Unlike the vertex form of a quadratic (y = a(x – h)² + k) which exposes the vertex of a parabola (h, k), the factored form (y = a(x – r)(x – s)) exposes the roots of the parabola – that is, if they exist. For example, if we have a factored form quadratic that looks like this:
y = 2(x – 3)(x + 2)
The x, y coordinates of the roots will (+3, y) and (–2, y). You can easily find the y coordinates by substituting the x’s back into the formula.
- Notice how –3 was written as +3, and +2 was written as a negative. This feature will be explained when we learn how to solve quadratics later on.
Once you find the x-intercepts, finding the average of these two numbers will give you the axis of symmetry, or simply the x-coordinate of the vertex. Here’s how to find the average: add the two numbers and divide the sum by 2, always.
+3 + (–2) =
+3 – 2 = +1
1 divided by 2 = 0.5
Therefore, the vertex would be (0.5, y). You can find y by substituting 0.5 into the original equation. All of these points can now be graphed on an x-y plane, connected together in a parabolic curve, thereby serving as your rough sketch of the equation. Take a look at the actual graph:
Let’s try another one, this time our equation is y = 2(x + 1)(x – 7). Before you watch the video underneath, see if you can write the coordinates of the roots and vertex, and the axis of symmetry.
Of course, if your parabola passes through the x-axis, you can derive an equation using some key features, namely the roots. Here’s a quick demonstration on how it’s done.