Grade 11 Maths · Functions · Paper 1

Hyperbola Grade 11

Real Hyperbola past paper questions and answers from 2023 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: 32 past paper questions
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Question 5.2 · November 2025 · Paper 1 · Provincial · 3 marks

The graphs of

f (x) = \frac{a}{x + p} + q

and

g (x) = \frac{x}{3} + c

are drawn below. C

(3; - 1)

is the point of intersection between the asymptotes of

f

. A and B are

x

- and

y

-intercepts of

f

and

g

respectively. E

(4; 2)

is a point on

f

in the first quadrant. $Diagram for this question$

5.2 Determine the equation of

f

.(3)

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

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More Hyperbola Grade 11 questions and answers

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4. Question 6.4.2 · November 2025 · Paper 1 · Provincial · 3 marks
The graphs of $f (x) = - x^{2} - 6 x$ and $g (x) = a x^{2} + b x$ are drawn below. $g$ is the reflection of $f$ along the $x$ -axis. A and B $(- 5; n)$ are symmetrical on $f$ along $x = - p$ , i.e. the axes of symmetry. $Diagram for this question$
6.4 Determine the value(s) of $k$ for which:
6.4.2 $f (x + k)$ will have two real roots that have different signs.(3)
Show the memo (2 ticks)Memo
1. ✓
  $x$ -values $- x^{2} - 6 x = 0$
  $x^{2} + 6 x = 0$
  $x (x + 6) = 0$
  $x = 0$ or $x = - 6$
  1 mark · Find the x-intercepts
2. ✓
  Answer $∴ - 6 < k < 0$
  2 marks · Shift a graph
5. Question 4.3.1 · November 2024 · Paper 2 · Provincial · 3 marks
In the diagram, $Δ$ ABC is drawn with vertices A $(- 7; 0)$ , B $(3; 8)$ and C. P is the midpoint of line AB. Q is a point on line AC. AQC is a line on the $x$ -axis. E $(- \frac{1}{2}; 2)$ is a point on line PQ. PQ $∥$ BC $Diagram for this question$
4.3 Calculate the:
4.3.1 Length of AC(3)
Show the memo (3 ticks)Memo
1. ✓
  $y = 0$ $- \frac{4}{3} x + 12 = 0$
  1 mark · Find the x-intercepts
2. ✓
  $x = 9$ $x = 9$
  1 mark · Find the x-intercepts
3. ✓
  AC = 16AC $= 9 - (- 7) = 16$
  1 mark · Find a horizontal or vertical length
6. Question 5.1 · November 2024 · Paper 1 · Provincial · 4 marks
The lines $y = - x + 4$ and $y = x - 2$ are the axes of symmetry of the function $f (x) = \frac{- 3}{x + p} + q$ .
5.1 Show that $p = - 3$ and $q = 1$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Equating $x - 2 = - x + 4$
  1 mark · Find axes of symmetry of hyperbola
2. ✓
  $x - 3 = 0$ $2 x = 6$
  $x = 3$
  $x - 3 = 0$
  $∴ p = - 3$
  1 mark · State the asymptotes
3. ✓
  Substituting $x$ $y = x - 2 = (3) - 2$
  OR $y = - x + 4 = - (3) + 4$
  1 mark · Find axes of symmetry of hyperbola
4. ✓
  $y = 1$ $= 1$
  $∴ q = 1$
  1 mark · State the asymptotes
7. Question 6.2 · November 2024 · Paper 1 · Provincial · 3 marks
Sketched below is the graph of $f (x) = x^{2} - 4 x - 5$ for $x \in (- \infty; 6]$ .

A and B are the $x$ -intercepts and C the $y$ -intercept of $f$ . $Diagram for this question$
6.2 Determine the coordinates of A and B.(3)
Show the memo (3 ticks)Memo
1. ✓
  Factors $x^{2} - 4 x - 5 = 0$
  $(x - 5) (x + 1) = 0$
  1 mark · Factorise a trinomial
2. ✓
  Coordinates of A $x = - 1$
  A $(- 1; 0)$
  1 mark · Find the x-intercepts
3. ✓
  Coordinates of B $x = 5$
  B $(5; 0)$
  1 mark · Find the x-intercepts
8. Question 5.1 · November 2023 · Paper 1 · Provincial · 2 marks
The diagram below shows the graph of $f (x) = \frac{a}{x + p} + q$ . The asymptotes of $f$ intersect at $(- 3; - 1)$ and $f$ cuts the $x$ -axis at $x = - 5$ . $Diagram for this question$
5.1 Write down the values of $p$ and $q$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  $p = 3$ $p = 3$
  1 mark · State the asymptotes
2. ✓
  $q = - 1$ $q = - 1$
  1 mark · State the asymptotes
9. Question 6.2 · November 2023 · Paper 1 · Provincial · 2 marks
Given: $f (x) = 2 (3^{x}) + 1$
6.2 Write down the equation of asymptote of $f$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Answer $y = 1$
  2 marks · State the asymptotes
10. Question 3.4 · November 2024 · Paper 2 · Provincial · 1 mark
In the diagram, P $(x; y)$ , Q $(- 2; 2)$ and R $(2; 3)$ are the vertices of triangle PQR. Line PR passes through the point of origin at O. The equation of line PQ is given as $y = 6 x + 14$ . S is the $x$ -intercept of line QP.

Note: the question text gives R $(2; 3)$ but the diagram labels R $(3; 4)$ . The official memo marks both versions: method 1 uses R $(3; 4)$ , method 2 uses R $(2; 3)$ . $Diagram for this question$
3.4 Determine the coordinates of S.(1)
Show the memo (1 tick)Memo
1. ✓
  y = 0 and x-coordinate $0 = 6 x + 14$
  $x = - \frac{7}{3}$
  S $(- \frac{7}{3}; 0)$
  1 mark · Find the x-intercepts

Hyperbola Grade 11 videos

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

Grade 9 Mathematics - Calculating the x and y intercepts of a straight lineMrs Townsend Mathematics - South Africa

Hyperbola grade 10: The BasicsKevinmathscience

Hyperbola Axis of Symmetry Grade 10: IntroductionKevinmathscience

Hyperbola Equation Given Graph | Grade 11 FunctionsKevinmathscience

Exponential Domain Range| Grade 11 FunctionsKevinmathscience

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