Consider the general function
[tex]g(x)=a(\frac{1}{2})^x[/tex]
Notice that, regardless of the value of a
[tex]\lim _{x\to\infty}g(x)=a\lim _{x\to\infty}(\frac{1}{2})^x=a\lim _{x\to\infty}\frac{1}{2^x}=a\cdot0=0[/tex]
Therefore, the graph cannot intersect the x-axis. Thus, the answer to the first blank is y-intercept.
Set b>4 and 1[tex]\begin{gathered} b(\frac{1}{2})^0=b\cdot1=b\to(0,b)\text{ y-intercept} \\ and \\ c(\frac{1}{2})^0=c\cdot1=c\to(0,c)\text{ y-intercept} \end{gathered}[/tex]Hence, the answer to the second blank is increase, and the answer to the third blank is decrease.
The descent of the graph depends on the derivative of the function; therefore,
[tex]g^{\prime}(x)=a(-2^{-x}\log 2)=-a\log 2(2^{-x})[/tex]
Notice that there is a 'a' term in the derivative; thus, the greater a is, the steeper is the descent of the graph.
Then, the answer to the fourth blank is a>4, and the answer to the fifth blank is 1
