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).