How do I write an equation from a parabola into factored form?

Question

A parabola has x-intercepts of 7 and 3 and it passes through the point (4, -9)

Answer 1

y = a(x-7)(x-3)

-9 =a(4-7)(4-3) = a(-3)(1)

a = -9/-3 = 3

f(x) = y = 3(x-7)(x-3) has x intercepts 3 and 7 and goes through the point (4,-9)

take each x intercept and change its sign, then add an x in front of it to construct the factor

x=7

x-7= 0

x=3

x-3 = 0

then multiply the factors together (x-7)(x-3)

find a coefficient a by plugging in the point (4,-9), y=-9, x = 4

into

y =a(x-7)(x-3)

-9=a(4-7)(4-3)

a=3

y=3(x-7)(x-3) = 3(x^2-10x +21) = 3x^2 -30x +63

the y intercept =63

y=3(x^2-10x+25) +63 - 75

y = 3(x-5)^2 -12 in vertex form, vertex and minimum point of the parabola is (5,12)

axis of symmetry is x=5, x intercepts are equidistant from x=5. 5-3 = 7-5 = 2