http://jwilson.coe.uga.edu/emt668/EMAT6680.Folders/Barron/unit/Lesson%206/6.html
The brief version follows:
Throughout this explanation, I will use x^2+18x+45 as an example equation
and will be working with the standard form of the parabola: f(x) = ax^2+by+c, which for our purposes is the same as y = ax^2+by+c.
Vertex: Lowest or highest point on the parabola (maximum or minimum)
Finding it: First get the line of symmetry, then plug its value in for x in the equation. The result is the y coordinate of the vertex.
Get the line of symmetry: FInd the value of -b/2a
In our example equation, -b = -18 and 2a = 2, so -b/2a = -9
Plug the value of the line of symmetry in for x in the equation.
-9^2 + -18 + 45 =
81 + -18 + 45 = 108
The vertex is located on the point (-9, 108). It is a minimum. Because a in the standard for of the equation is positive, the parabola opens upward (like a positive smile).
Y-intercept: the value of c in the equation ax^2+by+c=0
It is the point (0,c)
X-intercepts: solve by factoring
In an equation where a=1, factor a quadratic by finding the numbers that add to the coefficient b and multiply to c.
Example: x^2+18x+45
What number adds to 18 and multiplies to 45?
If you can’t figure this out off the top of your head, make a list of factor pairs for the c term, then check each to see if it adds to the b term.
45= 1x45 but 45+1 not= 18.
45=3*15 and 3+15 = 18
So the factors of x^2+18x+45 are (x+3)(x+15).
The value of x that will make the sum inside the parentheses 0 is the “root” or “zero” or x-intercept of the equation, in this case -3 and -15.
Try it. Plug -3 or -15 in for x in the equation x^2+18x+45 and you will get 0.
So the x intercepts are (-3, 0) and (-15,0)
