+212 The equation of a parabola is given.
y=−1/6x2+7x−80
What is the equation of the directrix of the parabola?
Enter your answer in the box.
i wrote the other one wrong sorry.
+258 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)
Usiing the form
4p (y - k) = ( x - h) we can write (1) as
4 (-3/2)(y - (-39/6) ) = ( x - 21)^2
The vertex = ( x, 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
+9673 First you need to complete the square.
\(y=-\dfrac{1}{6}x^2+7x-80\)
\(y = \dfrac{-1}{6}(x^2-42x+480)\\ y = \dfrac{-1}{6}((x-21)^2+39)\\ y = \dfrac{-1}{6}(x-21)^2-\dfrac{13}{2}\\ (y-(-\dfrac{13}{2})) = -\dfrac{1}{6}(x-21)^2\\ -6(y-(-\dfrac{13}{2})) = (x-21)^2\\ 4(\dfrac{-3}{2})(y-(-\dfrac{13}{2})) = (x-21)^2\\\)
We can see from the equation that vertex of parabola = (21,-13/2) and focus is 3/2 units below the vertex.
Therefore the equation of directrix is y = (-13/2 + 3/2), i.e., y = -5.
