Grade 10 Maths · Functions

State the asymptotes

Write down the lines a hyperbola or exponential graph gets close to but never touches (for example y = 1/x + 3 has asymptotes x = 0, y = 3).

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Question 4.1 · May/June 2025 · Paper 1 · National (DBE) · 2 marks

The graph of

f (x) = \frac{4}{x - 3} + 4

is drawn below. M is the point where the asymptotes of

f

intersect. C and D are the x- and y-intercepts respectively of

f

. A is the point on

f

that is closest to M. Diagram for this question

4.1 Write down the coordinates of M.(2)

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

More past paper questions: state the asymptotes

Work each one out first, then open the memo and check every tick.

3. Question 5.2 · November 2024 · Paper 1 · National (DBE) · 2 marks
Sketched below is the graph of $f (x) = \frac{a}{x + p} + q$ having the domain $(- \infty; 1) \cup (1; \infty)$ . The graph of $f$ cuts the y-axis at $(0; 1)$ . A line of symmetry of $f$ is given by $g (x) = x - 3$ .
5.2 Determine the equation of the horizontal asymptote of $f$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitution $y = x - 3$ passes through the point where the asymptotes meet, $(1; q)$ | $y = 1 - 3$
  1 mark · Find axes of symmetry of hyperbola
2. ✓
  Answer $y = - 2$
  1 mark · State the asymptotes
4. Question 3.1 · March 2024 · Paper 1 · Provincial · 2 marks
The diagram shows the graph of $f (x) = \frac{3}{x + 2} + 1$
3.1 Write down the equations of the asymptotes of $f$ .(2)
Show the memo (1 tick)Memo
1. ✓
  Identify vertical and horizontal asymptotes $x = - 2$
  $y = 1$
  2 marks · State the asymptotes
5. Question 6.2 · September 2024 · Paper 1 · Provincial · 2 marks
Sketched below are the graphs of $f (x) = - x + 4$ and $g (x) = \frac{2}{x + 1} + 2.$
6.2 Write down the equations of the asymptotes of g.(2)
Show the memo (2 ticks)Memo
1. ✓
  x asymptote $x = - 1$
  1 mark · State the asymptotes
2. ✓
  y asymptote $y = 2$
  1 mark · State the asymptotes
6. Question 6.1 · June 2024 · Paper 1 · Provincial · 2 marks
Given: $k (x) = \frac{- 4}{x + 3} + 1$ and $h (x) = 2^{- x} - 4$
6.1 Write down the equations of the asymptotes of k.(2)
Show the memo (2 ticks)Memo
1. ✓
  Vertical Asymptote $x = - 3$
  1 mark · State the asymptotes
2. ✓
  Horizontal Asymptote $y = 1$
  1 mark · State the asymptotes

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