If the graph of y = (ax - b)/(x - c) has a horizontal asymptote y=5 and a vertical asymptote x = − 2 , then b cannot be equal to what?
Wouldn't it be -10?
it would be -10
the vertical asymptote at x = -2 means that (x+2)=0 but (ax-b) is not 0.
the horizontal asymptote at y=5 means that a=5,so we have 5x-b is not zero at x = -2.
5(-2)+b not zero means b cannot be 10.
Well, b cannot be equal to "monkey's uncle" because that would just be bananas! But in all seriousness, b cannot be equal to c, because when c is equal to b, the denominator becomes zero, which means the fraction is undefined at that point. So, to avoid any mathematical mischief, b cannot equal c in this case!
To determine the restriction on the value of b, we need to understand the conditions for horizontal and vertical asymptotes.
A horizontal asymptote occurs when the values of y approach a constant value as x approaches positive or negative infinity. In this case, we are given that the horizontal asymptote is y = 5.
A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a particular value, causing a vertical line to be an asymptote. In this case, we are given that the vertical asymptote is x = -2.
To find the allowed values for b, we can analyze the equation of the function:
y = (ax - b)/(x - c)
For a horizontal asymptote to be y = 5, we need the function to approach 5 as x approaches positive or negative infinity. This can only happen if the degree of the numerator (ax - b) is the same as the degree of the denominator (x - c). In other words, b needs to be zero for the degrees to match.
However, we can't have b = 0 in this case because b is in the numerator and not being canceled by the denominator. Therefore, b cannot be equal to 0, or else the horizontal asymptote would not be y = 5.
In conclusion, b cannot be equal to 0 for the given conditions of the graph.