for g(x) = -3x^2 + 5.2x + 7, find the maximum value of the function

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 \( x = h \).

Learn how to find the maximum value of the quadratic function g(x) = -3x^2 + 5.2x + 7. Understand the vertex formula and its application to determine whether a quadratic function has a maximum or minimum. Suitable for high school students studying quadratic functions.

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