Practice Maths

Cubic Functions — Graphs and Features

Key Terms

A cubic function has degree 3: f(x) = ax³ + bx² + cx + d, where a ≠ 0.
End behaviour
if a > 0, the graph goes from bottom-left to top-right (↗); if a < 0, from top-left to bottom-right (↘).
y = x³
: passes through origin, point of inflection at (0, 0), odd function, no turning points.
y = a(x − h)³ + k
: point of inflection (not a turning point) at (h, k); translated h right and k up from y = x³.
y = a(x − x1)(x − x2)(x − x3)
: x-intercepts at x1, x2, x3; y-intercept at a(−x1)(−x2)(−x3).
A cubic can have 1, 2 or 3 x-intercepts (1 or 3 real roots; a repeated root creates a “touch”).
As x → ∞
y → ∞ if a > 0, y → −∞ if a < 0. As x → −∞ the opposite.

Example graph: y = x³ − 4x = x(x + 2)(x − 2)

Features: x-intercepts at −2, 0, 2; y-intercept at (0, 0); local max near x = −1.15, local min near x = 1.15.

x y −2 −1 1 2 1 2 −1 −2 (−2,0) (0,0) (2,0) local max local min y = x³ − 4x
FormKey features
y = x³Origin (0,0), inflection at origin, a > 0 ↗
y = a(x−h)³+kInflection at (h,k); stretch/reflect by a
y = a(x−x1)(x−x2)(x−x3)3 x-intercepts at x1, x2, x3
Repeated root (x−p)²(x−q)Touches at p, crosses at q
Hot Tip For y = a(x−h)³+k, the point (h, k) is an inflection point, not a turning point — the curve changes concavity there but the gradient is never zero (unless a = 0). This is different from the vertex of a parabola.

Worked Example 1 — Identifying Features from Factored Form

Question: For y = 2(x + 1)(x − 1)(x − 3), state the x-intercepts, y-intercept, and end behaviour.

x-intercepts: x = −1, 1, 3.

y-intercept: Set x = 0: y = 2(1)(−1)(−3) = 6.

End behaviour: a = 2 > 0, so y → −∞ as x → −∞ and y → ∞ as x → ∞.

Worked Example 2 — Translated Cubic

Question: Describe the graph of y = −(x − 2)³ + 3.

Solution: This is y = x³ reflected in the x-axis (a = −1), then shifted 2 right and 3 up.

Inflection point: (2, 3). End behaviour: a < 0, so ↘ (top-left to bottom-right).

y-intercept: y = −(0−2)³ + 3 = −(−8) + 3 = 11.

End Behaviour: Why Cubics Are Different From Parabolas

A parabola (even degree) has both ends going in the same direction — both up (a > 0) or both down (a < 0). A cubic (odd degree) has ends going in opposite directions. This is a fundamental consequence of the function y = x³ being an odd function: f(−x) = −f(x).

For the leading term ax³: as x → +∞, y → +∞ if a > 0, and y → −∞ if a < 0. As x → −∞, the sign flips. This means a cubic must cross the x-axis at least once — if one end goes to +∞ and the other to −∞, the curve must pass through y = 0 somewhere in between (by the intermediate value theorem).

Up to Three Roots: When and Why

A cubic polynomial of degree 3 can have 1, 2, or 3 real roots. It can have 3 distinct real roots (crosses x-axis three times), 1 real root and a repeated root (crosses once, touches once), or 1 real root only (the other two are complex, invisible in real graphs).

A repeated root x = p occurs in factored form as (x − p)². The graph is tangent to the x-axis at x = p: it touches but does not cross. This is visually similar to a parabola touching the x-axis at its vertex, but the cubic still crosses the axis at its other root.

When all three roots are the same (x = p)³, the graph has an inflection point on the x-axis and both touches and crosses at x = p.

The Inflection Point: A Change of Concavity

The inflection point is where the curve changes concavity — from concave up (curving upward like a smile) to concave down (curving downward like a frown), or vice versa. For y = x³, this happens at the origin: to the left of x = 0 the curve is concave down; to the right it is concave up.

For y = a(x − h)³ + k, the inflection point is exactly (h, k). This is analogous to the vertex of a parabola in vertex form, but with a critical difference: at the inflection point of a cubic, the gradient is not zero (unless a = 0). The curve is still passing through with a non-zero slope; it simply changes which way it bends.

The inflection point of y = ax³ + bx² + cx + d in general is found by setting the second derivative equal to zero: y′′ = 6ax + 2b = 0, so x = −b/(3a). This will be proved properly in calculus.

Factorised Form and What Repeated Roots Mean Graphically

In y = a(x − p)(x − q)(x − r), each factor contributes one x-intercept. The sign of the function changes at each simple root (crosses), but does not change at a double root (touches). This is because crossing requires an odd multiplicity and touching requires an even multiplicity.

