https://file.aiquickdraw.com/mathbot/file/1727062874698qb6djeyb.jpg

The function given in the image is \( f(x) = -3x^2 + 6x - 8 \).

### Part (a)
To determine if the function has a **minimum** or **maximum** value, we look at the coefficient of \( x^2 \) in the quadratic function \( f(x) = ax^2 + bx + c \).
- If \( a > 0 \), the parabola opens upward, and the function has a minimum.
- If \( a < 0 \), the parabola opens downward, and the function has a maximum.

In this case, \( a = -3 \), which is less than 0, so the function has a **maximum** value.

### Part (b)
To find the maximum value and where it occurs, we use the vertex formula:
\[
x = \frac{-b}{2a}
\]
Here, \( a = -3 \) and \( b = 6 \). Substituting these values into the formula:
\[
x = \frac{-6}{2(-3)} = 1
\]

Now substitute \( x = 1 \) back into the function \( f(x) \) to find the maximum value:
\[
f(1) = -3(1)^2 + 6(1) - 8 = -3 + 6 - 8 = -5
\]
So, the maximum value is \( -5 \), and it occurs at \( x = 1 \).

### Part (c)
- The **domain** of any quadratic function is all real numbers: \( (-\infty, \infty) \).
- The **range** of the function depends on the maximum value. Since the maximum value is \( -5 \), the range is \( (-\infty, -5] \).

### Summary:
- **(a)** The function has a **maximum** value.
- **(b)** The maximum value is \( -5 \), and it occurs at \( x = 1 \).
- **(c)** The domain is \( (-\infty, \infty) \), and the range is \( (-\infty, -5] \).

Would you like further details on any part of the solution?

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Here are 5 related questions:
1. How do you determine the vertex of a quadratic function?
2. What does the sign of the leading coefficient in a quadratic equation tell you about its graph?
3. Can a quadratic function have both a minimum and a maximum value?
4. How do you find the axis of symmetry for a quadratic function?
5. What is the general form of a quadratic equation?

**Tip:** Always check the sign of the \( x^2 \) coefficient to know if a quadratic function has a minimum or maximum.

Consider the function f(x) = -3x^2 + 6x - 8.
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 + 6x - 8. This step-by-step solution shows how to determine if the function has a maximum or minimum, calculate the maximum value, and find the domain and range. Ideal for students in Grades 9-10.

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