https://file.aiquickdraw.com/mathbot/file/1726630616235t0v5927f.jpg

Let's analyze the function \( f(x) \) based on the graph provided and fill in the blanks:

1. **Domain** (Using interval notation):
   - The domain refers to all possible x-values where the function exists. The graph shows that the function starts at \( x = -3 \) and extends up to \( x = 6 \). Therefore, the domain is:
   \[
   [-3, 6]
   \]

2. **Range** (Using interval notation):
   - The range refers to all possible y-values that the function takes. From the graph, the lowest value is \( y = -3 \) and the highest is \( y = 3 \). Thus, the range is:
   \[
   [-3, 3]
   \]

3. **\( f(-3) \)**:
   - Looking at the graph, the value of \( f(-3) \) corresponds to the y-value when \( x = -3 \), which is \( f(-3) = 1 \).

4. **Solve \( f(x) = 1 \)**:
   - The points where \( f(x) = 1 \) are where the graph intersects the line \( y = 1 \). These intersections occur at \( x = -3 \), \( x = -1 \), and \( x = 5 \). Therefore:
   \[
   x = -3, -1, 5
   \]

5. **Y-intercept** (Coordinates):
   - The y-intercept occurs where the graph crosses the y-axis. This happens when \( x = 0 \). From the graph, \( f(0) = -3 \), so the y-intercept is:
   \[
   (0, -3)
   \]

6. **X-intercepts** (List as decimal coordinates):
   - The x-intercepts occur where the graph crosses the x-axis. This happens at \( x = -2 \) and \( x = 4 \), so the x-intercepts are:
   \[
   (-2, 0), (4, 0)
   \]

7. **Zeros** (List of decimal numbers):
   - Zeros of the function are the x-values where \( f(x) = 0 \). These are the same as the x-intercepts, so:
   \[
   x = -2, 4
   \]

8. **Behavior over intervals**:
   - Over the interval \( [-3, 0] \), the function is **decreasing**.
   - Over the interval \( [0, 1] \), the function is **increasing**.
   - Over the interval \( [1, 4] \), the function is **decreasing**.
   - Over the interval \( [4, 6] \), the function is **increasing**.

9. **Minimum value**:
   - The minimum value of the function is the lowest point on the graph, which occurs at \( y = -3 \).

10. **Maximum value**:
    - The maximum value of the function is the highest point on the graph, which occurs at \( y = 3 \).

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Would you like to go over any specific details? Here are five related questions to explore this further:

1. How do you determine the intervals where a function is increasing or decreasing?
2. What is the difference between x-intercepts and zeros of a function?
3. How does the domain of a function relate to the behavior of the graph?
4. Can a function have multiple y-intercepts? Why or why not?
5. How do you solve for a specific function value graphically, like \( f(x) = 1 \)?

**Tip:** When analyzing a graph, always pay attention to where the curve crosses the axes, as these points often give key information about intercepts and zeros.

The function f(x) is graphed below. Using interval notation, the domain is: _____. Using interval notation, the range is: _____. Determine f(-3) = _____. Solve f(x) = 1 (enter as list of decimal numbers): _____. The y-intercept is at coordinates: _____. The x-intercepts are (enter as list of decimal coordinates): _____. The zeros are (enter as list of decimal numbers): _____. Over the interval [-3, 0], the function is _____. Over the interval [0, 1], the function is _____. Over the interval [1, 4], the function is _____. Over the interval [4, 6], the function is _____. The minimum value is: _____. The maximum value is: _____.

Explore the detailed analysis of a given function's graph to find its domain, range, intercepts, zeros, and intervals of increase and decrease. Understand how to interpret key characteristics from the graph of the function.

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