Find the minimum value of the function f, of, x, equals, x, squared, plus, 7, x, plus, 15, point, 3*f*(*x*)=*x*2+7*x*+15.3 *to the nearest hundredth*.

To find the minimum value of the quadratic function \( f(x) = x^2 + 7x + 15.3 \), we can use the following approach:

### 1. Identify the coefficients:
The quadratic function is in the form \( f(x) = ax^2 + bx + c \), where:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 7 \) (coefficient of \( x \))
- \( c = 15.3 \) (constant term)

### 2. Find the vertex:
The vertex of a parabola given by \( f(x) = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).

So, let's calculate the \( x \)-coordinate of the vertex:
\[
x = -\frac{b}{2a} = -\frac{7}{2 \times 1} = -\frac{7}{2} = -3.5
\]

### 3. Calculate the minimum value:
Substitute \( x = -3.5 \) back into the function \( f(x) \) to find the minimum value:
\[
f(-3.5) = (-3.5)^2 + 7(-3.5) + 15.3
\]
\[
f(-3.5) = 12.25 - 24.5 + 15.3 = 3.05
\]

### 4. Conclusion:
The minimum value of the function \( f(x) = x^2 + 7x + 15.3 \) is **3.05** to the nearest hundredth.

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the vertex of any quadratic function?
2. What does the vertex of a parabola represent in real-world contexts?
3. How can you determine if a quadratic function has a maximum or minimum value?
4. What are the properties of parabolas that open upwards versus downwards?
5. How does the discriminant of a quadratic equation relate to the graph of the function?

**Tip:** The vertex of a quadratic function gives the maximum or minimum value of the function, depending on the direction the parabola opens (upwards for a minimum, downwards for a maximum).

Find the minimum value of the function f(x) = x^2 + 7x + 15.3 to the nearest hundredth.

This problem focuses on determining the minimum value of the quadratic function f(x) = x^2 + 7x + 15.3. Learn how to apply the vertex formula to find the minimum point of a parabola. Suitable for students in Grades 9-10, with a step-by-step explanation provided.

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