June 2025 Paper 3 Q9

(a) Describe a sequence of two transformations which maps the graph with equation\[y = \frac{1}{x}\]onto the graph with equation\[y = \frac{3}{x - 4}\] [2 marks]

(b) State the equation of the vertical asymptote of the graph with equation\[y = \frac{3}{x - 4}\] [1 mark]

(c) A student is attempting to use a change of sign to determine if the equation\[\frac{3}{x - 4} = x\]has a solution between 3 and 5

The student correctly writes

\[\frac{3}{x - 4} = x \Leftrightarrow \frac{3}{x - 4} - x = 0\]\[\text{Let } \mathrm{f}(x) = \frac{3}{x - 4} - x\]\[\mathrm{f}(3) = -6 \lt 0 \ \text{ and } \ \mathrm{f}(5) = -2 \lt 0\]

The student then incorrectly states:

“Since there is no change of sign, there is no solution between 3 and 5”

Give two reasons why the student’s argument is invalid.

[2 marks]