For g(x)=-3x^2+5.2x+7 find the maximum value of the fuction

To find the maximum value of the quadratic function \( g(x) = -3x^2 + 5.2x + 7 \), we can use the vertex formula. The vertex of a parabola given by a quadratic function \( ax^2 + bx + c \) occurs at \( x = \frac{-b}{2a} \).

Here, the coefficients are:
- \( a = -3 \)
- \( b = 5.2 \)
- \( c = 7 \)

The x-coordinate of the vertex is:

\[
x = \frac{-b}{2a} = \frac{-5.2}{2(-3)} = \frac{-5.2}{-6} = \frac{5.2}{6} \approx 0.867
\]

Next, we substitute this \( x \)-value back into the original function to find the maximum value:

\[
g(0.867) = -3(0.867)^2 + 5.2(0.867) + 7
\]

Let's calculate it step by step:

1. Calculate \( (0.867)^2 \):

\[
(0.867)^2 \approx 0.751
\]

2. Substitute into the function:

\[
g(0.867) = -3(0.751) + 5.2(0.867) + 7
\]
\[
g(0.867) = -2.253 + 4.5084 + 7
\]
\[
g(0.867) \approx 9.2554
\]

Thus, the maximum value of the function \( g(x) = -3x^2 + 5.2x + 7 \) is approximately **9.255**.

Would you like further details or have any other questions?

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Here are 5 related questions:
1. How does the sign of the coefficient \( a \) in a quadratic function affect the shape of the parabola?
2. What is the significance of the vertex in the context of maximizing or minimizing a quadratic function?
3. How would the maximum value change if the coefficient of \( x^2 \) was positive?
4. Can you find the roots of the function \( g(x) = -3x^2 + 5.2x + 7 \)?
5. How would the function behave as \( x \) approaches infinity?

**Tip:** In a quadratic function, if the coefficient of \( x^2 \) is negative, the parabola opens downward, and the vertex represents the maximum point.

Learn how to find the maximum value of the quadratic function g(x) = -3x^2 + 5.2x + 7 using the vertex formula. This problem covers quadratic functions and their applications in finding maximum or minimum values. Suitable for Grades 10-12 students.

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