Quadratic functions are special types of math expressions written like this: (f(x) = ax^2 + bx + c). They have some interesting traits that shape their graphs.

1. Parabolic Shape: The graph looks like a U or an upside-down U. If (a) (the number before (x^2)) is greater than zero, it opens upward. If (a) is less than zero, it opens downward.

2. Vertex: The vertex is the highest or lowest point of the U shape. You can find it using this formula: (x = -\frac{b}{2a}). This point is really important because it gives us the maximum or minimum value of the function.

3. Intercepts: Quadratic functions can have different numbers of x-intercepts (the points where the graph crosses the x-axis). This is decided by something called the discriminant, written as (D = b^2 - 4ac). Here’s how it works:

   • If (D > 0), there are two points where the graph crosses the x-axis.
   • If (D = 0), there is just one point where it touches the x-axis.
   • If (D < 0), the graph never touches the x-axis.

4. End Behavior: As (x) gets really big or really small (like going to positive or negative infinity), the graph behaves in specific ways. If (a > 0), the function goes up towards infinity. If (a < 0), it goes down towards negative infinity.

These features help us understand how quadratic functions work and why they are important in math.