An exponential function of the form $f(x) = ab^x$ that passes through the points $(2,2)$ and $(4, \frac { 1 } { 2 })$.   

1. What is the equation of the function?

   Hint (a):

   Read Math Notes box in Lesson 8.2.2 on curve fitting an exponential function.

   Step 1(a):

   Create two equations using the points $(2,2)$ and $(4,\frac{1}{2})$.
   Equation $1$: $2 = ab^2$
   Equation $2$: $\frac{1}{2} = ab^4$

   Step 2(a):

   Isolate $a$ for both equations. Use the Equal Values Method to solve for $b$.

   Step 3(a):

   Substitute $b$ into one of the original equations, and solve for $a$.

2. Use the equation to sketch a graph of the function.