Changes in the numbers used in a quadratic function can really change how wide or narrow its parabola looks. A quadratic function is usually written like this:

$$f(x) = ax^2 + bx + c.$$

Important Points:

1. The 'a' Value:

   • The number 'a' affects how steep or flat the graph is.
   • If the absolute value of 'a' (written as $|a|$) is more than 1, the parabola is narrower. For example, if $a = 2$, the parabola is twice as steep.
   • If the absolute value of 'a' is between 0 and 1 (like $0 < |a| < 1$), the parabola gets wider. For example, if $a = 0.5$, the parabola is half as steep.

2. How Width Changes:

   • A wider parabola means that the $y$ values change more slowly when $x$ changes. This makes the graph look "flatter."
   • On the other hand, a narrower parabola shows that the $y$ values go up or down quickly, giving it a sharper curve.

3. Example Comparisons:

   • For the function $f(x) = 2x^2$, the parabola is narrower.
   • For the function $f(x) = 0.5x^2$, the parabola is wider.

In short, changing the number 'a' doesn't just make the parabola steeper or flatter; it also changes where it meets the x-axis. This all changes how the graph looks in a big way.