https://file.aiquickdraw.com/mathbot/file/1726799560809x31pw1bi.jpg

Use the graph to determine
a. open intervals on which the function is​ increasing, if any.
b. open intervals on which the function is​ decreasing, if any.
c. open intervals on which the function is​ constant, if any.

Let's analyze the graph step by step.

### a. **Open intervals where the function is increasing:**

The graph shows a parabola opening upwards. A function is increasing where the slope of the curve is positive (the graph rises as you move from left to right). From the graph:
- The function is increasing for \( x > 0 \), so the open interval where it is increasing is:
  \[
  (0, \infty)
  \]

### b. **Open intervals where the function is decreasing:**

A function is decreasing where the slope of the curve is negative (the graph falls as you move from left to right). From the graph:
- The function is decreasing for \( x < 0 \), so the open interval where it is decreasing is:
  \[
  (-\infty, 0)
  \]

### c. **Open intervals where the function is constant:**

A function is constant when the graph is a horizontal line, meaning the slope is zero. There are no horizontal sections in the graph, so:
- There are no open intervals where the function is constant.

Would you like more details on this or further clarifications?

Here are 5 follow-up questions to think about:
1. How does the vertex of the parabola relate to its increasing and decreasing behavior?
2. What would change in the intervals if the parabola opened downwards instead of upwards?
3. How would you find the exact equation of this parabola based on its graph?
4. How can you use the derivative of the function to confirm the increasing/decreasing intervals?
5. Could this function have any absolute minimum or maximum, and how can you identify them from the graph?

**Tip:** The vertex of a parabola plays a critical role in determining where the function increases or decreases.

Use the graph to determine a. open intervals on which the function is increasing, if any. b. open intervals on which the function is decreasing, if any. c. open intervals on which the function is constant, if any.

Analyze a quadratic function's graph to determine the open intervals where the function is increasing, decreasing, or constant. This problem focuses on interval notation and understanding the behavior of parabolas. Suitable for students in Grades 9-11.

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