​Determine, without​ graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value.
f left parenthesis x right parenthesis equals negative 3 x squared minus 12 x
Question content area bottom
Part 1
The quadratic function has a
maximum
value.
Part 2
The value is

negative 12.

To determine whether the quadratic function has a maximum or minimum value, we can examine the coefficient of the \( x^2 \) term. The function given is:

\[
f(x) = -3x^2 - 12x
\]

### Step 1: Check the sign of the quadratic term
- The coefficient of \( x^2 \) is \( -3 \), which is negative. 
- When the coefficient of \( x^2 \) is negative, the parabola opens downward, meaning the function has a **maximum** value.

### Step 2: Find the vertex
The vertex of a quadratic function in the form \( ax^2 + bx + c \) occurs at \( x = \frac{-b}{2a} \), where \( a \) and \( b \) are the coefficients of \( x^2 \) and \( x \), respectively.

For \( f(x) = -3x^2 - 12x \):
- \( a = -3 \)
- \( b = -12 \)

Using the vertex formula:
\[
x = \frac{-(-12)}{2(-3)} = \frac{12}{-6} = -2
\]

### Step 3: Find the value of the function at the vertex
Now, substitute \( x = -2 \) into the function to find the maximum value:
\[
f(-2) = -3(-2)^2 - 12(-2)
\]
\[
f(-2) = -3(4) + 24 = -12 + 24 = 12
\]

### Conclusion:
- The quadratic function has a **maximum** value.
- The maximum value is \( 12 \).

---

Would you like more details or have any further questions? Here are some related questions:

1. How do you find the vertex of a quadratic function in general form?
2. Why does a negative coefficient of \( x^2 \) indicate a maximum?
3. How would the graph of \( f(x) = -3x^2 - 12x \) look like?
4. What happens if the coefficient of \( x^2 \) is positive instead?
5. How can completing the square also help find the maximum or minimum?

**Tip:** The vertex formula \( x = \frac{-b}{2a} \) is key to quickly finding the maximum or minimum of any quadratic function!

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and then find the value. f(x) = -3x^2 - 12x

This problem explains how to determine the maximum value of the quadratic function f(x) = -3x^2 - 12x using the vertex formula. Key concepts covered include quadratic functions, parabolas, and how the sign of the x^2 coefficient determines the maximum or minimum. Ideal for students in Grades 9-11.

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