IB Maths AI HLFurther Functions & GraphsPaper 1 & 2Roots, turning points~8 min read
Cubic Functions & Graphs
A cubic y = ax3 + bx2 + cx + d always crosses the x-axis at least once and has at most three x-intercepts. The sign of a sets the end behaviour; the discriminant of the derivative decides whether there are two turning points or none at all.
📘 What you need to know
Form: y = ax3 + bx2 + cx + d with a ≠ 0.
End behaviour: a > 0 goes from bottom-left to top-right; a < 0 goes from top-left to bottom-right.
y-intercept: (0, d) — read off the constant term.
Roots: solutions of ax3 + bx2 + cx + d = 0; there are always 1, 2 or 3 real roots — never zero.
Turning points: 0 or 2 for a cubic. 0 ⇒ the function is monotonic; 2 ⇒ one local max and one local min.
A local max/min is not necessarily a global one — cubics extend to ±∞, so neither turning point is the overall maximum or minimum.
Shape and end behaviour
The leading coefficient a sets which way the ends of the cubic point. Positive a takes the curve from the bottom-left up to the top-right — in the long run y → +∞ as x → +∞. Negative a flips this: the curve falls from upper-left to lower-right. Because cubics go to ±∞ at the ends, they must cross the x-axis at least once.
Turning points and roots
A cubic either has two turning points (one local max, one local min) or none at all (then it is monotonic). The derivative f′(x) = 3ax2 + 2bx + c is a quadratic; whether it has real zeros decides which case you’re in. With two turning points the curve can cross the x-axis up to three times; without any turning points it crosses exactly once.
Both cubics have a local max and a local min plus a y-intercept and (here) three real roots; the sign of a just flips which end goes up.
Cubic graph at a glancey = ax3 + bx2 + cx + d; y-intercept (0, d)
number of real roots: 1, 2 or 3 · number of turning points: 0 or 2
Sketching a cubic with the GDC
The standard workflow is calculator-led: plot y = f(x), use the analyse-graph menu to read off any x-intercepts (roots), the y-intercept, and any local maxima or minima. Then redraw on paper with each labelled, taking care to capture the correct end behaviour from the sign of a.
Check the derivative to find out fast whether a cubic has two turning points or none: f′(x) is a quadratic; if its discriminant is negative the cubic is monotonic and has exactly one real root.
🧠Recipe — sketching a cubic
End behaviour from the sign of a: positive ⇒ rises right; negative ⇒ falls right.
y-intercept: (0, d).
Roots: use the GDC’s zero/intersect tool to read off all real x-intercepts.
Turning points: use the GDC’s max/min tool to find any local maxima and minima.
Plot the labelled points and draw a smooth curve with the correct end behaviour.
Worked examples
WE 1
End behaviour of a cubic
Describe the end behaviour of the graph of y = −2x3 + x2 − 3.
read the leading coefficienta = −2 < 0negative ⇒ falls from top-left to bottom-rightx → −∞: y → +∞ · x → +∞: y → −∞only the leading term matters at the extremes.
WE 2
y-intercept of a cubic
Find the y-intercept of y = 5x3 − 2x2 + 7x − 9.
substitute x = 0y = 5(0) − 2(0) + 7(0) − 9(0, −9)the y-intercept is always the constant term d.
WE 3
Roots and y-intercept from factored form
For y = (x − 1)(x + 2)(x − 3), find the roots and the y-intercept.
set each factor to zerox − 1 = 0 ⇒ x = 1x + 2 = 0 ⇒ x = −2x − 3 = 0 ⇒ x = 3y-intercept: x = 0y = (−1)(2)(−3) = 6roots: x = −2, 1, 3 · y-int (0, 6)factored form makes the roots immediate; expand only if needed.
WE 4
Monotonic cubic (no turning points)
For y = x3 + 3x − 2, show that the function is monotonic. Find the y-intercept and state how many real roots there are.
derivativef ′(x) = 3x² + 3discriminant of f′: 0² − 4(3)(3) = −36 < 0no real zeros of f′ ⇒ no turning pointsalso: 3x² + 3 ≥ 3 > 0 ⇒ always increasingy-intercept: (0, −2)monotonic + crosses from −∞ to +∞monotonic · (0, −2) · exactly 1 real root (≈ 0.60)a monotonic cubic always has exactly one real root.
WE 5
Cubic with two turning points
Sketch the key features of y = x3 − 3x + 1: local max and min, y-intercept and number of real roots.
derivative & critical pointsf ′(x) = 3x² − 3 = 0 ⇒ x = ±1local max at x = −1f(−1) = −1 + 3 + 1 = 3 ⇒ (−1, 3)local min at x = 1f(1) = 1 − 3 + 1 = −1 ⇒ (1, −1)y-intercept: (0, 1)roots: local max above x-axis, local min belowcurve crosses x-axis three times ⇒ 3 real rootsmax (−1, 3) · min (1, −1) · (0, 1) · 3 real roots (≈ −1.88, 0.35, 1.53)
WE 6
Applied: open-top box volume
An open box is made by cutting squares of side x cm from a 20 by 15 cm rectangle and folding up the sides. The volume is V(x) = x(20 − 2x)(15 − 2x) for 0 < x < 7.5. (a) Find V(2). (b) Find the value of x that maximises the volume and the maximum volume, to 2 d.p.
(a) V(2)V(2) = 2 · 16 · 11 = 352V(2) = 352 cm³(b) expand & differentiateV(x) = 4x³ − 70x² + 300xV′(x) = 12x² − 140x + 300solve V′(x) = 0 (or use GDC max)x = (35 ± 5√13)/6 ≈ 2.83 or 8.84only x ≈ 2.83 lies in (0, 7.5)V(2.83)≈ 379.04 cm³x ≈ 2.83 cm · max V ≈ 379.04 cm³
💡 Top tips
Sign of a first — the end behaviour is locked in before you do anything else.
Always at least one real root: a cubic can never miss the x-axis entirely.
Use the discriminant of f′ to predict whether two turning points exist before computing them.
For factored cubics the roots are immediate; for expanded ones, use the GDC.
“Local” max/min are not global on a cubic — the curve continues to ±∞ past them.
âš Common mistakes
Claiming a cubic might have no real roots — it always has at least one.
Sketching with the wrong end behaviour — if a > 0 the right end must go up.
Drawing one turning point — cubics have either 0 or 2, never 1.
Confusing roots with turning points — roots are x-intercepts; turning points are local max/min.
Treating a local max as the global max — a cubic has no global max or min on its full domain.
Next up: Exponential Functions & Graphs — horizontal asymptotes, growth and decay, and the role of the constant term.
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