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Functions and graphs

29 marks in our Paper 1 questions. 13 questions from 2024.

Practise this topic, marked

This topic tests how well you read and build graphs: straight lines, parabolas, exponential graphs and hyperbolas, with their intercepts, asymptotes, ranges and the lengths between two graphs. The 13 questions here are worth 29 marks in the November 2024 Paper 1 set. Of the 29 marks, 22 are for the correct final answer and 7 for method and working.

The 2024 Diagnostic Report says candidates lose marks in three ways. The horizontal asymptote is read off a line’s y-value instead of from where the vertical asymptote meets that line. The range of an exponential function is written as y not equal to a number. And a vertical length is never turned into an expression, so the candidate solves f(x) minus g(x) equal to zero as if it gave a maximum, when that only finds where the graphs meet. Draw the graph first, then write what each feature tells you.

Where marks are lost: in our words, from Department of Basic Education, NSC 2024 Diagnostic Report, Part 1 (Book 1).

Questions

These questions are not yet checked by a teacher.

g

g (x) = b^{x} + 2

b > 0

P (3; \frac{43}{8})

lies on the graph of

g

. Calculate the value of

b

(2)Based on Nov 2024 · Paper 1 · Q4.1

g (x) = b^{x} + 3

b > 0

, and the point

P (2; \frac{37}{4})

g

. State the set of all possible

y

g

(1)Based on Nov 2024 · Paper 1 · Q4.2

g

g (x) = b^{x} - 2

b > 0

P (2; - \frac{7}{4})

g

. Draw a neat sketch of

g

on your own set of axes, labelling where it cuts each axis and giving the equation of any asymptote.

(3)Based on Nov 2024 · Paper 1 · Q4.3

f (x) = {(\frac{1}{2})}^{x} - 1

is given. C is the point on the graph of

f

y = 7

. Determine the coordinates of

C ’

, the image of C when C is reflected in the line

y = x

(3)Based on Nov 2024 · Paper 1 · Q4.4

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

x \in R, x \neq 2

. Write down the value of

p

(1)Based on Nov 2024 · Paper 1 · Q5.1

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

(- \infty; 2) \cup (2; \infty)

f

y

(0; 2)

. The straight line

g (x) = x - 5

is a line of symmetry of

f

. Determine the equation of the horizontal asymptote of

f

(2)Based on Nov 2024 · Paper 1 · Q5.2

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

x \in R

x \neq 2

. The vertical asymptote is

x = 2

and the horizontal asymptote is

y = - 1

f

y

(0; 2)

. Calculate the value of

a

(2)Based on Nov 2024 · Paper 1 · Q5.3

f (x) = \frac{- 6}{x - 2} - 2

is drawn in the diagram below. Its domain is

x \in (- \infty; 2) \cup (2; \infty)

. The graph cuts the

y

(0; 1)

g (x) = x - 4

is a line of symmetry of

f

. Determine the values of

x

f (x) \geq 0

(3)Based on Nov 2024 · Paper 1 · Q5.4

f

is given in the information above. Graph

f

is transformed to a new graph

h

h

has the same domain and the same range as

f

• the gradient of

h

is negative at every point of its domain, that is

h ’ (x) < 0

. Describe one possible transformation of

f

h

(2)Based on Nov 2024 · Paper 1 · Q5.5

The diagram below shows the graph of

f (x) = - x^{2} + 6 x + 7

, a parabola with turning point B. Calculate the coordinates of B.

(3)Based on Nov 2024 · Paper 1 · Q6.1

The diagram below shows the parabola

f (x) = - x^{2} + 2 x + 8

and a straight line

g

g

cuts the parabola at point A on the left, which lies on the x-axis, and at point C(2;8) on the right. Prove that the straight line

g

has the equation

g (x) = 2 x + 4

(3)Based on Nov 2024 · Paper 1 · Q6.2

g

g (x) = - 4 x^{2}

. State one way to restrict the domain of

g

so that its inverse,

g^{- 1}

, is a function.

(1)Based on Nov 2024 · Paper 1 · Q8.3.2

g

g (x) = - 4 x^{2}

. Find the inverse relation of

g

and write it as

y = \dots

, choosing the branch for which

y \leq 0

(3)Based on Nov 2024 · Paper 1 · Q8.3.3

Updated 12 November 2026