What is the minimum value of f(x)=3x²+6x+7

To find the minimum value of the quadratic function \( f(x) = 3x^2 + 6x + 7 \), we follow these steps:

### Step 1: Identify the standard form of a quadratic function
The given function is already in the standard form:
\[
f(x) = ax^2 + bx + c
\]
where \( a = 3 \), \( b = 6 \), and \( c = 7 \).

### Step 2: Find the vertex of the parabola
The minimum (or maximum) value of a quadratic function occurs at the vertex. The x-coordinate of the vertex can be found using the formula:
\[
x = \frac{-b}{2a}
\]

Substitute \( b = 6 \) and \( a = 3 \) into the formula:
\[
x = \frac{-6}{2(3)} = \frac{-6}{6} = -1
\]

### Step 3: Calculate the minimum value
Now that we know \( x = -1 \), we can substitute this value back into the original function to find the minimum value of \( f(x) \):
\[
f(-1) = 3(-1)^2 + 6(-1) + 7
\]
\[
f(-1) = 3(1) - 6 + 7
\]
\[
f(-1) = 3 - 6 + 7 = 4
\]

### Conclusion:
The minimum value of the function \( f(x) = 3x^2 + 6x + 7 \) is **4**, and it occurs at \( x = -1 \).

Would you like more details or have any other questions?

Here are 5 questions for further practice:
1. How would you find the maximum value of a quadratic function?
2. Can the quadratic function \( f(x) = 3x^2 + 6x + 7 \) have a maximum value?
3. What happens to the graph of \( f(x) \) when the value of \( a \) is negative?
4. How would you solve \( f(x) = 0 \) for this function?
5. What is the axis of symmetry of \( f(x) = 3x^2 + 6x + 7 \)?

**Tip:** The sign of the coefficient \( a \) determines whether a quadratic function opens upwards (minimum point) or downwards (maximum point).

What is the minimum value of f(x)=3x²+6x+7?

Learn how to find the minimum value of the quadratic function f(x) = 3x² + 6x + 7 with a step-by-step solution. This problem demonstrates key concepts in algebra such as finding the vertex of a parabola and using the vertex formula. Suitable for students in Grades 9-11.

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