https://file.aiquickdraw.com/mathbot/file/1727868522276f7yo7fc2.jpg

Let's analyze the given task based on the visible image:

### Part a:
- **Range:** \( y \leq 5 \)
- **Axis of symmetry:** \( x = -2 \)
- **y-intercept:** \( (0, 1) \)

This describes a downward-opening parabola (since \( y \leq 5 \)) with the vertex at \( (-2, 5) \). The parabola passes through \( (0, 1) \), which is the y-intercept. The axis of symmetry is the vertical line \( x = -2 \).

### Part b:
- **Range:** \( y \geq -4 \)
- **x-intercepts:** The points \( (-2, 0) \) and \( (6, 0) \)
- **x-coordinate of the vertex:** \( x = 2 \)

This describes an upward-opening parabola (since \( y \geq -4 \)), and the vertex will have the form \( (2, -4) \). It passes through the x-intercepts \( (-2, 0) \) and \( (6, 0) \).

### Key Information for Graphing:
- For **a**: The parabola has its vertex at \( (-2, 5) \), opens downward, and passes through \( (0, 1) \) as one additional point. You could find another point by reflecting \( (0, 1) \) over the axis of symmetry or calculating based on the equation.
- For **b**: The parabola has its vertex at \( (2, -4) \), opens upward, and passes through the x-intercepts \( (-2, 0) \) and \( (6, 0) \).

Do you want me to provide the equations for these parabolas? Let me know if you need more details!

### Questions:
1. How do you determine the vertex of a parabola given its axis of symmetry and range?
2. How would you find additional points for the graph of a parabola based on its characteristics?
3. Why is the range of the parabola important for determining its direction of opening?
4. How do x-intercepts help in graphing the parabola?
5. What is the significance of the axis of symmetry in a parabola?

### Tip:
When graphing a parabola, the vertex and intercepts provide crucial reference points. Always ensure to check symmetry across the axis.

Sketch the graph of a parabola for each set of characteristics given below. Show at least two other points on each graph.

Learn how to sketch the graph of a parabola using characteristics like the axis of symmetry, intercepts, and range. This math problem covers key concepts in graphing quadratic functions, suitable for students in Grades 9-11.

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