Grade 12 Maths · Calculus · Paper 1

Cubic Graphs Grade 12

Real Cubic Graphs past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 10 past paper questions
skills: 5 skills
videos: 5 videos

Try a Cubic Graphs exam question

No sign-up. Takes 3 minutes.

Question 5.1 · May/June 2025 · Paper 1 · National (DBE) · 3 marks

The graph of

f (x) = a (x + p)^{2} + q

is drawn below.

C (- 1; 4)

is the turning point of

f

B (- 3; - 4)

is a point on

f

5.1 Show that

f (x) = - 2 x^{2} - 4 x + 2

.(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

Cubic Graphs skills

Each skill is one thing a memo gives a tick for. Tap one for its own questions, notes and videos.

More Cubic Graphs Grade 12 questions and answers

Try each one on paper before you open the memo.

4. Question 9.1 · May/June 2024 · Paper 1 · National (DBE) · 3 marks
The graph of $f (x) = a x^{3} + b x^{2} + c x - 5$ is drawn below. $E (- 1; 0)$ and $G (5; 0)$ are the $x$ -intercepts of $f$ .
9.1 Show that $a = 1$ , $b = - 3$ and $c = - 9$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Repeated factor at the turning point E $(x + 1)^{2}$
  1 mark · Find the equation of a graph
2. ✓
  Equation in factor form $f (x) = a (x + 1)^{2} (x - 5)$
  1 mark · Find the equation of a graph
3. ✓
  Substitution of $(0; - 5)$ $- 5 = a (0 + 1)^{2} (0 - 5) ⟹ a = 1$
  $f (x) = x^{3} - 3 x^{2} - 9 x - 5$
  1 mark · Find the equation of a graph
5. Question 3.1.3 · March 2025 · Provincial · 3 marks
3.1 Given: $f (x) = \frac{2 x + 3}{- x - 3}$

Given: $f (x) = \frac{2 x + 3}{- x - 3}$
3.1.3 Sketch the graph of $f$ . Clearly indicate the intercepts with both axes, as well as the asymptotes.(3)
Show the memo (3 ticks)Memo
1. ✓
  Draw asymptotesVertical asymptote: $x = - 3$
  Horizontal asymptote: $y = - 2$
  1 mark · State the asymptotes
2. ✓
  Plot interceptsx-intercept at $- 1, 5$ ( $- \frac{3}{2}$ )
  y-intercept at $- 1$
  1 mark · Find the x-intercepts
3. ✓
  Draw correct shapeHyperbola in 1st and 3rd quadrants relative to asymptotes (k > 0).
  1 mark · Sketch a graph from its equation
6. Question WE11 · Siyavula Grade 12 Mathematics, 5.4 Factor theorem, Worked example 11 · 3 marks
WE11 Factorise completely: $f (x) = x^{3} + x^{2} - 9 x - 9$ (3)
Show the memo (3 ticks)Memo
1. ✓
  Find a factor $f (1) = (1)^{3} + (1)^{2} - 9 (1) - 9 = - 16$ , so $(x - 1)$ is not a factor.
  $f (- 1) = (- 1)^{3} + (- 1)^{2} - 9 (- 1) - 9 = 0$ , so $(x + 1)$ is a factor.
  1 mark · Factorise a cubic polynomial
2. ✓
  Quadratic factor by inspection $x^{3} + x^{2} - 9 x - 9 = (x + 1) (x^{2} + ? x - 9)$
  $(+ 1) (x^{2})$ already gives the $x^{2}$ term, so the middle coefficient is $0$ .
  $f (x) = (x + 1) (x^{2} - 9)$
  1 mark · Factorise a cubic polynomial
3. ✓
  Factorise fully $f (x) = (x + 1) (x - 3) (x + 3)$
  1 mark · Factorise a difference of squares
7. Question WE12 · Siyavula Grade 12 Mathematics, 5.4 Factor theorem, Worked example 12 · 3 marks
WE12 Use the factor theorem to factorise $f (x) = x^{3} - 2 x^{2} - 5 x + 6$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Find a factor $f (1) = (1)^{3} - 2 (1)^{2} - 5 (1) + 6 = 0$ , so $(x - 1)$ is a factor.
  1 mark · Factorise a cubic polynomial
2. ✓
  Quadratic factor by inspection $x^{3} - 2 x^{2} - 5 x + 6 = (x - 1) (x^{2} + ? x - 6)$
  $(- 1) (x^{2}) = - x^{2}$ , so the middle coefficient must be $- 1$ to give $- x^{2} - x^{2} = - 2 x^{2}$ .
  $f (x) = (x - 1) (x^{2} - x - 6)$
  Check: $(x) (- 6) + (- 1) (- x) = - 5 x$
  1 mark · Factorise a cubic polynomial
3. ✓
  Factorise fully $f (x) = (x - 1) (x - 3) (x + 2)$
  1 mark · Factorise a trinomial
8. Question WE13 · Siyavula Grade 12 Mathematics, 5.5 Solving cubic equations, Worked example 13 · 4 marks
WE13 Solve: $6 x^{3} - 5 x^{2} - 17 x + 6 = 0$ (4)
Show the memo (4 ticks)Memo
1. ✓
  Find a factorLet $f (x) = 6 x^{3} - 5 x^{2} - 17 x + 6$
  $f (1) = 6 - 5 - 17 + 6 = - 10$ , so $(x - 1)$ is not a factor.
  $f (2) = 6 (2)^{3} - 5 (2)^{2} - 17 (2) + 6 = 48 - 20 - 34 + 6 = 0$ , so $(x - 2)$ is a factor.
  1 mark · Factorise a cubic polynomial
2. ✓
  Quadratic factor $6 x^{3} - 5 x^{2} - 17 x + 6 = (x - 2) (6 x^{2} + 7 x - 3)$
  1 mark · Factorise a cubic polynomial
3. ✓
  Factorise fully $6 x^{3} - 5 x^{2} - 17 x + 6 = (x - 2) (2 x + 3) (3 x - 1)$
  1 mark · Factorise trinomials with a leading coefficient
4. ✓
  Answer $(x - 2) (2 x + 3) (3 x - 1) = 0$
  $x = 2$ or $x = \frac{1}{3}$ or $x = - \frac{3}{2}$
  1 mark · Solve a quadratic by factorising
9. Question c · Siyavula Grade 12 Mathematics, 6.7 Applications of differential calculus, Worked example 23 · 1 mark
The height (in metres) of a golf ball $t$ seconds after it has been hit into the air, is given by $H (t) = 20 t - 5 t^{2}$ . Determine the following:
c The time at which the vertical velocity is zero.(1)
Show the memo (1 tick)Memo
1. ✓
  v = 0 $v (t) = 0$
  $20 - 10 t = 0$
  $t = 2$
  The velocity is zero after $2 s$ .
  1 mark · Find stationary points

Cubic Graphs Grade 12 videos

Short lessons from South African teachers first. Videos load only when you tap play, to save your data.

Parabola Grade 10: How to drawKevinmathscience

For Which Values of K | Grade 11 FunctionsKevinmathscience

Factorise Cubic Equation Grade 12: IntroductionKevinmathscience

Stationary Points - Corbettmathscorbettmaths

Inflection Point Grade 12Kevinmathscience

Free on Android

Know exactly which skill is costing you marks.

Mark past papers tick by tick in the NavyBlue app. It remembers every mark you lose, shows you the skills to fix first, and picks your next questions.

The NavyBlue app showing a learner's weak skills