Changing the a variable in a quadratic equation of the form y = ax2 + bx + c affects the "width" or "steepness" of the parabola and whether it opens upwards or downwards.
Here's a more detailed breakdown:
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The Sign of a:
- If a > 0 (positive), the parabola opens upwards, forming a U-shape.
- If a < 0 (negative), the parabola opens downwards, forming an inverted U-shape. This can be thought of as a reflection across the x-axis.
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The Magnitude (Absolute Value) of a:
- When |a| > 1 (the absolute value of a is greater than 1), the parabola becomes narrower or "steeper". The larger the absolute value of a, the more compressed the graph appears. Think of it as the parabola being "stretched" vertically.
- When 0 < |a| < 1 (the absolute value of a is between 0 and 1), the parabola becomes wider or "flatter". The closer the absolute value of a is to 0, the more stretched the graph appears horizontally.
| Value of a | Effect on Parabola | Description | Example |
|---|---|---|---|
| a > 0 | Opens Upwards | Parabola has a minimum point. | y = x2, y = 2x2 |
| a < 0 | Opens Downwards | Parabola has a maximum point. | y = -x2, y = -0.5x2 |
| a | > 1 | Narrower/Steeper | |
| 0 < | a | < 1 | Wider/Flatter |
| a = 0 | Not a Parabola | Results in a linear equation. | y = 0x2 + bx + c simplifies to y = bx + c |
In summary, changing the a value in a quadratic equation alters the shape of the parabola, controlling whether it opens upward or downward and how wide or narrow it appears. A larger absolute value of a makes the parabola steeper, while a smaller absolute value (between 0 and 1) makes it wider.
