ANSWER:
[tex]y=3\cdot4^x[/tex]STEP-BY-STEP EXPLANATION:
An exponential equation has the following form:
[tex]y=a\cdot b^x[/tex]We substitute each point to establish a system of equations:
[tex]\begin{gathered} 48=a\cdot b^2\rightarrow a=\frac{48}{b^2} \\ 192=a\cdot b^3\rightarrow a=\frac{192}{b^3} \end{gathered}[/tex]We can solve by equating both equations and solve for b:
[tex]\begin{gathered} \frac{48}{b^2}=\frac{192}{b^3} \\ 48\cdot b^3=192\cdot b^2 \\ \frac{b^3}{b^2}=\frac{192}{48} \\ b=4 \end{gathered}[/tex]Now, we can calculate the value of a by substituting in the previous equations:
[tex]\begin{gathered} a=\frac{48}{4^2}=\frac{48}{16} \\ a=3 \end{gathered}[/tex]Therefore, the exponential equation of the points would be:
[tex]y=3\cdot4^x[/tex]