Question

How do you graph `x^2-4(y+6)^2 = 9`?

Answer 1

`color(purple)("Note that graphing and sketching are two different things")`
`color(purple)("To sketch a curve you have to mathematically determine important points")`
Depends on what medium you use to plot it. A bit of specialist software could do so by entering the equation as presented.

However, if you wish to do this by hand from a table of values you need to manipulate the equation to give you the format of `y="something"`
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`4(y+6)^2=x^2-9`

`(y+6)^2=1/4x^2-9/4`

`y+6=+-sqrt(x^2-9)/2`

`y=+-sqrt(x^2-9)/2 -6`
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`color(red)("Split equation" -> y=+-(sqrt(x^2-9))/2-6)`

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`color(red)("Full equation" -> x^2-4(y+6)^2=9)`