For y = (x − 2)²(x + 1): at x = 2 (multiplicity 2), the graph touches the x-axis and bounces back; at x = −1 (multiplicity 1), it crosses. The y-intercept is found by setting x = 0: y = (0−2)²(0+1) = 4.

Sketching Strategy for Cubics

A systematic sketch requires: (1) Determine a for end behaviour. (2) Find x-intercepts (factorise or use the factor theorem). (3) Find the y-intercept (set x = 0). (4) Identify any repeated roots and note where the curve touches versus crosses the axis. (5) Check concavity: the inflection point is where the curve changes bend direction. A good sketch shows the characteristic S-shape (or reverse-S) of a cubic, clearly labelled with all intercepts and any inflection point.

Exam Tip: A cubic with a positive leading coefficient always has at least one real root. This is guaranteed by the intermediate value theorem: since the function goes from −∞ to +∞, it must pass through zero. If you apply the quadratic formula to the remaining quadratic after dividing out one factor and get Δ < 0, there is only one real root (the one you found), and the other two are complex.
Exam Tip: Do not confuse the inflection point of y = a(x − h)³ + k with a turning point. At (h, k), the gradient is not zero — it is the minimum gradient for the cubic (when a > 0), or maximum (when a < 0). If you calculate f′(h), you get 0 only when the cubic has a degenerate form where the whole curve has been compressed flat at that point, which is not the case for standard cubics.

Mastery Practice

See Answers ➔

  1. For each cubic, state: (i) the degree and leading coefficient, (ii) the y-intercept, (iii) the end behaviour (x → ±∞). Fluency

    1. (a) y = x³ − 5x + 2
    2. (b) y = −2x³ + x
    3. (c) y = 3(x − 1)³ + 4
    4. (d) y = −(x + 2)(x − 1)(x − 3)

  2. Expand each cubic from factored form and identify the x-intercepts and y-intercept. Fluency

    1. (a) y = (x − 1)(x + 2)(x − 3)
    2. (b) y = 2x(x + 1)(x − 4)
    3. (c) y = −(x + 3)(x − 2)(x + 1)
    4. (d) y = (x − 2)²(x + 1) (note the repeated root)

  3. For each cubic of the form y = a(x − h)³ + k, state the inflection point and sketch its shape. Fluency

    1. (a) y = (x − 3)³ + 1
    2. (b) y = −(x + 1)³ − 2
    3. (c) y = 2(x − 2)³
    4. (d) y = −½(x + 4)³ + 5

  4. Determine all key features and sketch each cubic. Fluency

    1. (a) y = x(x − 3)(x + 2)
    2. (b) y = −x(x − 2)(x + 3)
    3. (c) y = (x − 1)²(x + 2)

  5. Expand each product to obtain the cubic in the form ax³ + bx² + cx + d. Fluency

    1. (a) (x + 1)(x² + x − 2)
    2. (b) (x − 3)(x² − 4)
    3. (c) −2(x − 1)(x + 2)(x − 3)
    4. (d) (x + a)(x + b)(x + c) — expand and collect in terms of a, b, c.

  6. Working backwards from graph features. Understanding

    Finding the equation from given features.
    1. (a) A cubic has x-intercepts at −1, 2 and 4, and passes through (0, −8). Find its equation.
    2. (b) A cubic has inflection point (1, −3) and passes through (3, 5). Write its equation in the form y = a(x − h)³ + k.
    3. (c) A cubic has a double root at x = 2 and a single root at x = −1. Its leading coefficient is −2. Write its equation.

  7. Transformations of y = x³. Understanding

    Describe the transformation(s) applied to y = x³ to obtain each graph. State the inflection point.
    1. (a) y = (x − 4)³
    2. (b) y = −2x³ + 5
    3. (c) y = (x + 1)³ − 3

  8. Comparing cubics and parabolas. Understanding

    Understanding the differences in shape and features.
    1. (a) A student says “the point (h, k) on y = a(x − h)³ + k is a turning point, just like on a parabola.” Is this correct? Explain the difference between an inflection point and a turning point.
    2. (b) Explain why a cubic with a positive leading coefficient must have at least one x-intercept, whereas a quadratic with positive leading coefficient may have none.

  9. Sketching from expanded form. Problem Solving

    Factorising to find roots. Factorise each cubic, then identify all key features and sketch.
    1. (a) y = x³ − x² − 6x (factorise by taking out x first)
    2. (b) y = x³ − 3x + 2 (try x = 1 as a root, then factorise)
    3. (c) y = −x³ + x² + 4x − 4

  10. Intersections and models. Problem Solving

    Connecting cubic models to context.
    1. (a) The profit P (in thousands of dollars) of a business after t months is modelled by P(t) = t(t − 4)(t − 10) for 0 ≤ t ≤ 12. Find when the business is profitable (P > 0) in this interval.
    2. (b) Find where the cubic y = x³ − 4x and the line y = 3x − 6 intersect. (Set equal and solve.)