a vertical translation of f(x)=e^x changes its asymptote?
Sometimes, always or never??
never
Always.
A vertical translation of a function does not change its asymptote. The asymptote of an exponential function of the form f(x) = e^x is the horizontal line y = 0. Regardless of any vertical translation, the asymptote will always remain at y = 0. So the answer is "never".
To determine whether a vertical translation of the function f(x) = e^x changes its asymptote, we first need to understand what an asymptote is.
An asymptote is a line or curve that a function approaches but never touches or crosses. For the function f(x) = e^x, the graph approaches the x-axis but never actually intersects it. Therefore, the x-axis is the horizontal asymptote for this function.
Now, let's consider a vertical translation of the function f(x) = e^x. A vertical translation involves adding or subtracting a constant term to the function. For example, let's say we have a new function g(x) = e^x + c, where c is a constant.
To determine if the asymptote changes, we need to analyze the behavior of the translated function g(x) as x approaches positive or negative infinity.
When x approaches positive infinity: As x gets larger and larger, e^x also increases exponentially, and adding a constant c to it will not change this behavior. As a result, the graph of g(x) will still approach the same horizontal asymptote, which is the x-axis.
When x approaches negative infinity: As x gets smaller and smaller (tending towards negative infinity), e^x approaches 0, and again, adding a constant c to it will not alter this behavior. Hence, the graph of g(x) will still approach the x-axis as its horizontal asymptote.
Therefore, regardless of the value of the constant c in the vertical translation, the asymptote of the function f(x) = e^x remains the same. In other words, a vertical translation of f(x) = e^x will never change its asymptote.