https://file.aiquickdraw.com/mathbot/file/1730240550042uedg8pf2.jpg

To determine the correct matches between each graph and its corresponding \( a \)-value, let's analyze the effect of the \( a \)-value in the quadratic equation \( y = ax^2 + bx + c \):

1. **Sign of \( a \):**
   - If \( a > 0 \), the parabola opens upwards.
   - If \( a < 0 \), the parabola opens downwards.

2. **Magnitude of \( a \):**
   - The larger the absolute value of \( a \), the "narrower" the parabola.
   - The smaller the absolute value of \( a \), the "wider" the parabola.

### Analyzing the Graphs:

- **Left Graph (Parabola opening downward and narrow)**:
  This indicates a large negative \( a \)-value, which matches \( a = -4 \).

- **Middle Graph (Parabola opening downward and wide)**:
  This represents a smaller negative \( a \)-value, which matches \( a = -0.25 \).

- **Right Graph (Parabola opening upward and narrow)**:
  This indicates a large positive \( a \)-value, which matches \( a = 4 \).

### Solution:
- **Left Graph:** \( a = -4 \)
- **Middle Graph:** \( a = -0.25 \)
- **Right Graph:** \( a = 4 \)

Would you like further details or have any questions?

### Related Questions:
1. How does the \( b \)-value in \( y = ax^2 + bx + c \) affect the graph's symmetry?
2. What happens to the vertex of a parabola when \( a \) is changed while \( b \) and \( c \) are constant?
3. How can the vertex form of a parabola \( y = a(x - h)^2 + k \) help in graphing?
4. Why does a smaller \( |a| \) result in a wider parabola?
5. How can you determine the axis of symmetry from the equation \( y = ax^2 + bx + c \)?

### Tip:
When matching graphs to equations, always start by analyzing the direction of opening and the width of the parabola based on the sign and magnitude of \( a \).

Assuming all parabolas are of the form y = ax^2 + bx + c, drag and drop the graphs to match the appropriate a-value (if necessary). Given values are a = -4, a = -0.25, and a = 4.

Understand how to match graphs of parabolas with their corresponding 'a' values in quadratic functions of the form y = ax^2 + bx + c. Learn the effects of positive and negative 'a' values, and how the magnitude of 'a' affects the width of the parabola. Suitable for high school students.

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