Answer
$p(x)=20(1.0718)^x$
$q(x)=160(0.8706)^x $
Work Step by Step
Let $p(x)=a(b)^x $ for the increasing exponential function as shown in the figure. The value of $a$ can be read directly from the figure as $a= 20$. We now have $p= 20(b)^x$. Use the point (10,40) on the graph to find the value of $b$.
$$
\begin{aligned}
p(10) & =40 \\
20 b^{10} & =40 \\
b^{10} & =2 \\
b & =2^{1 / 10} \\
& =1.0718
\end{aligned}
$$ Hence,
$p(x)=20(1.0718)^x $
Let $q(x)=a(b)^x $ for the decreasing exponential function as shown in the figure. The values of $a$ and $b$ can be found from the points, $(10,40)$ and $(15, 20)$ . We make use of the ratio of $
q(10)=40$, and $q(15)=20$
$$
\begin{aligned}
\frac{a b^{15}}{a b^{10}} & =\frac{g(15)}{g(10)} \\
b^5 & =\frac{20}{40} \\
b & =\left(\frac{20}{40}\right)^{1 / 5} \\
& =(0.5)^{1 / 5}=0.8706
\end{aligned}
$$ and $$
\begin{aligned}
a\left((0.5)^{1 / 5}\right)^{10} & =40 \\
a & =\frac{40}{(0.5)^2} \\
& =160 .
\end{aligned}
$$
Hence,
$q(x)=160(0.8706)^x $
