If a quadratic line passes through (1,-2) and (2,-14)
what is the line that satisfies this information?
are there any others that satisfy this? how many?
The line passing through (1, -2) and (2, -14) is given by
(y - y1)/(y2 - y1) = (x - x1)/(x2 - x1). Here,
(y + 2)/-12 = (x - 1)/1
y+2 = -12x+12
y+12x = 10
or 12x + y = 10.
This is not a quadratic, it is a straight line.
This is the only equation of the line that passes through the two points.
Character is who you are when no one is looking.
We have y = ax^2 + bx + c.
Plugging in (1, -2), we get:
-2 = a + b + c
And plugging in (2, -14) we get:
-14 = 4a + 2b + c
Let's try as best we can to "solve" one of these. The first one, when multipled by -2, becomes:
4 = -2a - 2b - 2c
Adding this to the second one, we get:
-10 = 2a - c
And this seems to be the only restriction. So let's choose an a and c that works in the above equation, and then solve for b.
-10 = 2(-7) - (-4) = -10
So a = -7 and c = -4. Putting these back into the very first equation:
-2 = -7 + b - 4
b = 9
a = -7 b = 9 c = -4. But do these work?
-2 = -7(1)^2 + 9(1) - 4 which is true, so it goes through (1,-2)
-14 = -7(2)^2 + 9(2) - 4 which is true, so it goes through (2, -14)
But keep in mind, there are infinitely many solutions. This is just one of them.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."