A rational function, R(x) has the following characteristics: a vertical asymptote at x = 3,
a horizontal asymptote at y = 2,
and a hole at (2, −2).

Sketch the function and determine what it could be using the following steps:

Put in the factor that would account for the vertical asymptote at x = 3.
Add in the factors that would account for a hole at x = 2.
Determine what must be true about the numerator and denominator for there to be a horizontal asymptote at y = 2.
Add the factors that would account for the horizontal asymptote at y = 2.
Describe what you must do in order for the hole to appear at (2, −2).
Write the completed function.

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step-by-step explanation:

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step-by-step explanation:

let's first notice, the center of the hyperbola is at the origin, since the focus points are equidistantly 3 units away from the origin, that also means that the distance "c" from the center to either focus is 3.

it should also be notable that, the traverse axis for this hyperbola is vertical, or namely over the y-axis and therefore, the positive fraction will be the one with y-variable.

the traverse "a" component or distance from the center to either vertex is not 2, is more than 2, so we can safely put it at √5 ≈ 2.24.

it is ok do not worry about it my dude. have a good one