the graph of ( f(x)=2^{x} ) has which of the following features?\n a. a constant first difference\n b…

Explanation:

Step1: Analyze option a

The function (y = 2^{x}) is an exponential function. The first - difference of (y = 2^{x}) is (\Delta y=2^{x + 1}-2^{x}=2^{x}(2 - 1)=2^{x}), which is not a constant.

Step2: Analyze option b

For a function to be symmetric about the origin, (f(-x)=-f(x)). For (y = 2^{x}), (f(-x)=2^{-x}=\frac{1}{2^{x}}) and (-f(x)=-2^{x}). Since (2^{-x}\neq - 2^{x}) for all (x), the function (y = 2^{x}) is not symmetric about the origin.

Step3: Analyze option c

The function (y = 2^{x}) is an exponential function of the form (y = a^{x}(a>1)). It has no vertex. The general form of a function with a vertex is a quadratic function (y = ax^{2}+bx + c(a\neq0)) which has a vertex at (x =-\frac{b}{2a}).

Step4: Analyze option d

We know that (\lim_{x\rightarrow-\infty}2^{x}=0). The horizontal asymptote of a function (y = f(x)) is found by evaluating (\lim_{x\rightarrow\pm\infty}f(x)). For (y = 2^{x}), as (x\rightarrow-\infty), (y = 2^{x}) approaches (0). So, (y = 0) is a horizontal asymptote.

Answer:

d. A horizontal asymptote at (y = 0)