Given the criteria below create a graph/ function

-bounded above
-domain (-infinity, 7]
-Range (-infinity, 9]
-constant from [-3,1]
-relative maximum at -5
-absolute maximum at 3
-increases at least twice
-decreases at least once

To create a function that meets all the given criteria, let's break down each requirement and design a function step by step.

### Criteria Breakdown:

1. **Bounded above**: The function should have a highest value, meaning it doesn't go to infinity as \( x \) increases. This suggests a horizontal asymptote or a maximum point.

2. **Domain (-∞, 7]**: The function is defined from negative infinity up to and including 7.

3. **Range (-∞, 9]**: The output values (y-values) should go up to 9 but not exceed it. This implies the absolute maximum y-value is 9.

4. **Constant from [-3,1]**: The function remains flat (constant) on this interval.

5. **Relative maximum at -5**: There is a peak at \( x = -5 \), but it's not the highest point.

6. **Absolute maximum at 3**: The highest point of the function occurs at \( x = 3 \) with \( y = 9 \).

7. **Increases at least twice**: The function should have two distinct intervals where it rises.

8. **Decreases at least once**: The function should have at least one interval where it falls.

### Proposed Function:
Let’s consider a piecewise function that fits these criteria.

\[
f(x) =
\begin{cases}
-x^2 + 4x + 1 & \text{for } x \leq -5 \text{ (decreasing)} \\
9 & \text{for } x = 3 \text{ (absolute maximum)} \\
5 & \text{for } x \in [-3, 1] \text{ (constant interval)} \\
\text{Linear function } ax+b & \text{for } -5 < x < -3 \text{ (increasing interval)} \\
\text{Linear function } cx+d & \text{for } 1 < x \leq 7 \text{ (increasing interval)} \\
\end{cases}
\]

### Function Characteristics:
- **Bounded above**: By defining an absolute maximum at \( x = 3 \), with \( f(3) = 9 \), the function is bounded above.
- **Domain**: Defined for all \( x \) in the interval \( (-\infty, 7] \).
- **Range**: The function takes values up to 9.
- **Constant on [-3, 1]**: The function is constant at \( y = 5 \) over this interval.
- **Relative maximum at -5**: The quadratic segment of the function has a peak at \( x = -5 \).
- **Absolute maximum at 3**: The function reaches its highest point at \( x = 3 \), with \( y = 9 \).
- **Increases twice**: The function increases on \( (-\infty, -5) \) and from \( (1, 7] \).
- **Decreases once**: The function decreases on \( (-5, -3] \).

### Example Graph:
I'll plot this function next.

Learn how to create a function that meets specific criteria including being bounded above, having defined domains and ranges, constants intervals, relative and absolute maxima, and specific increasing and decreasing behaviors. Suitable for high school students learning about advanced functions.

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