Question

Is {(-2,4), (1,5),(5,-2)} a function?

Answer 1

Yes. It would not be a function if there were `(a,b), (a,c), b \ne c`

Explanation:

Three points define a parabola `y = ax^2 + bx + c`

First point:
`4 = a(-2)^2 -2 b + c \Rightarrow c(a,b) = 4 - 4a + 2b`

Second point:
`5 = a(1)^2 + b + c`

`\Rightarrow 5 = a + b + 4 - 4a + 2b`

`\Rightarrow 1 = -3a + 3b`

`\Rightarrow 1/3 + a = b(a) \Rightarrow c(a) = 4 - 4a + 2(1/3 + a)`

`\Rightarrow c(a) = 14/3 - 2a`

Third point:
`-2 = a(5)^2 + 5b + c`

`\Rightarrow -2 = 25a + 5(1/3 + a) + 14/3 - 2a`

`\Rightarrow -6 = 75a + 5(1 + 3a) + 14 - 6a`

`\Rightarrow -20 = 69a + 5 + 15a`

`\Rightarrow -25/84 = a`

`\Rightarrow b = 1/3 - 25/84 = 3/84`

`\Rightarrow c = 14/3 - 2(-25/84) = (28 * 14 + 50)/84 = 442/84`

`y = f(x) = (-25x^2 + 3x + 442)/84`

Exercise: Determine `a,b,r` in the circle `(x - a)^2 + (y - b)^2 = r^2`

The three points are in the upper semicircle or in the downer one?