Quadratic graphs have a special shape that is called a “U,” which we also call a parabola. They are symmetrical because of their math structure.

Let’s look at a simple quadratic equation, like $y = ax^2 + bx + c$.

In this graph, the line of symmetry is always a straight line that goes up and down through a point called the vertex. This means if you could fold the graph in half along this line, both sides would match perfectly.

One cool thing about the vertex is found with the formula $x = -\frac{b}{2a}$. This vertex shows us where the graph is balanced. If you choose numbers on either side of the vertex, their $y$ values will be the same. It’s sort of like looking in a mirror!

Now, what happens when we start transforming the graph? When we move the graph to the left or right, or up and down—like when we write $y = a(x - h)^2 + k$—we are shifting the entire graph. The symmetry is still there, but the line of symmetry moves too!

If we stretch the graph up or squish it down (by changing the value of $a$ in $y = ax^2$), the shape will still be symmetrical around the vertex. However, this “U” can become wider or narrower.

These transformations are interesting because they keep the symmetry while changing the shape. This is what makes quadratic graphs cool and helpful in real life!