Grade 10 Maths · Functions

Find axes of symmetry of hyperbola

Find the two lines of symmetry of a hyperbola through the point where its asymptotes cross (for example y = x + c and y = −x + c).

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Hyperbola Axis of Symmetry Grade 10: IntroductionKevinmathscience

Hyperbola Axis of symmetry equations Grade 11+ (mathdou)Dou Kleefstra

Hyperbola 4: The Axis of SymmetryRaeesa Manjoo

11 HYPERBOLA Axes of symmetryGrace Maths

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Question 4.3 · 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

4.3 If

y = x + t

is the equation of a line of symmetry of

f

, calculate the value of

t

.(2)

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

More past paper questions: find axes of symmetry of hyperbola

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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 3.1.4 · March 2025 · Provincial · 2 marks
3.1 Given: $f (x) = \frac{2 x + 3}{- x - 3}$

Given: $f (x) = \frac{2 x + 3}{- x - 3}$
3.1.4 Determine the equation of the axis of symmetry of $f$ , which is a decreasing function.(2)
Show the memo (2 ticks)Memo
1. ✓
  Identify gradient and pointDecreasing axis of symmetry has $m = - 1$ .
  Passes through intersection of asymptotes $(- 3; - 2)$ .
  1 mark · Find axes of symmetry of hyperbola
2. ✓
  Find equation $y = - x + c$
  $- 2 = - (- 3) + c \to c = - 5$
  $y = - x - 5$
  1 mark · Find axes of symmetry of hyperbola
5. Question 5.1 · June 2025 · Paper 1 · Provincial · 2 marks
Given: $f (x) = - x + 1$ and $g (x) = \frac{- 2}{x} + 2$ , which are not drawn to scale. R and S are points of intersection of the graphs.

DBQ is a line perpendicular to the x-axis with D and B on $g$ and $f$ respectively. The coordinates of Q are $(- 1; 0)$ .
5.1 Determine the equation of the line of symmetry of $g$ with a negative gradient.(2)
Show the memo (1 tick)Memo
1. ✓
  Determine the equation of the line of symmetry $y = - (x - p) + q$
  $y = - (x - 0) + 2$
  $y = - x + 2$
  2 marks · Find axes of symmetry of hyperbola
6. Question 3.5 · March 2024 · Paper 1 · Provincial · 2 marks
The diagram shows the graph of $f (x) = \frac{3}{x + 2} + 1$
3.5 $y = x + c$ is the equation of one of the axes of symmetry of $g$ . Determine the value of $c$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute intersection of asymptotes into symmetry formulaAsymptotes of $g$ : $x = 2$ and $y = 3$
  Intersection point $(2; 3)$
  $3 = 2 + c$
  1 mark · Find axes of symmetry of hyperbola
2. ✓
  Solve for $c$ $c = 1$
  1 mark · Find axes of symmetry of hyperbola

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