Describe the transformation to the parent function in order to graph the following function, then write the equations of the Vertical and Horizontal Asymptote.

Y = _1__ + 5
x- 3

To describe the transformation to the parent function and find the equations of the vertical and horizontal asymptotes, let's first identify the parent function. The given function has the form Y = 1/(x - 3) + 5, which is a variation of the reciprocal function f(x) = 1/x.

1. Vertical Shift: The "+ 5" term in the given function indicates a vertical shift. The parent function has its graph passing through the point (0, 1), and adding 5 to the function shifts it up by 5 units. Hence, the graph of the given function is shifted vertically upward by 5 units.

2. Horizontal Shift: The "-3" term in the denominator indicates a horizontal shift. In the parent function, the graph passes through the y-axis (x = 0). Subtracting 3 from x in the denominator of the given function shifts the graph horizontally to the right by 3 units.

3. Vertical Asymptote: In the parent function f(x) = 1/x, the vertical asymptote is x = 0 because the function is undefined at x = 0 (division by zero). The "+ 5" term in the given function does not affect the vertical asymptote, so the vertical asymptote remains x = 0.

4. Horizontal Asymptote: The parent function f(x) = 1/x has a horizontal asymptote y = 0. Since the additional terms + 5 and -3 in the given function do not change the behavior of the function as x approaches positive or negative infinity, the horizontal asymptote remains y = 0.

In summary:
- The graph of the given function is shifted 5 units upward and 3 units to the right from the parent function.
- The equation of the vertical asymptote is x = 0.
- The equation of the horizontal asymptote is y = 0.

To describe the transformation to the parent function and find the equations of the vertical and horizontal asymptotes for the given function, we need to determine the changes made to the parent function.

The parent function for this question is f(x) = 1/x.

1. Vertical Transformation:
To describe the vertical transformation, we need to focus on the part of the function that involves the variable 'x'. In this case, the function has (x - 3) in the denominator.

The number inside the parentheses, in this case, 3, represents the horizontal shift to the right if the number is positive or to the left if the number is negative. So, the function is shifted 3 units to the right.

2. Horizontal Transformation:
In this case, there is no horizontal transformation mentioned because there is no multiplication or addition/subtraction involving x.

3. Vertical Asymptote:
The vertical asymptote of a rational function occurs where the denominator of the function becomes zero, resulting in undefined values. In this case, the denominator is (x - 3). To find the vertical asymptote, we set the denominator equal to zero and solve for x:

x - 3 = 0
x = 3

So, the equation of the vertical asymptote is x = 3.

4. Horizontal Asymptote:
To find the horizontal asymptote of a rational function, we need to consider the degrees of the numerator and denominator.

In this case, the numerator is a constant, which can be considered as a polynomial of degree 0. The denominator is (x - 3), which can be considered as a polynomial of degree 1.

When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. Therefore, the equation of the horizontal asymptote for this function is y = 0.

In summary:
The vertical transformation is a right shift of 3 units.

The equation of the vertical asymptote is x = 3.

The equation of the horizontal asymptote is y = 0.

Finally, using this information, you can graph the function Y = (1/(x - 3)) + 5 by starting with the parent function y = 1/x, shifting it 3 units to the right, and then shifting it up 5 units.