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It looks like you've uploaded an image that contains a mathematical problem related to a quadratic function \( f(x) = ax^2 + bx + c \) and \( g(x) = d\left( x - e \right)^2 + f \). There are some graphs and specific points on the curves.

Here is how I interpret the problem based on the visible part of the image:

1. **Given equations and graphs:** The problem seems to involve quadratic functions with specified points. The graph shows the curves for \( f(x) \) and \( g(x) \), with turning points and intersections labeled.
2. **Task (7.1):** You are asked to determine the values of \( a \) and \( b \) for \( f(x) \), given that the turning point occurs at \( x = -2 \), and another point on the graph passes through \( f(-4) = -10 \).

### Steps to Solve:
1. **Quadratic form and turning point:** The turning point form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the turning point. Since the turning point is given as \( (-2, 2) \), we can write:
   \[
   f(x) = a(x + 2)^2 + 2
   \]
2. **Substitute \( f(-4) = -10 \):** Using the fact that \( f(-4) = -10 \), substitute \( x = -4 \) into the equation and solve for \( a \):
   \[
   -10 = a(-4 + 2)^2 + 2
   \]
   Simplify:
   \[
   -10 = a(2)^2 + 2
   \]
   \[
   -10 = 4a + 2
   \]
   \[
   4a = -12
   \]
   \[
   a = -3
   \]
3. **Final form of the function \( f(x) \):** Now that we know \( a = -3 \), substitute it back into the equation:
   \[
   f(x) = -3(x + 2)^2 + 2
   \]
   Expanding this equation:
   \[
   f(x) = -3(x^2 + 4x + 4) + 2
   \]
   \[
   f(x) = -3x^2 - 12x - 12 + 2
   \]
   \[
   f(x) = -3x^2 - 12x - 10
   \]
   Therefore, the quadratic function \( f(x) = -3x^2 - 12x - 10 \).

Would you like me to go into further detail or help with the next part of the question?

### Follow-up questions:
1. How is the turning point of a quadratic function related to its vertex form?
2. Can you derive the general quadratic function from a different pair of points on the graph?
3. What effect does changing the value of \( a \) have on the shape of a parabola?
4. How do we calculate the intersection points of two quadratic functions?
5. Can you explain the significance of the second curve \( g(x) \) in the diagram?

### Tip:
When working with quadratic functions, it's useful to recognize both the standard form \( ax^2 + bx + c \) and the vertex form \( a(x - h)^2 + k \), as each provides different insights into the function's behavior.

Determine the values of a and b if the turning point of f(x) is at (-2, 2) and f(x) passes through the point f(-4) = -10.

Learn how to solve a quadratic equation given the vertex at (-2, 2) and the point f(-4) = -10. This solution covers key algebraic concepts such as vertex form, standard form, and properties of quadratic functions. Suitable for high school students in Grades 10-12.

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