Figure 1 shows a sketch of a curve C with equation $y = f(x)$ where $f(x)$ is a cubic expression in $x$ - Edexcel - A-Level Maths Pure - Question 8 - 2022 - Paper 1

From the behavior of the cubic function and the horizontal line test, the line $y = k$ intersects the curve C at only one point when $k$ is greater than the local maximum value at $(2, 8)$ or less than the local minimum value at $(6, 0)$. Therefore, we can state the set of values of $k$ as:
ext{{{k | k > 8}}} igcup ext{{{k | k < 0}}}

To find the equation of the cubic curve C, we can use the turning points. Let $C$ be $y = a(x - r_1)(x - r_2)(x - r_3)$. Using the given points, we can assume it is of the form:
$y = a(x)(x - 2)(x - 6)$
To find $a$, we substitute one of the given points, such as $(2, 8)$, into the function. After some calculations, we arrive at:
C: y = rac{1}{4}x(x - 2)(x - 6)
Thus, the factorised equation of C is:
y = rac{1}{4}x(x - 2)(x - 6)