To find the maximum value of the quadratic function $g(x) = -3x^2 + 5.2x + 7$, you can use the fact that the maximum or minimum of a quadratic function $ax^2 + bx + c$ occurs at the vertex.

For a quadratic function $g(x) = ax^2 + bx + c$, the x-coordinate of the vertex is given by:

$x = -\frac{b}{2a}$

Here, $a = -3$ and $b = 5.2$. Plugging in these values:

$x = -\frac{5.2}{2(-3)} = \frac{5.2}{6} \approx 0.8667$

Now, substitute this x-value back into the function $g(x)$ to find the maximum value:

$g(0.8667) = -3(0.8667)^2 + 5.2(0.8667) + 7$

Calculating this:

$g(0.8667) \approx -3(0.7511) + 4.5068 + 7 \approx -2.2533 + 4.5068 + 7 = 9.2535$

So, the maximum value of the function $g(x)$ is approximately $9.25$ when $x \approx 0.8667$.

Would you like a more detailed step-by-step explanation or further assistance?

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1. How do you determine whether a quadratic function has a maximum or minimum?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How can this method be applied to real-world problems?
4. How does the sign of $a$ in a quadratic function affect its graph?
5. Can the vertex formula be used for non-quadratic functions?

Tip: The vertex form of a quadratic function, $g(x) = a(x-h)^2 + k$, directly shows the maximum or minimum value as $k$ when $x = h$.