Question

Write an equation of the parabola in intercept form.
(3,4) (4,0) (2,0)

Answer 1

Nope, not possible
the robot tutor would have a root of (3,0).

since (4,0) and (2,0) are x-intercepts, they must be roots.
so
y = a(x - 4)(x - 2)
with (3,4) on ...
4 = a(-1)(1)
a = -4

y = -4(x-4)(x-2)
or
y = -4x^2 + 24x - 32

Answer 2

since the roots are at 2 and 4, the vertex must lie on the line x=3.
Since the point (3,4) lies on the parabola, it must be the vertex, so
y = a(x-3)^2 + 4
since +2 is a root,
a+4 = 0, so
y = -4(x-3)^2 + 4
which agrees with the solution above

Answer 3

y = -2(x - 3)(x - 4)