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Let's go through each part of the problem step-by-step.

Given:
1. The functions are \( f(x) = ax^2 + q \) and \( g(x) = \frac{3}{x} - 1 \), with \( x > 0 \).
2. Point \( A \) is the \( x \)-intercept of \( f \) and also the intersection point of both functions.
3. Point \( B(0, 9) \) is the turning point of \( f \), and line \( CD \) is vertical.

### 3.1 Write down:

#### 3.1.1 Range of \( f \)
Since \( B(0, 9) \) is the turning point of \( f(x) \), and the parabola opens downward (as seen in the graph), the maximum value of \( f(x) \) is 9. Therefore, the range of \( f(x) \) is:
\[
(-\infty, 9]
\]

#### 3.1.2 Domain of \( g \)
The function \( g(x) = \frac{3}{x} - 1 \) has a domain restriction where \( x > 0 \). Thus, the domain of \( g \) is:
\[
(0, \infty)
\]

#### 3.1.3 Equation of the horizontal asymptote of \( g \)
As \( x \to \infty \), the term \( \frac{3}{x} \to 0 \), so \( g(x) \to -1 \). Therefore, the horizontal asymptote of \( g \) is:
\[
y = -1
\]

#### 3.1.4 Values of \( x \) such that \( f \) is decreasing
Since \( f(x) = ax^2 + q \) opens downward, \( a < 0 \). \( f \) will be decreasing to the right of the turning point \( B \), which is at \( x = 0 \). So, \( f \) is decreasing for:
\[
x > 0
\]

### 3.2 Determine the values of \( a \) and \( q \)

Given that \( B(0, 9) \) is the turning point, we know:
1. \( f(0) = 9 \), so substituting \( x = 0 \) gives \( f(0) = q = 9 \).
   \[
   q = 9
   \]

2. Since \( A \) is the \( x \)-intercept, \( f(x) = 0 \) at \( x = a \). The values of \( a \) and \( q \ 

The graphs of f(x) = ax^2 + q and g(x) = 3/x - 1; x > 0 are drawn below. A is the x-intercept and point of intersection of both graphs. B(0 ; 9) is the turning point of f and CD is a vertical line. Write down:

1. Range of f
2. Domain of g
3. Equation of the horizontal asymptote of g
4. Value(s) of x such that f is decreasing

Determine the values of a and q.

If h(x) = -f(x) - 3, write down the:
1. Equation of h
2. Coordinates of the turning point of h.

Learn to analyze quadratic and rational functions with parameters, including finding range, domain, horizontal asymptote, and the conditions under which the function decreases. This problem covers key topics in algebra and calculus, suitable for high school students in Grades 10-12.

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