Expert: Josh - 1/8/2005

Question
y=xsquared-12x+20

Answer
Hi Alex,

I am very sorry for the lateness of this reply.
I thought I had entered the right date in the system, but apparently questions still got through while I was on vacation.

To answer this type of question, here is what you need to remember -- the general form of the equation for a parabola. If you can do that, everything basically follows.

For an upright parabola (by that, I mean one which looks like the alphabet "U", its equation is given by
y=a*(x-b)^2. ...[#0]

Let me define what these letters represent.
y, as you know, represent values in the direction of the vertical axis. Similarly, x represent values in the direction of the horizontal axis.
"a" is just a scaling factor which controls the shape of the parabola. "x=b" is understood to be the line of symmetry for the parabola. This line x=b always passes through the vertex.

Case 1: If "a" is positive (i.e., a>0),
then, the parabola is upright (looks like a "U").....[#1]
Case 2: If "a" is negative (i.e., a<0),
then, the parabola is up-side-down (looks somewhat like a hat or "^") .....[#2]
In case 1: The vertex (also called the turning point) represents the minimum value on the parabola.
In case 2: The vertex represents the maximum value on the parabola.

Useful thing to remember:
Basically, any equation involving two variables, with one depending on the square of the other is a parabola.

To flip the parabola on its side, we simply replace x with y and replace y with x.

Using the convention for "a" in [#1] and [#2], such a parabola has the general form, x=a*(y-b)^2,
notice how similar this is to [#0]
a parabola which looks like "(" has a>0,
a parabola which looks like ")" has a<0.

To analyze the parabola y=x^2-12x+20,
factorize it into the form y=a*(x+r)(x=q).
You can do this in various ways.

Method 1: You can apply the quadratic formula.
WHereby, the solutions to a*x^2+b*x+c=0 are
x=[-b+sqrt(b^2-4ac)]/(2a),[-b-sqrt(b^2-4ac)]/(2a).
Method 2: Factorize it using a trial-and-error approach,
Method 3: Obtain the factorization without thinking with enough experience.
Clue: 20 = r*q. Possible values for r and q are {1,2,4,5,10,20}. Very soon, the numbers 2 and 10 will catch your attention. And you will find that if r=10,q=2, then, (x-r)(x-q) = x^2+(p+q)+pq = x^2-12x+20.

In any case, you get y=x^2-12x+20=a*(x-10)(x-2), where a=1.
What this says, is that the parabola is upright (looks like "U", because a is positive) and it crosses the x-axis at x=2 and x=10 (from the solutions to the equation of y). Right in the middle is the axis of symmetry, x=(2+10)/2=6.

Now, the axis of symmetry must pass through the vertex.
To find the coordinate of the vertex, substitute x=6 into the parabolic equation, and it will give you the value of y.

The other things we have already worked out.