The shape of a parabola in quadratic equations is mostly explained by how we write the equation and some important parts of it. A quadratic equation usually looks like this:

$$y = ax^2 + bx + c$$

In this equation, (a), (b), and (c) are numbers. The number (a) cannot be zero. The value of (a) is very important because it tells us which way the parabola opens.

Important Parts of Parabolas

1. Direction of Opening:

   • Upwards: If (a > 0), the parabola opens upwards. This means the vertex, or the lowest point of the parabola, is the smallest point on the graph.
   • Downwards: If (a < 0), the parabola opens downwards. This makes the vertex the highest point on the graph.

2. Vertex:

   • To find the vertex, we can use this formula for the x-coordinate:
   $$x = -\frac{b}{2a}$$

   To get the y-coordinate of the vertex, we put the x-coordinate back into the original equation:

   $$y = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c$$

3. Axis of Symmetry:

   • A parabola is symmetrical, meaning it looks the same on both sides. The line that divides it in half is called the axis of symmetry. This can be described with the equation:
   $$x = -\frac{b}{2a}$$

4. Y-intercept:

   • The y-intercept is where the parabola crosses the y-axis. This happens when (x = 0):
   $$y = c$$

   So the point ((0, c)) shows where the parabola touches the y-axis.

5. X-intercepts (Roots):

   • The points where the parabola crosses the x-axis can be found with the quadratic formula:
   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

   The expression (b^2 - 4ac) helps us understand the roots:

   • If (b^2 - 4ac > 0), there are two different real roots.
   • If (b^2 - 4ac = 0), there is one real root (the parabola just touches the x-axis).
   • If (b^2 - 4ac < 0), there are no real roots (the parabola does not cross the x-axis).

Summary

In short, a parabola in quadratic equations is shaped by the number (a), which tells us which way it opens. Key points like the vertex, axis of symmetry, y-intercept, and x-intercepts help us understand the graph better. By looking at the values of (a), (b), and (c), we can draw an accurate graph of any quadratic equation.