Mathematics

Problem

Consider the two exponential functions

y=2^{x}

and

y=\left(\tfrac12\right)^{x}

drawn on the same set of axes. (i) Explain, without calculation, which of the two graphs is increasing and which is decreasing as

x

increases, giving reasons for your answer. (ii) Describe how the positions of the two graphs differ with respect to the

y

-axis and the horizontal asymptote. (iii) Both graphs pass through the same point

(0,1)

. Give a brief explanation for this fact.