Solution
The general form of an exponential function is
[tex]y=a(b)^x+c[/tex]
The graph of the function is inverted and shifted by 7 units to give
[tex]y=-a(b)^x+7[/tex]
Where
[tex]x=0,y=5[/tex]
Substitute for x and y in the general form of an exponential function
[tex]\begin{gathered} y=-a(b)^x+7 \\ 5=-a(b)^0+7 \\ 5=-a(1)+7 \\ 5=-a+7 \\ \text{Collect like terms} \\ a=7-5 \\ a=2 \end{gathered}[/tex]
Where x = 1, y = 0.5, substitute into the function
[tex]\begin{gathered} y=-a(b)^x+7 \\ 0.5=-2(b)^1+7 \\ 0.5=-2b+7 \\ \text{Collect like terms} \\ 2b=7-0.5 \\ 2b=6.5 \\ \text{Divide both sides by 2} \\ \frac{2b}{2}=\frac{6.5}{2} \\ b=\frac{13}{4} \end{gathered}[/tex]
The exponential function is
[tex]y=-2(\frac{13}{4})^x+7[/tex]
Hence,
The coefficient, a, is -2.
The base is, b, 13/4
The exponent is x
The constant, c, we adding to our function is 7
