Vertex Form and Graphing

The Vertex Form of a quadratic equation is written as $y = a(x - h)^2 + k$. This form is a 'cheat code' for graphing because it tells you exactly where the 'turning point' or vertex of the parabola is located: $(h, k)$. Note the minus sign before the $h$—this means the horizontal shift is often the opposite of what you see! If $a$ is positive, the parabola opens upward like a smile; if $a$ is negative, it opens downward like a frown. The value of $a$ also determines if the graph is narrow (stretched) or wide (compressed).

To move from Standard Form ($y = ax^2 + bx + c$) to Vertex Form, we use Completing the Square. This involves creating a perfect square trinomial within the equation. We look at the $x$ terms, take half of the coefficient of $x$, square it, and then add and subtract that value. This keeps the equation balanced while allowing us to factor part of it into the $(x - h)^2$ format. This transformation reveals the hidden horizontal and vertical shifts of the parent function $y = x^2$.

Graphing is the final step. Once you have the vertex $(h, k)$, plot it first. Then, use the Axis of Symmetry, which is the vertical line $x = h$. Because parabolas are perfectly symmetrical, any point you find on one side can be 'mirrored' to the other. Pick a simple $x$-value near the vertex (like $h+1$), solve for $y$, plot it, and then reflect it across the axis of symmetry to get your third point. This 'mirror' technique ensures your graph is perfectly balanced.