to bottom-right. Don’t draw it the same way as a positive cubic.
- Reading roots backwards from factored form: y = (x + 1)(x − 2)(x − 4) has roots at x = −1, 2, 4 — flip the sign inside each bracket.
- Ignoring the y-intercept when finding a: with three roots given, the curve y = (x − p)(x − q)(x − r) is just ONE of infinitely many cubics through those roots. You need the extra point to find a.
Up next: Exponential Functions & Graphs. Exponential curves break the polynomial pattern — instead of crossing the x-axis multiple times, they approach a horizontal asymptote at one end and shoot off to infinity at the other. The key features are the y-intercept, the asymptote, and whether the curve is growing or decaying.
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