https://file.aiquickdraw.com/mathbot/file/17290495481051u0d6zhb.jpg

Let's analyze the function \( f(x) = 3x^2 - 24x - 4 \), as requested in the image.

### Part a: Determine if the function has a minimum or maximum value.
This is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \( a = 3 \), \( b = -24 \), and \( c = -4 \).

For a quadratic function:
- If \( a > 0 \), the parabola opens upwards and the function has a **minimum** value.
- If \( a < 0 \), the parabola opens downwards and the function has a **maximum** value.

Since \( a = 3 > 0 \), the function has a **minimum** value.

### Part b: Find the minimum value and determine where it occurs.
To find the minimum value, we first need to find the vertex of the parabola, which occurs at \( x = \frac{-b}{2a} \).

\[
x = \frac{-(-24)}{2(3)} = \frac{24}{6} = 4
\]

Now, substitute \( x = 4 \) back into the function to find the minimum value:

\[
f(4) = 3(4)^2 - 24(4) - 4 = 3(16) - 96 - 4 = 48 - 96 - 4 = -52
\]

Thus, the minimum value is \( -52 \), and it occurs at \( x = 4 \).

### Part c: Identify the function's domain and range.
- The **domain** of a quadratic function is all real numbers, i.e., \( (-\infty, \infty) \).
- The **range** depends on the vertex and the direction of the parabola:
  - Since the parabola opens upwards (because \( a > 0 \)), the range starts from the minimum value \( -52 \) and goes to infinity.
  - Therefore, the **range** is \( [-52, \infty) \).

### Summary:
- a. The function has a **minimum** value.
- b. The minimum value is \( -52 \) at \( x = 4 \).
- c. Domain: \( (-\infty, \infty) \), Range: \( [-52, \infty) \).

Would you like further details or explanations? 

Here are 5 related questions you might find useful:
1. How do you determine the vertex of any quadratic function?
2. What happens when the leading coefficient \( a \) is negative in a quadratic function?
3. How do you find the axis of symmetry for a quadratic equation?
4. Can quadratic functions have both a minimum and maximum value?
5. How do transformations affect the graph of a quadratic function?

**Tip**: Remember, the sign of the leading coefficient \( a \) determines whether a quadratic function has a minimum or maximum!

Consider the function f(x) = 3x^2 - 24x - 4. Determine, without graphing, whether the function has a minimum value or a maximum value. Find the minimum or maximum value and determine where it occurs. Identify the function's domain and its range.

Learn how to analyze the quadratic function f(x) = 3x^2 - 24x - 4. Determine whether it has a minimum or maximum value, find the vertex, and identify the function's domain and range. Suitable for high school students studying quadratic functions in Grades 9-12.

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