consider the function y=(ax+b)/(cx+d), where a,b,c,and d, are constants, a,c cannot equal 0. determine the horizontal and vertical asymptotes of the graph

horizontal: y = a/c

vertical: x = -d/c (if b/a ≠ d/c -- why must this be so?)

To determine the horizontal and vertical asymptotes of the given function, we need to analyze its behavior for large values of x and as x approaches certain values.

1. Vertical Asymptotes:
Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a certain value.

A vertical asymptote occurs at a value x = k if the denominator of the function becomes zero at that point and the numerator doesn't also become zero.

In this case, for the given function y = (ax + b)/(cx + d), the vertical asymptote(s) occur when the denominator (cx + d) becomes zero. Solving cx + d = 0 for x, we get x = -d/c.

Therefore, the vertical asymptote of the graph is x = -d/c.

2. Horizontal Asymptotes:
Horizontal asymptotes occur when the function approaches a certain value as x becomes very large or very small.

To determine the horizontal asymptote, we need to compare the degrees of the numerator and the denominator.

If the degree of the numerator is less than the degree of the denominator (i.e., the degree of the denominator is greater), the horizontal asymptote is y = 0.

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator. In this case, the horizontal asymptote is y = a/c.

If the degree of the numerator is greater than the degree of the denominator (i.e., the degree of the numerator is greater), there is no horizontal asymptote.

Therefore, the horizontal asymptote of the graph is y = a/c if the degree of the numerator is less than or equal to the degree of the denominator.

In summary:
- The vertical asymptote of the graph is x = -d/c.
- The horizontal asymptote of the graph is y = a/c if the degree of the numerator is less than or equal to the degree of the denominator.

To determine the horizontal and vertical asymptotes of the graph of the function y = (ax + b)/(cx + d), we need to analyze the behavior of the function as x approaches positive or negative infinity.

1. Vertical Asymptotes:
Vertical asymptotes occur when the function approaches positive or negative infinity as x approaches a certain value. In this case, vertical asymptotes occur when the denominator, cx + d, equals zero.

Set cx + d = 0 and solve for x:
cx + d = 0
cx = -d
x = -d/c

Therefore, the vertical asymptote is x = -d/c.

2. Horizontal Asymptotes:
Horizontal asymptotes occur when the function approaches a constant value as x approaches positive or negative infinity. To determine the horizontal asymptote, we can observe the degrees of the numerator and denominator polynomials.

Case 1: Degree of numerator < Degree of denominator
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. This means that the function approaches zero as x approaches positive or negative infinity.

Case 2: Degree of numerator = Degree of denominator
If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.
Let p and q be the leading coefficients of the numerator and denominator, respectively.
Horizontal asymptote is y = p/q.

Case 3: Degree of numerator > Degree of denominator
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, the function may have slant asymptotes or no asymptote at all.

In summary:
- Vertical asymptote: x = -d/c
- Case 1: Degree of numerator < Degree of denominator: Horizontal asymptote is y = 0.
- Case 2: Degree of numerator = Degree of denominator: Horizontal asymptote is y = p/q.
- Case 3: Degree of numerator > Degree of denominator: No horizontal asymptote.

Note: The cases mentioned above assume that the function is well-behaved and does not have any removable discontinuities or other special conditions.