Answer:
d.
Explanation:
various possibilities.
first suggestion : remember the general form of a parabola :
y = (x - h)² + k
with (h, k) being the the vertex (the maximum or minim point of the parabola).
as we can see, the vertex is (3, -1).
so, the correct equation is
y = (x - 3)² - 1
second suggestion : we have 4 points :
2 zeroes (2, 0) and (4, 0)
the y-intercept (0, 8)
and the vertex again (3, -1)
in general, the equating equation can be built by factorization around the zeros :
y = a×(x - zero1)(x - zero2)
in our case that is
y = a×(x - 2)(x - 4)
let's do the multiplication :
y = a×(x² - 2x - 4x + 8) = (x² - 6x + 8)
"a" we get by using the 3rd point like the y-intercept :
8 = a×(0² - 6×0 + 8) = 8a
a = 1
a > 0 confirms what we are seeing : the parabola opens upwards.
to bring
y = x² - 6x + 8
into the standard form
y = (x - h)² + k
we need to complete the square in the expression :
(x - h)² = x² - 2hx + h²
now compare this to our real expression
x² - 2hx + h² = x² - 6x + 8
x² = x² check
-2hx = -6x
-2h = -6
-h = -3
h = 3
h² = 9
8 = 9 - 1 = h² - 1
so,
y = x² - 6x + 8 = (x - 6x + 9) - 1 = (x - 3)² - 1