https://file.aiquickdraw.com/mathbot/file/17286908940202sowgj08.jpg

Let's analyze the graph of the function to answer the questions.

1. **The function has a maximum/minimum at \( x = \)**:
   - The graph shows a parabola opening downward, indicating it has a **maximum** point at the vertex.
   - The vertex occurs at \( x = -6 \) with a value of \( y = 4 \).
   - So, the function has a **maximum** at \( x = -6 \).

2. **The function is increasing on the interval(s):**
   - A function increases when moving from left to right if the slope is positive.
   - The graph increases on the left side of the vertex, i.e., from \( x = -\infty \) to \( x = -6 \).
   - The interval where the function is increasing is \( (-\infty, -6] \).

3. **The function is decreasing on the interval(s):**
   - A function decreases when moving from left to right if the slope is negative.
   - The graph decreases on the right side of the vertex, i.e., from \( x = -6 \) to \( x = \infty \).
   - The interval where the function is decreasing is \( [-6, \infty) \).

4. **The domain of the function is:**
   - The domain of a quadratic function is all real numbers since there are no restrictions on the \( x \)-values.
   - The domain is \( (-\infty, \infty) \).

5. **The range of the function is:**
   - The range is the set of all possible \( y \)-values.
   - Since the maximum value is \( y = 4 \) and the parabola opens downward, the range is \( (-\infty, 4] \).

### Summary of Answers:
- **Maximum** at \( x = -6 \)
- **Increasing** on \( (-\infty, -6] \)
- **Decreasing** on \( [-6, \infty) \)
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (-\infty, 4] \)

Would you like further clarification or additional details on any of these? Here are five related questions you could explore:

1. How can you find the vertex of a quadratic function algebraically?
2. Why does the parabola open downward in this case?
3. How would the domain change for a different type of function (e.g., a square root function)?
4. How can you determine the intervals of increase or decrease using calculus (derivatives)?
5. What real-world scenarios could this quadratic model represent?

**Tip**: Remember that the vertex form of a quadratic function, \( f(x) = a(x-h)^2 + k \), directly gives the vertex as \( (h, k) \).

Consider the function in the graph to the right. The function has a maximum or minimum at x = ? The function is increasing on the interval(s): ? The function is decreasing on the interval(s): ? The domain of the function is: ? The range of the function is: ?

Learn how to analyze a quadratic function graph to determine the vertex, intervals of increase and decrease, domain, and range. This problem focuses on understanding the vertex of a parabola, identifying intervals where the function is increasing or decreasing, and finding the domain and range. Suitable for high school students in Grades 10-12.

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