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We are asked to compare the maximum values of two functions: \( f(x) = -2(x + 4)^2 + 1 \) and \( g(x) \) (represented graphically).

### 1. Analyze \( f(x) \):
- The function \( f(x) = -2(x + 4)^2 + 1 \) is a quadratic equation in vertex form. The general form is \( a(x - h)^2 + k \), where \( (h, k) \) is the vertex.
- In this case, the vertex occurs at \( x = -4 \), and the maximum value of \( f(x) \) is \( k = 1 \), because the parabola opens downwards (since \( a = -2 \), which is negative).

Thus, the maximum value of \( f(x) \) is **1** at \( x = -4 \).

### 2. Analyze \( g(x) \):
- From the graph, the maximum value of \( g(x) \) occurs at \( x = -3 \), and the maximum y-value is **3**.

### 3. Conclusion:
- \( f(x) \) has a maximum value of 1, and \( g(x) \) has a maximum value of 3.
- Therefore, the function **\( g \) has a greater maximum than \( f \)**.

The correct answer is:
- \( g \) has a greater maximum than \( f \).

Would you like more details or have any questions? Here are some related questions to expand on this:

1. How can we determine the vertex of a parabola in vertex form?
2. How do changes in the coefficient of the quadratic term \( a \) affect the graph?
3. What is the importance of the vertex in finding maximum and minimum points?
4. How would you find the maximum or minimum of a quadratic function in standard form?
5. What is the domain and range of \( f(x) \) and \( g(x) \) based on their equations and graphs?

**Tip:** In vertex form, the sign of the coefficient \( a \) determines whether a parabola opens upwards (minimum point) or downwards (maximum point).

Which function has a greater maximum? f(x) = -2(x + 4)^2 + 1.

Learn how to compare the maximum values of two quadratic functions, f(x) = -2(x + 4)^2 + 1 and g(x), using graph analysis and properties of vertex form. Suitable for students in Grades 9-11.

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