Question

Write an exponential function whose graph passes through the points $\left(0,0.2\right)\text{and}\left(4,51.2\right)$.

Short answer

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

Step-by-step solution

Step 1. Write down the given information.

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

Step 2. Concept used.

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

Step 3. Satisfying the given points in exponential function.

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

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

Similarly,

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

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.

Step 4. Conclusion.

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