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.

vertical asymptote at x = 3

y = a/(x-3)

horizontal asymptote at y = 2
y = 2x/(x-3)
y(2) = 4/-1 = -4
so let's use
y = 2(x-1)/(x-3)
y(2) = 2/-1 = -2

a hole at (2, −2)
y = 2(x-1)(x-2) / (x-2)(x-3)

Step 1: To account for the vertical asymptote at x = 3, we need a factor of (x - 3) in the denominator.

Step 2: To account for the hole at (2, -2), we need a factor of (x - 2) in both the numerator and denominator. However, because the hole is at (2, -2), we can cancel out this factor, resulting in a hole.

Step 3: For there to be a horizontal asymptote at y = 2, the degree of the numerator should be less than or equal to the degree of the denominator. This means that the numerator should have a lower degree or be a constant term.

Step 4: To account for the horizontal asymptote at y = 2, we can add a constant term of 2 in the numerator or a constant factor of 2 in the denominator.

Step 5: To make the hole appear at (2, -2), we need to have canceled out the (x - 2) factor in both the numerator and denominator.

Step 6: The completed function would be R(x) = (x - 2) / (x - 3). This rational function has a vertical asymptote at x = 3, a horizontal asymptote at y = 2, and a hole at (2, -2).

To determine the function that satisfies the given characteristics, follow the steps mentioned:

1. Vertical asymptote at x = 3:
The function would have a vertical asymptote at x = 3 if there is a factor in the denominator of the form (x - 3). Therefore, include (x - 3) as a factor in the denominator.

2. Hole at (2, -2):
The hole is a point that is not on the graph but is significant for the function. It occurs when both the numerator and denominator have a common factor that cancels out at a particular point. To account for the hole at x = 2, include (x - 2) as a factor in both the numerator and denominator so that it cancels out:
(x - 2)/(x - 2) = 1

3. Horizontal asymptote at y = 2:
For a rational function, the degree of the numerator should be less than or equal to the degree of the denominator for there to be a horizontal asymptote. Since a horizontal asymptote is given at y = 2 (constant value), the numerator can be a constant value (related to y = 2) and the denominator can be a polynomial of any degree.

For simplicity, consider the numerator as a constant value of 2. Thus, the function will be:
R(x) = 2/(x - 3)(x - 2)

4. Vertical asymptote, hole, and horizontal asymptote factors:
As stated earlier, the factors in the function are: (x - 3)(x - 2) in the denominator to account for the vertical asymptote and hole, and a numerator of 2 for the horizontal asymptote.

R(x) = 2/(x - 3)(x - 2)

5. Description of the hole:
The hole appears at (2, -2) because both the numerator and denominator have a factor of (x - 2). This factor cancels out at x = 2, creating a hole or point of discontinuity.

6. Completed function:
R(x) = 2/(x - 3)(x - 2)

By following these steps, you can determine the rational function that satisfies the given characteristics and sketch it accordingly.