Grade 12 Maths · Calculus · Paper 1

Graph Interpretation Grade 12

Real Graph Interpretation 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
skills: 10 skills
videos: 10 videos

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Question 5.2 · May/June 2025 · Paper 1 · National (DBE) · 2 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

5.2 Determine the values of

k

for which

h (x) = f (x) + k

will have no real roots.(2)

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

Graph Interpretation skills

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More Graph Interpretation Grade 12 questions and answers

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4. Question 7.4.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = 2 cos x + 1$ and $g (x) = sin 2 x$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
7.4 Use the graphs to determine the values of x, in the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ for which:
7.4.1 $g (x) \cdot f^{'} (x) < 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Analyze the conditions for the product to be negativeFor $g (x) \cdot f^{'} (x) < 0$ , the signs of $g (x)$ and $f^{'} (x)$ must be different (+ and - OR - and +).
  $f^{'} (x)$ represents the gradient of $f$ . It is positive when $f$ is increasing ( $- 18 0^{\circ} < x < 0^{\circ}$ ) and negative when $f$ is decreasing ( $0^{\circ} < x < 18 0^{\circ}$ ).
  
  Condition 1: $g (x) > 0$ and $f^{'} (x) < 0$
  $g (x)$ is positive for $(0^{\circ}; 9 0^{\circ}) \cup (- 18 0^{\circ}; - 9 0^{\circ})$ .
  $f^{'} (x)$ is negative for $(0^{\circ}; 18 0^{\circ})$ .
  Intersection: $x \in (0^{\circ}; 9 0^{\circ})$
  
  Condition 2: $g (x) < 0$ and $f^{'} (x) > 0$
  $g (x)$ is negative for $(9 0^{\circ}; 18 0^{\circ}) \cup (- 9 0^{\circ}; 0^{\circ})$ .
  $f^{'} (x)$ is positive for $(- 18 0^{\circ}; 0^{\circ})$ .
  Intersection: $x \in (- 9 0^{\circ}; 0^{\circ})$
  1 mark · Solve product inequality from graphs
2. ✓
  State final union of intervals $x \in (- 9 0^{\circ}; 0^{\circ}) \cup (0^{\circ}; 9 0^{\circ})$
  
  OR
  
  $- 9 0^{\circ} < x < 0^{\circ}$ or $0^{\circ} < x < 9 0^{\circ}$
  1 mark · Solve product inequality from graphs
5. Question 6.4 · May/June 2024 · Paper 1 · National (DBE) · 2 marks
The graphs of $f (x) = x^{2} - 2 x - 3$ and $g (x) = m x + c$ are drawn below. D and E are the $x$ -intercepts and P is the $y$ -intercept of $f$ . The turning point of $f$ is T $(1; - 4)$ . The graphs of $f$ and $g$ intersect at P and E.
6.4 Write down the values of x for which $f (x) - g (x) > 0$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify interval left of first intersection $x < 0$
  1 mark · Solve an inequality from a graph
2. ✓
  Identify interval right of second intersection $x > 3$
  1 mark · Solve an inequality from a graph
6. Question 3.1.5(a) · March 2025 · Provincial · 2 marks
3.1 Given: $f (x) = \frac{2 x + 3}{- x - 3}$

