The parent function of the graph is
[tex]y=2^x[/tex]
The function of the grapg will be
[tex]y=2^{x\text{ + a}}\text{ + b}[/tex]
at x equals infinity y equals 1
Therefore
[tex]\begin{gathered} \text{x =}\infty\text{ then y = 1} \\ \text{this implies} \\ b\text{ = 1} \end{gathered}[/tex]
Therefore the function will now be
[tex]y=2^{x\text{ + a}}\text{ + 1}[/tex]
from the graph
x = 2 when y = 2
Therefore,
[tex]\begin{gathered} 2=2^{2\text{ + a}}\text{ + 1} \\ 2-1=2^{2\text{ + a}} \\ 1=2^{2\text{ + a}} \\ \text{recall 1 = 2}^0 \\ 2^0=2^{2\text{ + a}} \\ 2\text{ + a = 0} \\ a\text{ = - 2} \end{gathered}[/tex]
Therefore, the function is
[tex]y=2^{x\text{ - 2}}\text{ + 1}[/tex]
