Sketched below are the graphs of functions defined by $f(x) = ax^2 + bx + c$ and h(x) = \frac{k}{x} + q$ with U(1; 10) one of the points of intersection of f and h - NSC Technical Mathematics - Question 4 - 2021 - Paper 1

The coordinates of P, which is the x-intercept of the function $f$, are given by its intersection with the x-axis. From the diagram, P is located at: $P(-4; 0)$

To determine the equation of the function $f$, given that it is a quadratic function and knowing its turning point and intercepts, we can use the vertex form. The points given allow us to formulate: $f(x) = a(x + 4)^2 - S$ From the axis of symmetry and the turning point, we derive the equation: f(x) = -2(x + 1)^2 + 18\n$$

The equation of the function $h$ is influenced by the asymptote's equation $y = 9$. From this we find: $h(x) = \frac{k}{x} + 9$ where the specific value of k can be found using the additional given points.

To find the length of segment RV, we determine the coordinates of R and V first. Given the coordinates: $R(-1; 9) \quad V(\text{on h and axis}) = (\text{value from analysis})$ The length is computed as: $RV = \sqrt{(x_r - x_v)^2 + (y_r - y_v)^2}$

The function \frac{h(x)}{f(x)} is undefined when $f(x) = 0$. Therefore, we can determine: $x = -4, 2$ These points occur when the denominator equals zero.