https://file.aiquickdraw.com/mathbot/file/1730265150803en6ps0yt.jpg

Let's analyze the statements based on the graph provided:

1. **Statement (a): The domain is \((-∞, 4]\)**  
   - The graph shown appears to be a parabola, opening downwards, which suggests a domain that covers all \(x\)-values from \(-∞\) to \(+∞\). Hence, statement (a) is **incorrect**.

2. **Statement (b): The range is \((-∞, 4]\)**  
   - The maximum \(y\)-value of the graph is 4, and it extends downward to \(-∞\). This makes the range \((-∞, 4]\). Thus, statement (b) is **correct**.

3. **Statement (c): The graph is positive on the interval \((0, ∞)\)**  
   - The graph is positive (above the \(x\)-axis) only from \(x = -2\) to \(x = 2\). Since this does not match the interval \((0, ∞)\), statement (c) is **incorrect**.

4. **Statement (d): The graph is decreasing on the interval \((-1, ∞)\)**  
   - The graph begins decreasing from the vertex at \(x = 0\) to \(\infty\). However, it is not decreasing on \((-1, ∞)\) because it increases from \(-∞\) to \(x = 0\). Therefore, statement (d) is **incorrect**.

### Summary:
Only **statement (b)** is correct.

Would you like a more detailed breakdown of each step or have any specific questions?

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Here are some follow-up questions to deepen your understanding:

1. What does it mean for a function to be positive on an interval?
2. How can we determine the range of a function by looking at its graph?
3. Why is the domain of a parabola generally all real numbers?
4. How can we identify intervals where a graph is increasing or decreasing?
5. What would change in the answer if the parabola opened upwards instead of downwards?

**Tip:** When analyzing the domain and range of a function from its graph, remember the domain covers all possible \(x\)-values, while the range covers all possible \(y\)-values.

Select all that apply for the given parabola graph: 

a. The domain is (-∞, 4].
b. The range is (-∞, 4].
c. The graph is positive on the interval (0, ∞).
d. The graph is decreasing on the interval (-1, ∞).

Learn how to analyze the domain, range, and intervals of increase and decrease in a quadratic function's graph. This example helps in understanding key concepts in quadratic functions, including interpreting graph intervals and identifying function behavior.

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