To determine which function has a greater maximum, we need to compare the maximum values of the given functions $f(x)$ and $g(x)$.

1. For the function $f(x) = -5(x + 4)^2 + 8$:

This is a quadratic function in vertex form:
$f(x) = a(x - h)^2 + k$
where $a = -5$, $h = -4$, and $k = 8$.

• The vertex of this parabola is at $(-4, 8)$, which is the maximum point because the parabola opens downwards (since $a = -5$ is negative).
• Thus, the maximum value of $f(x)$ is 8.

2. For the function $g(x)$ (from the graph):

The graph of $g(x)$ shows a parabola that opens downward. The vertex, or the maximum point, appears to be at $(x = -7, y = 9)$.

• So, the maximum value of $g(x)$ is 9.

Conclusion:

Since $g(x)$ has a maximum of 9 and $f(x)$ has a maximum of 8, $g(x)$ has a greater maximum than $f(x)$.

The correct answer is B: $g$ has a greater maximum than $f$.

Would you like further explanation or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you determine the vertex of a quadratic function in vertex form?
2. Why does the parabola open downward when the coefficient of the squared term is negative?
3. How can you find the maximum or minimum of a quadratic function using its standard form?
4. How does changing the coefficient $a$ in a quadratic function affect its graph?
5. How can you convert a quadratic function from standard form to vertex form?

Tip: The vertex form of a quadratic equation easily reveals the maximum or minimum value, which occurs at the vertex.