Graphing is a great way to understand how quadratic equations connect to their parabolas.

Important Parts of Quadratic Equations:

1. Standard Form: A quadratic equation looks like this: $y = ax^2 + bx + c$. Here’s what each part means:
   • $a$ decides if the parabola opens up or down. If $a$ is more than 0, it opens up. If $a$ is less than 0, it opens down.
   • $b$ changes where the highest or lowest point (called the vertex) is along the x-axis.
   • $c$ shows where the parabola touches the y-axis.

How They Relate to Parabolas:

• When you graph a quadratic equation, you get a shape called a parabola. This graph has some important features:
  • Vertex: This is the highest or lowest point of the parabola. You can find it using the formula $x = -\frac{b}{2a}$.
  • Axis of Symmetry: This is a vertical line at $x = -\frac{b}{2a}$ that divides the parabola into two equal halves.
  • Roots/Zeros: These are the points where the parabola meets the x-axis. You can find them by solving the equation $ax^2 + bx + c = 0$. You can use different methods like factoring, completing the square, or the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Fun Fact:

• About 30% of quadratic equations have real roots, which means they touch the x-axis. The other 70% might not have real roots, meaning they don’t touch the x-axis at all.

Using graphing calculators or special computer programs can help you see these relationships clearly. They let you watch how the graph changes when the numbers change.