Grade 12 Maths · Calculus · Paper 1

Tangents to Graphs of Functions Grade 12

Real Tangents to Graphs of Functions 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: 8 past paper questions
skills: 2 skills
videos: 2 videos

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Question 8.3 · May/June 2025 · Paper 1 · National (DBE) · 6 marks

8.3 Given that

y = 4 x - 14

is a common tangent to

f (x) = 2 x^{2} - 4 x - 6

and

g (x) = a x^{2} + b x - 18

, calculate the values of

a

and

b

.(6)

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

Tangents to Graphs of Functions skills

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More Tangents to Graphs of Functions Grade 12 questions and answers

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4. Question 6.6 · May/June 2024 · Paper 1 · National (DBE) · 5 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.
6.6 Given: $k (x) = g (x) - n$ . Determine $n$ if $k$ is a tangent to $f$ .(5)
Show the memo (5 ticks)Memo
1. ✓
  Equate derivative of f to gradient of g $2 x - 2 = 1$
  1 mark · Find equation of a tangent
2. ✓
  Find x-coordinate of tangency $x = 1, 5$
  1 mark · Find equation of a tangent
3. ✓
  Find y-coordinate on f $y = (1, 5)^{2} - 2 (1, 5) - 3 = - 3, 75$
  1 mark · Find equation of a tangent
4. ✓
  Substitute point into k(x) equation $- 3, 75 = 1, 5 - 3 - n$
  1 mark · Find equation of a tangent
5. ✓
  Solve for n $n = 2, 25$
  1 mark · Find equation of a tangent
5. Question 8.3 · May/June 2024 · Paper 1 · National (DBE) · 4 marks
8.3 The equation of the tangent to $f (x) = 3 x^{2} + b x + c$ at $x = 1$ is given by $y = 9 x - 9$ . Determine the values of $b$ and $c$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Gradient of the tangent at x = 1 equated to 9 $f^{'} (x) = 6 x + b$
  $f^{'} (1) = 6 + b = 9$
  1 mark · Find equation of a tangent
2. ✓
  Value of b $b = 3$
  1 mark · Find equation of a tangent
3. ✓
  Point of contact substituted into f $y = 9 (1) - 9 = 0$
  $f (1) = 3 + 3 + c = 0$
  1 mark · Find equation of a tangent
4. ✓
  Value of c $c = - 6$
  1 mark · Find equation of a tangent
6. Question 9.2 · September 2024 · Paper 1 · Provincial · 2 marks
9.2 Given: $h (x) = 4 x^{3} + 5 x$ . Substantiate whether it is possible to draw a tangent to the graph of h that has a negative gradient.(2)
Show the memo (2 ticks)Memo
1. ✓
  Derivative $h^{'} (x) = 12 x^{2} + 5$
  1 mark · Differentiate using the power rule
2. ✓
  ExplanationAlways positive
  1 mark · Find equation of a tangent
7. Question 10 · September 2024 · Paper 1 · Provincial · 8 marks
The diagram below shows the plan for a verandah which is to be built onto the corner of a cottage. A railing ABCDE is to be constructed around the four edges of the verandah. It is given that $A B = D E = x$ and $BC = C D = y$ , and the length of the railing must be 30 metres.
10 Calculate the value of x and y for which the veranda will have a maximum area.(8)
Show the memo (8 ticks)Memo
1. ✓
  Expression for FE and FA $FE = F A = y - x$
  1 mark · Set up an expression or equation
2. ✓
  Expression for y $2 x + 2 y = 30$ $y = 15 - x$
  1 mark · Set up an expression or equation
3. ✓
  Substitute into the area formula $A (x) = y^{2} - (y - x)^{2} = 2 x y - x^{2}$
  1 mark · Calculate area or perimeter
4. ✓
  Substitute for y $A (x) = 2 x (15 - x) - x^{2}$
  1 mark · Substitute into a formula
5. ✓
  Expression for the area $A (x) = 30 x - 3 x^{2}$
  1 mark · Collect like terms
6. ✓
  Derivative = 0For a maximum: $A^{'} (x) = 30 - 6 x = 0$
  1 mark · Find stationary points
7. ✓
  Value of x $x = 5$ m
  1 mark · Find stationary points
8. ✓
  Value of y $y = 15 - 5 = 10$ m
  1 mark · Substitute into a formula

Tangents to Graphs of Functions Grade 12 videos

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Equation of Tangent Calculus Grade 12Kevinmathscience

Grade 11 Analytical geometry: angle of inclinationKevinmathscience

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