Question

What is the vertex of `y^2+4y+3x-4=0`?

Answer 1

`" "x=1/3(y+2)^2-8/3`

Explanation:

This is a quadratic expressed in terms of y instead of terms in x. Consequently the graph will be of shape type `sub` instead of type `nn`.
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`color(blue)("Manipulating the equation to give the required format")`

Given:`" "y^2+4y+3x-4=0`

`color(brown)("Subtract "3x" from both sides")`

`" "y^2+4y+0-4=-3x`

`color(brown)("Divide both sides by 3")`

`" "1/3y^2 +4/3y-4/3=x`

`" "color(blue)(x=1/3y^2 +4/3y-4/3)` ........................(1)

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`color(blue)("Converting to Vertex Form")`

Write as `x=1/3(y^2+4y)-4/3`

`color(brown)("Changing the structure into vertex form and jumping a")`
`color(brown)("number of steps.")`

`" "x=1/3(y+2)^2-4/3 +k`

But `k=-4/3` giving

`" "x=1/3(y+2)^2-8/3`..........................(2)

`color(red)("If you need further explanation go to my profile page")` `color(red)("and leave me a message. You must also give me a link to this page.")`

Link `->` http://socratic.org/s/aszzseH6
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`color(green)("Observe that the plots of equation (1) and equation (2) coincide.")`
`color(green)("This demonstrates that they are the same thing but just look ")` `color(green)("different!")`

Also notice the reversal of where you obtain the vertex coordinates
If the equation form had been y=... then you would have `y=-8/3` but in this case it is `x=-8/3` so also in this case `y=(-1)xx2=-2`

Vertex`" "->(x,y)" "-> (-8/3,-2)`

Tony B