The vertex form is a transformation of the parent function y=x^2.
f(x) = a(x-h)^2 +k
In this form, the vertex is located at (h,k). Moreover, a is a parameter that can stretch or compress the parabola vertically, or perform a reflection across the x-axis. The location of the vertex can be seen explicitly. Let's see an example.
f(x) = (x-3)^2+2 ⇔ f(x) = 1(x-3)^2+ 2
We can see that a=1, h=3 and k=2. Therefore, its shape should be the same parabola as the parent function, translated in such way that its vertex is at (3,2).
As we can see, finding the vertex is direct when working with the vertex form.
Finding the Vertex Using the Standard Form
The standard form of a quadratic function is shown below.
f(x) = ax^2+bx+c
In this form a, b, and c are real numbers, and a≠0. In contrast to the vertex form, here the information of the vertex location is hidden. Instead of appearing explicitly in the equation, we have calculate the x- and y-coordinates of the vertex. We can do this using the vertex formulas shown below.
h=-b/2a k=f(-b/2a)
Notice that we are using the previous notation for the vertex (h,k). Let's consider an example.
f(x) = x^2-6x +11
⇕
f(x) = 1x^2+( - 6)x +11
Notice that, at first sight, it is hard to say where the vertex will be. Let's start by calculating its x-coordinate.
The vertex will be located at (3,2). This is the same parabola as before! This time finding the vertex took more effort, because the equation was given in standard form.
