Part 1)
we have that
y=−1/6x2+7x−80
y= (−1/6)x^2+7x−80 multiply both sides by -6-6y = x^2 - 42x + 480 subtract 480 from both sides -6y - 480 = x^2 - 42x take (1/2) of 42 = 21.....square this = 441 and add to both sides -6y - 480 + 441 = x^2 - 42x + 441 simplify the left, factor the right -6y - 39 = (x - 21)^2 factor the left side as -6 (y + 39/6) = ( x - 21)^2 (1) Using the form 4p (y - k) = ( x - h)^2 we can write (1) as 4 (-3/2)(y - (-39/6) ) = ( x - 21)^2 The vertex = ( h, k) = ( 21, -39/6) and p = -3/2 And the directrix is given by y = k - p → y = -39/6 - (-3/2) = -39/6 + 3/2 = -39/6 + 9/6 =-30/6 = - 5-------> y=-5
the answer Part 1) isy=-5
Part 2) (y+2)^2= 129X-5 What is the equation of the directrix of the parabola?
(y+2)²= 129X-5 -----> 129*(x-5/129)=(y+2)² (1)
Using the form 4p (x - h) = ( y - k)^2 we can write (1) as 4*(129/4)*(x-5/129)=(y+2)² The vertex = ( h, k) = ( 5/129, -2) and p = 129/4 And the directrix is given by x = h - p → x = 5/129 - (129/4) = (20-16641)/516-----> -16621/516
x=-32.21
the answer part 2) is x=-32.21