A negative number in front of the x² in a quadratic function really changes how the graph looks and which way it points. A quadratic function usually looks like this:

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

Here, $a$, $b$, and $c$ are just numbers that help shape the function. The number $a$ is especially important because it tells us which direction the curve (called a parabola) will go:

1. Direction:

   • If $a$ is greater than 0 (like 1, 2, or 3), the parabola opens upwards, like a smile.
   • If $a$ is less than 0 (like -1, -2, or -3), the parabola opens downwards, like a frown.

   For example, if $a$ is -2, then the function $f(x) = -2x^2 + 3x + 1$ will create a frown-shaped graph.

2. Vertex:

   • The top point of the frown (also called the vertex) is the highest point when $a$ is negative. We can find this highest point using the formula $x = -\frac{b}{2a}$. When $a$ is negative, this point is a maximum, meaning it's the biggest value.

3. Axis of Symmetry:

   • The graph will be symmetrical, which means it looks the same on both sides, around the line $x = -\frac{b}{2a}$. This is true whether $a$ is negative or positive.

4. Y-Intercept:

   • The graph will cross the vertical y-axis at the point where $f(0) = c$, no matter if $a$ is negative or positive.

In short, when the coefficient ($a$) is negative, the parabola turns upside down, has a maximum point at the top, and still looks the same on both sides. This really changes how the graph looks compared to one with a positive coefficient!