Given: $f (x) = \frac{2 x + 3}{- x - 3}$
3.1.5 Determine the value(s) of $x$ for which:
3.1.5(a) $f$ is decreasing.(2)
Show the memo (2 ticks)Memo
1. ✓
  $x \in R$ $x \in R$
  1 mark · State increasing and decreasing intervals
2. ✓
  $x \neq = - 3$ $x \neq = - 3$
  1 mark · State increasing and decreasing intervals
7. Question 6.2.2 · March 2025 · Provincial · 4 marks
6.2 In the diagram, the graphs of $f (x) = sin 2 x$ and $g (x) = cos (x - 4 5^{\circ})$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
6.2.2 Write down the coordinates of A and B.(4)
Show the memo (4 ticks)Memo
1. ✓
  A: x-coordinate $g (x)$ has its minimum $- 1$ where $x - 4 5^{\circ} = - 18 0^{\circ}$ , so $x = - 13 5^{\circ}$
  1 mark · Find trig graph amplitude and range
2. ✓
  A: y-coordinate $y = - 1$
  $A (- 13 5^{\circ}; - 1)$
  1 mark · Find trig graph amplitude and range
3. ✓
  B: x-coordinateFrom 6.1: $x = 4 5^{\circ} + k \cdot 12 0^{\circ}$ . For B ( $x < 0$ ), $k = - 1$ : $x = - 7 5^{\circ}$
  1 mark · Find solutions in an interval
4. ✓
  B: y-coordinate $y = sin (2 (- 7 5^{\circ})) = - \frac{1}{2}$
  $B (- 7 5^{\circ}; - \frac{1}{2})$
  1 mark · Find solutions in an interval
8. Question 4.4 · March 2024 · Paper 1 · Provincial · 5 marks
The sketch below shows the graphs of $h (x) = - x^{2} + 2 x + p$ and $t (x) = m x + 5$ . $D$ is the $x$ -intercept, and $B$ the $y$ -intercept of $t$ . $A$ is the $y$ -intercept of $h$ . $C$ is the $x$ -intercept of $h$ . $F$ is the turning point of $h$ . $E$ is a point on $t$ , such that $EC$ is parallel to the $y$ -axis. $A B$ is 2 units and $C D$ is 7 units.
4.4 Determine the length of $EC$ .(5)
Show the memo (4 ticks)Memo
1. ✓
  Determine coordinates of $D$ $C (3; 0)$ and $C D = 7 ⟹ D (10; 0)$
  1 mark · Find a horizontal or vertical length
2. ✓
  Calculate gradient of line $t$ $B (0; 5)$ and $D (10; 0)$
  $m = \frac{0 - 5}{10 - 0} = - \frac{1}{2}$
  2 marks · Calculate gradient between two points
3. ✓
  Find $y$ -coordinate of $E$ Equation of $t$ : $y = - \frac{1}{2} x + 5$
  Since $EC ∥ y$ -axis, $x_{E} = x_{C} = 3$
  $y_{E} = - \frac{1}{2} (3) + 5 = 3, 5$
  1 mark · Substitute into a formula
4. ✓
  Calculate length $EC$ $EC = 3, 5 - 0 = 3, 5$ units
  1 mark · Find a horizontal or vertical length
9. Question 8.4 · March 2024 · Paper 1 · Provincial · 2 marks
In the diagram below, the graphs of $f (x) = cos x + m$ and $g (x) = sin (x + n)$ are drawn on the same set of axes for $x \in [- 24 0^{\circ}; 24 0^{\circ}]$ . $A$ is an $x$ -intercept of $f$ and has coordinates $(6 0^{\circ}; 0)$ . $B$ is a turning point of $g$ and has coordinates $(6 0^{\circ}; 1)$ .
8.4 For which values of $x$ will $f (x) \cdot g (x) \leq 0$ in the interval $x \in [0^{\circ}; 24 0^{\circ}]$ ?(2)
Show the memo (1 tick)Memo
1. ✓
  Analyze signs of both graphs $6 0^{\circ} \leq x \leq 15 0^{\circ}$
  2 marks · Solve product inequality from graphs
10. Question 4.1 · March 2023 · Provincial · 4 marks
Given: $f (x) = \frac{2}{x}$ and $g (x) = x - 1$
4.1 Determine the coordinates of the point(s) where the two graphs intersect.(4)
Show the memo (4 ticks)Memo
1. ✓
  Equate functions $\frac{2}{x} = x - 1$
  $2 = x^{2} - x$
  $x^{2} - x - 2 = 0$
  1 mark · Find where two graphs intersect
2. ✓
  Factors $(x - 2) (x + 1) = 0$
  1 mark · Factorise a trinomial
3. ✓
  x-values $x = 2$ or $x = - 1$
  1 mark · Solve a quadratic by factorising
4. ✓
  y-values $y = 2 - 1 = 1$ ; $y = - 1 - 1 = - 2$
  $(2; 1)$ and $(- 1; - 2)$
  1 mark · Find where two graphs intersect

Graph Interpretation Grade 12 videos

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

Vertical Distance Grade 11 FunctionsKevinmathscience

Functions : Horizontal Translations - Horizontal Shift - y = f(x - c) Examples and ExplanationRadford Mathematics

Hyperbola Equation Given Graph | Grade 11 FunctionsKevinmathscience

Parabola + Straight Line| Grade 11 FunctionsKevinmathscience

f(x) bigger than 0 | Grade 11 FunctionsKevinmathscience

Hyperbola Increasing Decreasing | Grade 11 FunctionsKevinmathscience

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