Sketch 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 point $P$ are given as $P(-4; 0)$. Thus, the coordinates are:

$P(-4; 0)$

The equation of the parabola given the axis of symmetry $x = -1$ can be expressed. Since we have the point $P(-4; 0)$, we can use this to establish the equation:

Starting from the vertex form, we have:

$f(x) = a(x + 1)^2 + k$

Substituting $P(-4; 0)$ into the equation leads to:

$0 = a(-4 + 1)^2 + k$ This will help us determine the coefficients once $k$ is known.

Given the asymptote $y = 9$, we establish that:

$h(x) = \frac{k}{x} + 9$

We also know that $h(1) = 10$ from the point of intersection $U(1; 10)$. Thus:

$10 = \frac{k}{1} + 9 \Rightarrow k = 1$

So the equation of $h$ becomes:

$h(x) = \frac{1}{x} + 9$

To find the length of segment $RV$, we first need the coordinates of point $V$. Given that $V$ is on the axis of symmetry and also on $h$, we substitute the $x$ coordinate from the axis line:

Using $x = -1$, we find:

$V = \left(-1, \frac{1}{-1} + 9 \right) =\left(-1, 8 \right)$

Now the coordinates for $R$ (as the $x$-intercept for $f$) are $R(2;0)$. Thus:

The length of $RV$ can be calculated using the distance formula:

$RV = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ $RV = \sqrt{(2 - (-1))^2 + (0 - 8)^2} = \sqrt{(3)^2 + (-8)^2} = \sqrt{9 + 64} = \sqrt{73}$

The expression \frac{h(x)}{f(x)}$is undefined when$f(x)$equals zero. From the earlier derived information, we find solutions based on the roots of$f(x)$. Setting this equal to zero will help find the$x$ intercepts which can be identified graphically or algebraically depending on provided information.