Question

Is y = 0 the asymptote of all functions of the form f(x) = ab^x? Explain your reasoning.

Answer 1

Y=0 would be a horizontal line.  An asymptote is a line that a function approaches, but never reaches.  Exponential functions such as these are a smooth curve.  If both numbers are positive numbers greater than or equal to 1, the curve increases.  If at least one of the numbers is a positive number between 0 and 1, the curve decreases.  If a is a negative number, the curve decreases as well.  If either a or b is zero, then the graph would stay constant at 0.  However, as long as neither a nor b is zero, then this graph will never touch that point.  The only way to get an answer of y=0 is to multiply by 0.  If neither a nor b is zero, this won't happen.