91Ó°ÊÓ

The given points are $\left(0,0.2\right)\text{and}\left(4,51.2\right)$.

The exponential function is of the form $y=a{b}^x$ where $a≠0,b≠1\text{and}b>0$.

On satisfying the points $\left(0,0.2\right)\text{and}\left(4,51.2\right)$in exponential function $y=a{b}^x$.

$\begin{array}{c}y=a{b}^x\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„}\mathrm{....}\left(1\right)\\ \left(0.2\right)=a{b}^{\left(0\right)}\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„    }\mathrm{....}\left(\text{Forthepoint}\left(0,0.2\right)\right)\\ a=0.2\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„       }\mathrm{....}\left(2\right)\end{array}$

Similarly,

role="math" $\begin{array}{c}y=a{b}^x\\ \left(51.2\right)=a{b}^{\left(4\right)}\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„    }\mathrm{....}\left(\begin{array}{l}\text{Forthepoint}\\ \left(4,51.2\right)\end{array}\right)\\ \left(51.2\right)=\left(0.2\right)b^{\left(4\right)}\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„    }\mathrm{....}\left(\begin{array}{l}\text{From}\left(2\right)\\ a=0.2\end{array}\right)\\ b=4\text{ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„ â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„â¶Ä„        }\mathrm{....}\left(âˆ\mu b>0\right)\end{array}$

Plugging $a=0.2,b=4$ in (1) gives $y=0.2{\left(4\right)}^x$ which is the required exponential function passing through the given points.

The exponential function passing through the given points is $y=0.2{\left(4\right)}^x$.