Question

How do you find the vertex of `(x + 6)^2 = -36(y − 3)`?

Answer 1

vertex`=(-6,3)`

Explanation:

1. Expand both sides of the equation.

`(x+6)^2=-36(y-3)`

`x^2+12x+36=-36y+108`

2. Isolate for y.
Recall that general equation for a quadratic equation in standard form is `y=ax^2+bx+c`. Thus, isolate for `y`.

`x^2+12x+36-108=-36y`

`x^2+12x-72=-36y`

`y=-1/36x^2-1/3x+2`

3. Factor -1/36 from the first two terms.
To find the vertex, we must complete the square. We can do this by first factoring `-1/36` from the first two terms.

`y=-1/36(x^2+12x)+2`

4. Rewrite the bracketed terms as a perfect square trinomial.
The value of `c` in a perfect square trinomial is `(b/2)^2`. Thus, divide `12` by `2` and square the value.

`y=-1/36(x^2+12x+((12)/2)^2)+2`

`y=-1/36(x^2+12x+36)+2`

5. Subtract 36 from the perfect square trinomial.
We cannot just add `36` to the perfect square trinomial, so we must subtract `36` from the `36` we just added.

`y=-1/36(x^2+12x+36` `color(red)(-36))+2`

6. Multiply -36 by -1/36 to move -36 out of the brackets.

`y=-1/36(x^2+12+36)+2(-36)*(-1/36)`

7. Simplify.

`y=-1/36(x^2+12+36)+2[(-color(red)cancelcolor(black)36)*(-1/color(red)cancelcolor(black)36)]`

`y=-1/36(x^2+12+36)+2+1`

`y=-1/36(x^2+12+36)+3`

8. Factor the perfect square trinomial.
The final step to finding the vertex is to factor the perfect square trinomial. This will tell you the `x` coordinate of the vertex. The `y` coordinate of the vertex, `3`, has already been found.

`y=-1/36(x+6)^2+3`

`:.`, the vertex is `(-6,3)`.