graph the parabola. find the vertex.
y = 2(x + 3)(x - 1)
Multiply using FOIL
y = 2(x² - x + 3x - 3)
Combine like terms:
y = 2(x² + 2x - 3)
Distribute to remove parentheses:
y = 2x² + 4x - 6
Factor coefficient of x² out of first
two terms only:
y = 2(x² + 2x) - 6
On scratch paper,
multiply the coefficient
of x inside the parentheses, which is -2,
by one-half (always 1/2):
-2 times 1/2 equals -1
Now, still on scratch paper, square -1.
(-1)² = -1 times -1 equals +1, so add 1,
then subtract 1 inside the parentheses,
like this:
y = 2(x² + 2x + 1 - 1) - 6
This does not change the value because
adding 1 and then subtracting 1 amounts
to adding 0.
Change the parentheses to brackets because
we are going to have parentheses inside of
parentheses next:
y = 2[x² + 2x + 1 - 1] - 6
Now factor the first three terms inside the
brackets as a trinomial:
y = 2[(x+1)(x+1) - 1] - 6
Notice that the two factors are exactly
alike so we can write (x+1)(x+1) as (x+1)²
y = 2[(x+1)² - 1] - 6
Now remove the brackets by distributing.
Multiply the 2 by the (x+1)² leaving the
(x+1)² intact, then multiply the 2 by
the -1
y = 2(x+1)² - 2 - 6
Combine the -2 and the -6 as -8
y = 2(x+1)² - 8
Now it is in the standard form:
y = a(x-h)² + k
where a = 2, h = -1 and k = -8
The vertex is the point (h,k) = (-1,-8)
It also goes through the points
(h-1,k+a) and (h+1,k+a)
which are
(-1-1, -8+2) and (-1+1, -8+2)
or
(-2, -6) and (0, -6)
So the graph looks like this:
Edwin
