A parabola has a step pattern of 4, 12, 20, ... and is reflected over the x-axis. What is the 'a' value of the parabola?

To determine the 'a' value of the parabola, let's analyze the given step pattern and the reflection over the x-axis.

### Step Pattern Analysis

1. **Step Pattern Sequence**: The step pattern given is 4, 12, 20, ...

   These values represent the vertical distances between consecutive points on the parabola as you move horizontally from the vertex. 

2. **Quadratic Growth**: Notice that the step pattern increases quadratically:
   - The difference between 12 and 4 is 8.
   - The difference between 20 and 12 is also 8.
   
   This consistent second difference (which is constant) suggests a quadratic relationship.

### Quadratic Relationship

For a parabola \( y = ax^2 \):
- The step pattern of the points represents the vertical change \( \Delta y \) as you move horizontally.

Given the sequence 4, 12, 20, ..., let's break this down:
- The first difference in the sequence (4, 12, 20) is 8.
- The second difference (8, 8) is constant.

This implies the sequence follows a quadratic relationship: \( \Delta y = 4n + k \), where \( n \) is the step number (starting from 0), and \( k \) is a constant.

For a standard parabola \( y = ax^2 \):
- At \( x = 1 \), \( y = a(1)^2 = a \).
- At \( x = 2 \), \( y = a(2)^2 = 4a \).
- At \( x = 3 \), \( y = a(3)^2 = 9a \).

The step pattern is:
- Between \( x = 0 \) and \( x = 1 \): \( a \)
- Between \( x = 1 \) and \( x = 2 \): \( 4a - a = 3a \)
- Between \( x = 2 \) and \( x = 3 \): \( 9a - 4a = 5a \)

Given the step pattern starts at 4, then 12, then 20:
- First step: \( 4a = 4 \Rightarrow a = 1 \)

However, the parabola is reflected over the x-axis, thus \( a \) becomes negative:
\[ a = -1 \]

### Conclusion

The 'a' value of the parabola is \(-1\).

Would you like more details or have any questions? Here are 5 questions you might ask next:

1. How do step patterns work in more complex parabolas?
2. Can you explain how the second difference indicates a quadratic relationship?
3. What happens if the parabola is shifted horizontally or vertically?
4. How do we derive the quadratic equation from a given set of points?
5. What are some applications of parabolas in real-world scenarios?

**Tip:** Always check the symmetry and reflection properties of the graph when dealing with parabolas and their transformations.

Explore how to determine the 'a' value of a parabola reflected over the x-axis with a step pattern sequence of 4, 12, 20. Understand the quadratic relationship between step patterns and parabolas. Suitable for advanced high school students.

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