Find the maximum value of the function f, of, x, equals, minus, 1, point, 8, x, squared, minus, 16, point, 1, x, minus, 38, point, 5f(x)=−1.8x 
2
 −16.1x−38.5 to the nearest hundredth.

To find the maximum value of the quadratic function:

\[
f(x) = -1.8x^2 - 16.1x - 38.5
\]

we start by noting that this is a parabola that opens downward (since the coefficient of \(x^2\) is negative), so it has a maximum point, which occurs at the vertex.

The formula for the x-coordinate of the vertex for a quadratic function \( ax^2 + bx + c \) is:

\[
x = -\frac{b}{2a}
\]

For the function \( f(x) = -1.8x^2 - 16.1x - 38.5 \), we have:
- \( a = -1.8 \)
- \( b = -16.1 \)

Now, let's calculate the x-coordinate of the vertex:

\[
x = -\frac{-16.1}{2(-1.8)} = \frac{16.1}{3.6}
\]

I'll calculate this value for you.The x-coordinate of the vertex is approximately \( x = -4.47 \).

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

\[
f(x) = -1.8x^2 - 16.1x - 38.5
\]

I'll calculate the maximum value \( f(-4.47) \).The maximum value of the function is approximately \( f(x) = -2.50 \) to the nearest hundredth.

Do you have any questions or would you like further details?

Here are 5 related questions:
1. How do we determine whether a quadratic function has a maximum or minimum?
2. What is the significance of the vertex in a quadratic function?
3. How do the coefficients \(a\), \(b\), and \(c\) in a quadratic function affect its graph?
4. Can a parabola have more than one maximum or minimum point?
5. What other methods can be used to find the maximum value of a quadratic function?

**Tip:** Always check the sign of the \(x^2\) coefficient to determine whether a quadratic opens upwards (minimum) or downwards (maximum).

Find the maximum value of the function f(x) = -1.8x^2 - 16.1x - 38.5 to the nearest hundredth.

Learn how to find the maximum value of the quadratic function f(x) = -1.8x^2 - 16.1x - 38.5 by calculating the vertex using the formula x = -b / 2a. This solution is suitable for students in Grades 9-10 and covers key algebraic concepts like the vertex of a parabola and quadratic functions.

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