Find axis of symmetry, focus, directirx , and vertex for -33=x^2-12y-6x

Answer 1

Greetings! Lets solve this shall we ?

So, we must solve this equation for the axis of symmetry, focus, directrix and vertex. This means this equation will end up as an equation of a parabola after we Complete the Square. Then, we have the equation,

-33 = x² - 12y - 6x........We add 12y to the left side first such that

12y - 33 = x² - 6x..........Then we complete the square on the right side by dividing -6 by two to get -3, then squaring -3 to get 9 such that

12y - 33 = x² - 6x + 9.........We must add 9 to the left side too to get

12y - 24 = x² - 6x + 9.............We then factor both sides to obtain

12(y - 2) = (x - 3)²........This shows the equation of a parabola to be vertical in the form 4p(y - k) = (x - h)². Next, we see that the vertex in this form is V = (3,2). The axis of symmetry is the x-value of of the vertex such that x = 3. To find the directrix, we let 4p = 12 to obtain p = 3. Since p > 0, the parabola will open upwards and the directrix is y = -p = -3. Finally, the focus has the form of (0, p) = (0, 3).

I hope this helped!