Cubic and quartic graphs and models | Year 11

01 / LEARN

The idea, step by step

Degree, sign and multiplicity

A positive cubic falls on the left and rises on the right. A negative cubic reverses this. A positive quartic rises at both ends; a negative quartic falls at both ends. Lower-degree terms shape the middle.

Odd-multiplicity roots are crossings; even-multiplicity roots are touches. Mark roots, y-intercept and end behaviour first. Extra points or a graphing tool then refine a sketch without contradicting those features.

Use structure and context

For $x^{4} - 5 x^{2} + 4 = 0$ , set u=x squared and solve a quadratic in u. Return to x carefully: a positive u gives two real x-values, while a negative u gives none.

A box-volume model can be cubic even though only a short interval of positive cut sizes is practical. State the domain from positive dimensions before interpreting graph values.

A common trap

From x squared =4, keeping only x=2.

Instead: Both 2 and -2 square to 4; keep both unless the context excludes one.

02 / WORKED EXAMPLES

See the thinking

Example 1

A repeated cubic root

Describe $y = (x + 2) {(x - 1)}^{2}$ .

Roots are -2 simple and 1 double.
It crosses at -2, touches at 1, and has y-intercept 2.
The positive cubic leading term gives left end down/right end up.

Crosses (-2,0), touches (1,0), passes (0,2).

Example 2

Quadratic in x squared

Solve $x^{4} - 5 x^{2} + 4 = 0$ .

Let u=x squared: $(u - 1) (u - 4) = 0$ .
Thus x squared is 1 or 4.
Use both square-root branches.

x = - 2, - 1, 1, 2

Check your understanding

Does $y = {(x - 3)}^{2} {(x + 1)}^{2}$ cross at x=3?

Show answer

No; it touches.

The factor x-3 has even multiplicity.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Describe crossing/touching roots of $y = (x + 1) {(x - 2)}^{2}$ .
2 marks
State end behaviour of $y = - 2 x^{4} + 3 x - 1$ .
2 marks
Solve $x^{4} - 10 x^{2} + 9 = 0$ .
3 marks
Solve $x^{4} + x^{2} - 6 = 0$ for real x.
3 marks
Cut squares of side x cm from the corners of a 12 cm by 8 cm sheet and fold a box. Give volume and domain.
3 marks
A box has volume $V = x (12 - 2 x) (8 - 2 x)$ cubic centimetres. Find its volume when $x = 1$ and when $x = 2$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Describe crossing/touching roots of $y = (x + 1) {(x - 2)}^{2}$ .
2 marks
Use multiplicities one and two.
Crosses at -1; touches at 2.
State end behaviour of $y = - 2 x^{4} + 3 x - 1$ .
2 marks
The leading degree is even and coefficient negative.
Both ends fall.
Solve $x^{4} - 10 x^{2} + 9 = 0$ .
3 marks
Let u=x squared: $(u - 1) (u - 9) = 0$ .
Use both roots of each positive u.
$x = - 3, - 1, 1, 3$
Solve $x^{4} + x^{2} - 6 = 0$ for real x.
3 marks
The u-equation is $(u + 3) (u - 2) = 0$ .
Reject u=-3 for real x.
$x = \pm \sqrt{2}$
Cut squares of side x cm from the corners of a 12 cm by 8 cm sheet and fold a box. Give volume and domain.
3 marks
The three dimensions must be positive.
Multiply height, length and width.
$V = x (12 - 2 x) (8 - 2 x)$ cubic centimetres; $0 < x < 4$ .
A box has volume $V = x (12 - 2 x) (8 - 2 x)$ cubic centimetres. Find its volume when $x = 1$ and when $x = 2$ .
2 marks
At 1: 1(10)(6)=60.
At 2: 2(8)(4)=64.
60 cubic centimetres and 64 cubic centimetres.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics topics 2.1–2.2: quadratic relationships, cubic and quartic polynomials. Original Ascent teaching sequence.

Official curriculum source ↗

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