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The image contains a math problem involving two graphs \( f(x) = -(x - 2)^2 + 4 \) and \( g(x) = a \left(x + \frac{1}{2} \right)^2 + 4 \). Below are the details of each sub-question:

1. **3.16.1**: Write down the equation of the asymptote of \( g \).
2. **3.16.2**: Write down the coordinates of \( D \).
3. **3.16.3**: Write down the range of \( f \).
4. **3.16.4**: Calculate the length of \( HB \).
5. **3.16.5**: Determine the values of \( x \) for which \( f \) is increasing.
6. **3.16.6**: Calculate the average gradient between points \( A \) and \( B \).
7. **3.16.7**: Graph \( h \) is obtained by reflecting \( g \) about the x-axis. Write down the range of \( h \).
8. **3.16.8**: If \( p(x) = f(x) + 2 \), write down the coordinates of the turning point of \( p \).
9. **3.16.9**: Determine the value of \( k \) for which the straight line \( y = 2x + k \) will be a tangent to \( f \).

Would you like me to solve any specific sub-questions or provide explanations for all of them?

The diagram shows the graphs of f(x) = -(x - 2)^2 + 4 and g(x) = a(x + 1/2)^2 + 4. The problem involves various sub-questions, including writing equations, determining coordinates, calculating range, determining values for which f(x) is increasing, and more.

This problem involves solving various sub-questions related to quadratic graphs f(x) = -(x - 2)^2 + 4 and g(x). Topics include determining the equation of the asymptote, calculating average gradients, finding the range of functions, and determining values for which f(x) is increasing. Suitable for Grade 10-12 students.

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