Answer:
[tex]y=\dfrac{3}{2}\cdot 2^x[/tex]
Step-by-step explanation:
The general equation of the exponential function is
[tex]y=a\cdot b^x[/tex]
If the graph of the exponential function passes through the points (-2, 0.375) and (7, 192), then their coordinates satisfy the equation:
[tex]0.375=a\cdot b^{-2}\\ \\192=a\cdot b^7[/tex]
Divide the second equation by the first:
[tex]\dfrac{192}{0.375}=\dfrac{a\cdot b^7}{a\cdot b^{-2}}=\dfrac{b^7}{b^{-2}}\\ \\512=b^9\\ \\b=\sqrt[9]{512}=2[/tex]
Substitute it into the second equation:
[tex]192=a\cdot 2^7\\ \\192=a\cdot 128\\ \\a=\dfrac{192}{128}=\dfrac{96}{64}=\dfrac{48}{32}=\dfrac{6}{4}=\dfrac{3}{2}[/tex]
So, the equation of the exponential function is
[tex]y=\dfrac{3}{2}\cdot 2^x[/tex]
