Grade 10 Maths · Functions

Find equation of exponential graph

Find the equation of an exponential graph using its asymptote and a point on it (for example y = a·2ˣ + q, where q comes from the asymptote).

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Exponential Equation Given Graph | Grade 11 FunctionsKevinmathscience

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Question 6.1 · May/June 2025 · Paper 1 · National (DBE) · 4 marks

The graphs of

f (x) = p^{x} + q

and

g (x) = m x + c

are drawn below.

A (3; 4)

is the point of intersection of

f

and

g

B (0; - 3)

is the y-intercept of

f

. E is the x-intercept of

g

6.1 Calculate the values of

p

and

q

.(4)

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

More past paper questions: find equation of exponential graph

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3. Question 5.2 · May/June 2024 · Paper 1 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = lo g_{a} x$ and $g$ are drawn. Graph $g$ is the reflection of $f$ in the line $y = x$ . Graph $f$ passes through the point P $(4; 2)$ . Q is the $x$ -intercept of $f$ and R is the $y$ -intercept of $g$ .
5.2 Show that $a = 2$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute coordinates into equation $2 = lo g_{a} 4$
  1 mark · Find equation of exponential graph
2. ✓
  Convert to exponential form $a^{2} = 4 ⟹ a = 2$
  1 mark · Convert between log and exponential form
4. Question 7.2 · November 2025 · Paper 1 · Provincial · 4 marks
7.2 An exponential function with equation $h (x) = a \cdot k^{x} + r$ has the following properties:
• The range is $y \in (- 2; \infty)$
• The points $(0; 0)$ and $(2; 6)$ lie on the graph

Determine the equation of $h$ , in the form $y = ...$ (4)
Show the memo (4 ticks)Memo
1. ✓
  $y = a \cdot k^{x} - 2$ $y = a \cdot k^{x} + r$
  $y = a \cdot k^{x} - 2$
  1 mark · Find equation of exponential graph
2. ✓
  $a = 2$ Subst. $(0; 0)$ : $0 = a \cdot k^{0} - 2$
  $2 = a$
  1 mark · Find equation of exponential graph
3. ✓
  Substitution of $(2; 6)$ $6 = 2 k^{2} - 2$
  1 mark · Find equation of exponential graph
4. ✓
  Equation of $h$ $8 = 2 k^{2}$
  $4 = k^{2}$
  $k = \pm 2$
  $∴ k = 2$ ; $k > 0$
  $h (x) = 2 \cdot 2^{x} - 2$
  1 mark · Find equation of exponential graph
5. Question 4.2.3 · March 2025 · Provincial · 2 marks
4.2 Consider $g (x) = (\frac{1}{3})^{x}$ .
4.2.3 The graph of $h (x) = a (\frac{1}{3})^{x} + 7$ passes through the point $(- 2; 10)$ . Calculate the value of $a$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute point $10 = a (\frac{1}{3})^{- 2} + 7$
  $10 = a (3^{2}) + 7 = 9 a + 7$
  1 mark · Find equation of exponential graph
2. ✓
  Solve for a $3 = 9 a \to a = \frac{1}{3}$
  1 mark · Find equation of exponential graph
6. Question 5.2.2 · September 2024 · Paper 1 · Provincial · 2 marks
5.2 The graph of $h (x) = a^{x}$ is sketched below. $A (- 1; \frac{1}{2})$ is a point on the graph of h.
5.2.2 Calculate the value of a.(2)
Show the memo (2 ticks)Memo
1. ✓
  Substitute point A $a^{- 1} = \frac{1}{2}$
  1 mark · Find equation of exponential graph
2. ✓
  Answer $a = 2$
  1 mark · Find equation of exponential graph

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