Question

sketch each parabola using the given information. vertex (2,3) point (6,9)

Answer 1

The canonical form of the parabola is
y=a(x-h)²+k
where (h,k) is the vertex, and a is a constant.

For a vertex of (2,3), we have
y=a(x-2)²+3
where a is a constant to be found from the other point, (6,9)
=>
9=a(6-2)²+3
=>
a=3/8
The complete equation of the parabola is therefore
y=(3/8)(x-2)²+3


Check:
y(6)=9 OK
y(2)=3 OK
y'(x)=(3(x-2)^2)/8+3
y'(2)=0 OK

For sketching, it would help to know the zeroes (if any), so put
y(x)=(3/8)(x-2)²+3=0
=>
(3/8)(x-2)²=-3
=>
there are no real zeroes.