(Algebra II) A rational function, R(x) has the following characteristics:

Question

A rational function, R(x) has the following characteristics:

1. a vertical asymptote at x = 3,
2. a horizontal asymptote at y = 2,
3. and a hole at (2, −2).

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

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

Answer 1

(x-3)(y) = 4 or any constant

x cannot =3 because the left side would =0 with the right side = 4

(x-2)(x-3)y/(x-2) = 4

while the x-2 factors cancel, x=2 makes the denominator =0, leaving "hole" a discontinuity

(x-2)(x-3)(y-2)/(x-2) = 4 means y cannot = 0, it can only get increasingly closer to zero,

for a discontinuity or hole at (2,-2)

(x-2)(x-3)(y-2)(y+2)/(x-2)(y+2) = 4

while (x-2)/(x-2) = 1 and could be left out, its presence creates a hole

same with (y+2)/(y+2),

the equation is a hyperbola with two branches, with center or midpoint between the 2 branches at (3,2)

to get a "hole" at (2,-2) the right side constant must be 4. Unlike a standard hyperbola, this hyperbola is rotated 45 degrees counterclockwise, and there are no x^2 and y^2 terms in the equation.

plug x=2, y=-2 into the equation and it gives 4=4

(x-3)(y-2)=

(2-3)(-2-2) =

-1(-4) = 4