what is the equation of the horizontal asymtpote of f(x)=-4(5^(x-3))+1
To find the equation of the horizontal asymptote of a function, we need to examine the behavior of the function as x approaches positive infinity and negative infinity.
In this case, let's analyze the behavior of the function as x approaches positive infinity. We have:
f(x) = -4(5^(x-3)) + 1
As x approaches positive infinity, the term 5^(x-3) grows significantly. Since the base 5 is larger than 1, it will continue to increase without bound. As a result, the negative sign in front of the term will cause the whole expression to tend towards negative infinity. Hence, as x approaches positive infinity, the function f(x) approaches negative infinity.
Now let's consider the behavior of the function as x approaches negative infinity. Again, we have:
f(x) = -4(5^(x-3)) + 1
As x approaches negative infinity, the term 5^(x-3) diminishes significantly. Since the base 5 is larger than 1, the term approaches zero. The negative sign in front of the term causes the whole expression to approach zero. Thus, as x approaches negative infinity, the function f(x) approaches zero.
Since the function approaches negative infinity as x approaches positive infinity and approaches zero as x approaches negative infinity, the equation of the horizontal asymptote is y = 0.
Therefore, the equation of the horizontal asymptote of f(x) = -4(5^(x-3)) + 1 is y = 0.