Question

What is the vertex form of `3y=-5x^2 - x +7`?

Answer 1

`y = -5/3(x-(-1/10))^2+141/60`

Explanation:

Given:

`3y = -5x^2-x+7`

Divide both sides by `3` to get `y` on the left hand side, then complete the square...

`y = 1/3(-5x^2-x+7)`

`color(white)(y) = -5/3(x^2+1/5x-7/5)`

`color(white)(y) = -5/3(x^2+2(1/10)x+1/100-141/100)`

`color(white)(y) = -5/3((x+1/10)^2-141/100)`

`color(white)(y) = -5/3(x+1/10)^2+141/60`

`color(white)(y) = -5/3(x-(-1/10))^2+141/60`

The equation:

`y = -5/3(x-(-1/10))^2+141/60`

is in the form:

`y = a(x-h)^2+k`

which is vertex form for a parabola with vertex `(h, k) = (-1/10, 141/60)` and multiplier `a = -5/3`

graph{3y = -5x^2-x+7 [-4.938, 5.06, -1.4, 3.6]}