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The image shows a problem asking for the maximum value of the quadratic function \( f(x) = -0.6x^2 - 3.7x + 1 \) to the nearest hundredth.

### Step 1: Determine the Vertex of the Quadratic Function
For the quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using:
\[
x = -\frac{b}{2a}
\]
Given:
- \( a = -0.6 \)
- \( b = -3.7 \)

Substituting these values into the formula:
\[
x = -\frac{-3.7}{2(-0.6)} = \frac{3.7}{-1.2} \approx -3.0833
\]

### Step 2: Find the Maximum Value of the Function
Substitute \( x \approx -3.0833 \) back into the function to find the corresponding y-coordinate, which is the maximum value since the parabola opens downwards (because \( a \) is negative):
\[
f(-3.0833) = -0.6(-3.0833)^2 - 3.7(-3.0833) + 1
\]
\[
f(-3.0833) \approx -0.6(9.5088) + 11.2082 + 1
\]
\[
f(-3.0833) \approx -5.7053 + 11.2082 + 1 \approx 6.5029
\]

Rounding to the nearest hundredth:
\[
f(-3.0833) \approx 6.50
\]

### Conclusion
The maximum value of the function \( f(x) = -0.6x^2 - 3.7x + 1 \) is approximately **6.50**.

Would you like further explanation or have any questions?

Here are 5 related questions to further your understanding:
1. Why does the function \( f(x) = -0.6x^2 - 3.7x + 1 \) have a maximum value instead of a minimum?
2. How does the value of \( a \) affect the steepness of the parabola?
3. What is the significance of the vertex in determining the extremum of a quadratic function?
4. Can you find the x-intercepts of the function \( f(x) = -0.6x^2 - 3.7x + 1 \)?
5. How do the values of \( b \) and \( c \) influence the position of the vertex on the graph?

**Tip:** Always double-check your calculations when rounding to ensure accuracy, especially in exams or assessments where precision matters.

Learn how to find the maximum value of the quadratic function f(x) = -0.6x^2 - 3.7x + 1 to the nearest hundredth. Step-by-step solution included. Suitable for high school students in Grades 10-12.

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