Understanding Quadratic Equations

Quadratic equations are written in this format:

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

Here, $a$, $b$, and $c$ are constants, meaning they are fixed values, and $a$ cannot be zero.

When you graph a quadratic equation, it forms a U-shaped curve called a parabola. This curve has some important parts:

1. Vertex: This is the highest or lowest point of the parabola. You can find the vertex using these coordinates:

   $$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$

   Depending on whether $a$ is positive or negative, this point can be the peak (maximum) or the bottom (minimum) of the curve.

2. Axis of Symmetry: This is an imaginary vertical line that splits the parabola into two equal halves. You can find it using:

   $$x = -\frac{b}{2a}$$

3. Roots: The roots are where the graph crosses the x-axis. They can also be called solutions or x-intercepts. You can find them using this formula:

   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

   The type of roots you get depends on a value called the discriminant ($D = b^2 - 4ac$):

   • If $D > 0$, there are two different real roots.
   • If $D = 0$, there is one real root.
   • If $D < 0$, there are no real roots.

4. Transformations: Quadratic functions can change positions or flip while still keeping their parabolic shape. For example, the equation $f(x) = a(x-h)^2 + k$ shows a shift by $(h, k)$.

Understanding how quadratic equations work and how they look on a graph is really important. They appear in many fields like science, engineering, and economics.